A Case Study in Horned Dinosaur Evolution

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Advances in Quantitative Methods in Dinosaur Palaeobiology: A Case Study in Horned Dinosaur Evolution

Caleb Marshall Brown

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Ecology and Evolutionary Biology University of Toronto

Advances in Quantitative Methods in Dinosaur Palaeobiology: A Case Study in Horned Dinosaur Evolution

Caleb Marshall Brown

Doctor of Philosophy

Graduate Department of Ecology and Evolutionary Biology University of Toronto

2013

Abstract

Discerning modes and rates of biological evolution and speciation are some of the primary objectives of evolutionary biology. Much palaeobiological work has focused on developing robust methods for testing and fitting evolutionary models to samples of fossils across a stratigraphic or temporal axis, with most analyses centering on marine invertebrates. Recent extensive sampling of dinosaur deposits now allows for testing of evolutionary modes in this clade, a first for large-bodied terrestrial vertebrates. Within dinosaur palaeobiology, the relative roles of anagenesis and cladogenesis in diversification, particularly for horned dinosaurs, are hotly debated. Due to their large sample sizes, well-documented stratigraphic positions, highly diagnostic ornamentation, and monodominant bonebeds (representing populations), centrosaurine dinosaurs from the Belly River Group of Alberta make an ideal model system for testing the predictions of these two divergent evolutionary modes.

Despite this unparalleled fossil record, it (as well as most fossil records) is limited by missing data, small sample size, taphonomic biases, and stratigraphic error. In this thesis, I

ii present case studies that attempt to quantify and better understand these limitations, and inform best practices for overcoming them. The first four chapters, utilizing data sets for crocodilians

(extant archosaurs) and a model geological system (upper Belly River Group), allow for a better- constrained quantitative evolutionary analysis of the Belly River Group centrosaurines in chapter five. Correlations and time-series analyses of morphology and stratigraphic position of

Centrosaurus apertus and Styracosaurus albertensis are used to test for directional trends and evolutionary model fitting. Evolutionary results are robust to multiple simulations of stratigraphic uncertainty, and overlap between the taxa depends on a single locality. Results find no support for anagenesis, and rather are consistent with taxonomic turnover due to punctuated evolutionary events or, more likely, ecological replacement due to habitat tracking.

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Acknowledgments

Firstly I would like to thank my supervisors David Evans and Robert Reisz. David provided a research lab full of like-minded researchers always ready to collaborate on projects, work on developing new ideas or challenge existing ones. He constantly pushed the idea of hypothesis testing in palaeobiology, a powerful tool that has greatly shaped my research program. Robert provided the needed experience and ability to see the big picture in seemingly trivial details.

Both supervisors also gave me enough freedom to pursue productive side projects along with my thesis research.

Michael Ryan has been a longtime mentor and served on both the thesis and appraisal committee. His extensive experience with all things Centrosaurus was of great asset to the development of the ideas in the thesis. The other members of my thesis and appraisal committee,

Peter Dodson, Don Jackson, Deborah McLennan, Mary Silcox, and Denis Walsh, provided encouragement and support, and poignantly illustrated gaps in my knowledge when necessary.

David Eberth provided more than one crash course in sedimentology, was the most qualified

‘field assistant’ I will ever have, and was always willing to challenge my ideas as devils advocate.

The research and technical crew at the Royal Ontario Museum, both staff and students, provided an amazing environment to grow as a scientist and person. Kevin Seymour was the man responsible for making sure things kept running and took care of numerous administrative hoops, many of which I am sure I am still unaware of. Ian Morrison, Brian Iwama, Shino Sugimoto,

Peter Fenton, Janet Waddington, and Jean-Bernard Caron provided a positive research and social environment at the museum. David’s lab (Nicolás Campione, Collin VanBuren, Chris McGarrity, iv

Derek Larson, Kirstin Brink, Kentaro Chiba, Ryan Schott, Jessica Hawthorn, and Matt Vavrek), were great collaborators, editors, debaters, and friends. Together they provided an immense amount of feedback and proofreading of my thesis and papers. Jessica Arbour, Nicolás

Campione, and Matt Vavrek provided assistance with coding in R. Research discussions with

Don Brinkman, Phil Currie, Don Henderson, Jordan Mallon, Tony Russell, Darren Tanke,

François Terrien, and Jessica Theodor proved fruitful and greatly benefited the thesis.

The specimens utilized for this thesis are housed a numerous institutions across North

America and Europe, and access to these specimens is acknowledged in the respective chapters of the thesis. The staff of the Royal Tyrrell Museum and Royal Ontario Museum deserve special mention for their long history of openness and assistance with my research, and both feel like my academic homes. None of this research could have been possible without the collection and research of generations of scientists who have come before me.

I am grateful for the financial support that I have been fortunate to receive including:

NSERC CGS D - Alexander Graham Bell Canada Graduate Scholarship, Ontario Graduate

Scholarship, numerous EEB and University of Toronto fellowships, and funding from the

Dinosaur Research Institute and Jurassic Foundation.

Finally I would like to family and friends, for their constant support during my thesis. My parents Jim and Deborah Brown encouraged my interest in science and natural history, and without their support I would not have pursued research. Most of all, I thank my partner, Lorna

O’Brien, for her tireless support and encouragement. She has proofread and formatted countless manuscripts, listened patiently to all my crazy ideas, and put up with the stress of finishing not only one, but two theses.

Table of Contents

ABSTRACT...... ii

ACKNOWLEDGMENTS ...... iv

TABLE OF CONTENTS...... vi

LIST OF TABLES...... xvi

LIST OF FIGURES ...... xix

LIST OF APPENDICES...... xxvi

GENERAL INTRODUCTION...... xxviii

Thesis Design and Overview...... xxx

Chapter One...... xxx

Chapter Two ...... xxxii

Chapter Three ...... xxxiii

Chapter Four...... xxxv

Chapter Five ...... xxxvi

Contributions to Co-authored Chapters...... xxxvii

References ...... xxxvii

CHAPTER 1: Testing of the Effect of Missing Data Estimation and Distribution in

Morphometric Multivariate Data Analyses ...... 1

Abstract...... 2

Introduction ...... 3

Institutional Abbreviations ...... 6

Materials ...... 6

Variables...... 7

Methods ...... 8

Missing Data Input ...... 8

Random ...... 9

Anatomic Bias...... 9

Taxonomic Bias...... 12

Missing Data Analysis...... 13

Pairwise Deletion ...... 13

Gower Distance Matrix ...... 14

Substitution of Mean ...... 14

A priori Size Regression ...... 14

Correlated Variable Regression ...... 15

Bayesian PCA Missing Value Estimator (BPCA) ...... 15

Multivariate Explorative Data Analysis ...... 16

Results ...... 18

Relative Performance ...... 18

Effect of the Distribution of Missing Data ...... 20

Taxonomic Bias...... 21

Anatomic Bias ...... 22

Discussion...... 24

Relative Performance of Missing Data Estimation Methods ...... 25

Threshold of Missing Data Estimation...... 27

Distribution of Missing Data ...... 29

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Conclusion...... 32

Funding...... 33

Acknowledgements ...... 33

Supplementary Material ...... 34

References ...... 34

Tables ...... 40

Figures ...... 41

Appendices ...... 47

CHAPTER 2 : Evidence for Taphonomic Size Bias in the Dinosaur Park Formation (Campanian,

Alberta), a Model Mesozoic Terrestrial Alluvial Paralic System...... 54

Abstract...... 55

Introduction ...... 56

Institutional Abbrevatiations ...... 59

Methods ...... 59

Dinosaur Park Formation Faunal Data Set ...... 59

Body Mass Estimation...... 60

Taxonomic Completeness Index...... 62

Taphonomic Mode...... 63

Centrosaurus and Hadrosaurid Specimen Size Distribution...... 63

Statistical Analyses...... 64

Results ...... 64

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Correlation of Body Mass and Completeness ...... 64

Correlation of Body Mass and Taphonomic Mode ...... 67

Correlation of Body Mass and Time to Description/Discovery ...... 68

Description and Discovery Rates of Small and Large Taxa...... 70

Size Distribtuion of Specimens Withing a Taxon ...... 71

Correlation of Body Mass and Diversity...... 71

Discussion...... 72

Taphonomic Bias and Compositional Fidelity of the DPF...... 72

Historical Collection Bias...... 76

Rates of Discovery in the Dinosaur Park Formation...... 76

Implications for Other Non-marine Systems...... 77

Implications for Global Dinosaur Diversity ...... 79

Acknowledgements ...... 80

References ...... 81

Tables ...... 96

Figures ...... 102

Appendices ...... 126

CHAPTER 3 : Testing the Limits of Small Sample Size in Allometric Analyses; Implications for

Palaeobiology...... 132

Abstract...... 133

Introduction ...... 134

Institutional Abbrevations ...... 137

Materials ...... 137

Literature Survey ...... 137

Empirical Dataset...... 138

Methods ...... 139

Regression...... 139

Subsampling ...... 140

Random Subsample (Without Replacement) ...... 140

Binned Subsamples ...... 140

Even Length Binned Subsample ...... 140

Occupancy Binned Subsample...... 141

Adult Biased Subsamples...... 141

Adults Plus One...... 141

Subsampling Intervals...... 142

Comparison of Subsamples to Whole Sample...... 142

Results ...... 143

Distribtutions of Sample Size in Allometry...... 143

Minimum Sample Size for Determination of Allometry...... 144

Slope of Allometric Relationship Versus Sample Size...... 145

Correlation of Slope and Minimum Number of Specimens...... 145

Correlaion of Variation and Minimum Number of Specimens...... 146

Comparison of Minimum Sample Required for OLS and RMA ...... 147

Error Rate as a Function of Sample Size...... 148

Discussion ...... 149

Extinct and Extant Sample Sizes ...... 149

Relative Performance of OLS and RMA at Small Sample Size...... 150

Type I and II Error and Sample Size ...... 151

Allometric Nomenclature and the False Dominance of Isometry ...... 152

Acknowledgements ...... 153

References ...... 154

Tables ...... 163

Figures ...... 168

Appendices ...... 189

CHAPTER 4 : Quantifying Stratigraphic Accuracy and Precision for Dinosaur Quarries in the

Upper Belly River Group: A Case Study for Centrosaurinae ...... 210

Abstract...... 211

Introduction ...... 212

Sources of Stratigraphic Error in Alluvial-Paralic Systems ...... 213

The Belly River Group ...... 215

Materials and Methods ...... 219

Results ...... 221

Accuracy of Stratigraphic Position from Differential GPS ...... 221

Host Rhythm Thickness and Stratigraphic Resolution...... 222

Comparison of Stratigraphic Precision to Estimation Accuracy ...... 223

Discusssion ...... 224

References ...... 228

Tables ...... 236

Figures ...... 239

Appendices ...... 257

CHAPTER 5 : Morphological Variation and Evolutionary Trends in Centrosaurine Ceratopsids from the Belly River Group (Campanian) Alberta ...... 268

Abstract...... 269

Introduction ...... 271

Evolutionary Modes...... 271

Evolutionary Modes in Dinosaurs ...... 273

Belly River Group Centrosaurines as an Evolutionary Model System ...... 274

Hypotheses...... 276

Morphology ...... 277

Variation ...... 277

Asymmetry...... 278

Allometry...... 278

Multivariate Morphometrics ...... 278

Stratigraphy...... 279

Stratigraphic Correlation ...... 279

Stratigraphic Overlap ...... 280

Evolution...... 280

Morphological Correlation ...... 281

Time-Series Analysis ...... 281

Institutional Abbreviations ...... 281

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Materials ...... 282

Individual Skull Dataset ...... 283

Bonebed Datasets...... 284

Nasal Dataset ...... 284

Postorbital Dataset...... 285

Parietal Dataset ...... 285

Methods ...... 286

Completeness...... 286

Variation ...... 287

Asymmetry ...... 287

Allometry...... 288

Multivariate Analysis...... 289

Stratigraphic Position...... 289

Simulation of Stratigraphic Uncertainty...... 290

Lower ...... 291

Upper...... 291

Method 1 – Normal Distribution Around a Mean...... 291

Method 2 – Unifrom Distribtuion Within Bounds...... 292

Correlation of Morphology and Stratigraphy ...... 292

Time Series Anlysis of Evolutionary Modes...... 293

Results ...... 294

Morphological Variation and Asymmetry...... 294

Completeness of Articulated Skulls ...... 294

Variation within Centrosaurus Crania ...... 294

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Asymmetry Within Skulls...... 296

Correlaion of Variation with Asymmetry ...... 296

Regression and Allometry ...... 297

Nasal Allometry ...... 297

Postorbital Allometry ...... 298

Parietal Allometry ...... 299

Multivariate and Morphospace Analysis...... 300

Nasal...... 300

Postorbital...... 300

Parietal...... 301

Stratigraphy...... 302

Stratigraphic Correlation...... 302

Stratigraphic Overlap ...... 302

Evolution...... 302

Correlation of Morphology and Stratigraphy...... 303

Nasal...... 303

Postorbital...... 303

Parietal...... 304

Time-Series Analysis of Evolutionary Modes ...... 304

Nasal...... 304

Postorbital...... 305

Parietal...... 305

Discussion ...... 306

Completeness and Element Representation...... 306

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Variation and Taxonomic Utility...... 307

Variation and Sexual Display ...... 308

Asymmetry Within Skulls ...... 308

Correlation of Variation and Asymmetry ...... 309

Allometry ...... 310

Multivariate Morphmetrics ...... 312

Stratigraphy...... 312

Morphological Explanation of Overlap ...... 314

Stratigraphic Explanation of Overlap ...... 315

Evolution...... 315

References ...... 320

Tables ...... 334

Figures ...... 357

Appendices ...... 427

List of Tables

Chapter 1......

Table 1.1 – Summary of best-fit lines of missing data and estimanation error ...... 40

Chapter 2......

Table 2.1 – Dinosaur Park Formation dinosaur faunal list ...... 96

Table 2.2 – Correlation coefficients for body mass and skeletal completeness ...... 97

Table 2.3 – Correlation coefficients and significance values for body mass and ranked

taphonomic mode...... 98

Table 2.4 – Correlation coefficients for estimated body mass and both discovery and

description dates ...... 99

Table 2.5 – Average years to discovery and description of the small and large-bodied

dinosaur taxa in the Dinosaur Park Formation ...... 100

Table 2.6 – Goodness of fit between accumulated discovery and description and linear,

logarithmic and power functions ...... 101

Chapter 3......

Table 3.1 – Description of the 23 cranial measurements used in this study...... 163

Table 3.2 – Summary statistics for the sample of published sample sizes in studies of

intraspecies allometry ...... 164

Table 3.3 – The results of a Kolmogorov-Smirnov test of the allometric study sample

sizes...... 165

Table 3.4 – Results of allometric analysis for OLS and RMA regression of the 22

cranial skull variables in A. mississippiensis ...... 166

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Table 3.5 – Results of model fitting of equations 1.1 and 1.2 to the empirical data for

the minimum number of specimens required for determination for

allometry ...... 168

Chapter 4......

Table 4.1 – Summary statistics of the differences between the GPS stratigraphic

estimates and the measured position...... 236

Table 4.2 – Summary statistics for the thickness of sandstone palaeochannels ...... 237

Table 4.3 – Summary statistics and stratigraphic data for exemplar studies of

evolutionary modes...... 238

Chapter 5......

Table 5.1 – List of measurements used for morphometric analyses of centrosaurine

skulls grouped by element ...... 334

Table 5.2 – Results for completeness, variation, and asymmetry for each linear

measurement of centrosaurine skulls ...... 336

Table 5.3 – Results of allometric analyses of the height of the parietal ornamentation for

Centrosaurus apertus and Styracosaurus albertensis ...... 339

Table 5.4 – PC loadings, and proportion of variance of nasal shape...... 340

Table 5.5 – Results of Kolmogorov-Smirnov test on nasal PC scores ...... 341

Table 5.6 – PC loadings, and proportion of variance of postorbital shape...... 342

Table 5.7 – Results of Kolmogorov-Smirnov test on postorbital PC scores ...... 343

Table 5.8 – PC loadings, and proportion of variance of parietal shape ...... 344

Table 5.9 – Results of Kolmogorov-Smirnov test on parietal PC scores ...... 345

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Table 5.10 – Results of Kendall rank correlation of taxonomy and statigraphic order of

centrosaurine specimens ...... 346

Table 5.11 – Occurrence and proportion for replicates, for which stratigraphic overlap

occurs between taxa...... 347

Table 5.12 – Results of Pearson correlation between stratigraphic positions and the first

three PC axes of nasal shape...... 348

Table 5.13 – Results of Pearson correlation between stratigraphic positions and the first

three PC axes of postorbital shape...... 349

Table 5.14 – Results of Pearson correlation between stratigraphic positions and the first

three PC axes of parietal shape...... 350

Table 5.15 – Model-fitting results of time-series analysis of the first three PC axes of

the nasal for entire series ...... 351

Table 5.16 – Model-fitting results of time-series analysis of the first three PC axes of

the nasal for C. apertus and S. albertensis...... 352

Table 5.17 – Model-fitting results of time-series analysis of the first three PC axes of

the postorbital for entire series ...... 353

Table 5.18 – Model-fitting results of time-series analysis of the first three PC axes of

the postorbital for C. apertus and S. albertensis...... 354

Table 5.19 – Model-fitting results of time-series analysis of the first three PC axes of

the parietal for entire series...... 355

Table 5.20 – Model-fitting results of time-series analysis of the first three PC axes of

the parietal for C. apertus and S. albertensis...... 356

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List of Figures

Chapter 1......

Figure 1.1 – The 23 linear morphometric measurements used in this study illustrated on

the skull of Alligator mississippiensis ...... 41

Figure 1.2 – Estimation errors obtained as a function of the proportion of missing data

in the dataset introduced A) randomly, B) with a taxonomic bias and C)

with an anatomic bias ...... 43

Figure 1.3 – Estimation error introduced by the different methods as a function of both

proportion of missing data and distribution of that missing data ...... 45

Chapter 2......

Figure 2.1 – Histogram illustrating the distribution of body masses of the dinosaur fauna

of the Dinosaur Park Formation ...... 102

Figure 2.2 – Plots of skeletal completeness as a function of estimated mass...... 104

Figure 2.3 – All DPF dinosaur taxa ranked by body mass showing an increase in

skeletal completeness and taphonomic mode, as body mass increases...... 106

Figure 2.4 – All DPF dinosaur taxa ranked by body mass showing an increase in

completeness as body mass increases, with significane illustrated ...... 108

Figure 2.5 – DPF dinosaur species segregated into A) ornithischian and B) theropod

taxa ranked by estimated body mass showing an increase in skeletal

completeness and taphonomic mode ...... 110

Figure 2.6 – Size distributions of DPF species for the taphonomic modes ...... 112

Figure 2.7 – Discovery year and description year of DPF dinosaur taxa plotted against

body mass ...... 114

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Figure 2.8 – Cumulative taxonomic diversity of DPF dinosaur taxa through time based

on time of discovery and time of description ...... 116

Figure 2.9 – Discovery and description curves of rates of cumulative taxonomic

diversity of DPF dinosaur taxa ...... 118

Figure 2.10 – Histograms illustrating the abundance of size classes of Centrosaurus

specimens based on occipital condyle diameter ...... 120

Figure 2.11 – Histograms illustrating the abundance of size classes of articulated

hadrosaur skulls based on skull length and quadrate height...... 122

Figure 2.12 – Correlation between mean body mass of each dinosaur family and number

of species per family...... 124

Chapter 3......

Figure 3.1 – The 23 linear morphometric variables of Alligator mississippiensis skulls

used in this study ...... 169

Figure 3.2 – Diagram illustrating the five subsampling techniques utilized ...... 171

Figure 3.3 – Distribution of 542 sample sizes in published studies examining

intraspecies allometry ...... 173

Figure 3.4 – Comparison of allometric sample size distributions between extant and

extinct taxa, and invertebrates and vertebrates...... 175

Figure 3.5 – Allometric power plots illustrating the effect of sample size on the

allometric trend of three representative variables...... 177

Figure 3.6 – Minimum sample size required for correct identification of allometric trend

as a function of slope ...... 179

Figure 3.7 – Correlation between the absolute deviation of the slope from one and the

correlation coefficient...... 181

Figure 3.8 – Relationship between the slope and the difference in the minimum sample

size for allometry between OLS and RMA ...... 183

Figure 3.9 – The effect of sample size on the frequency of Type I error or false

allometry, Type II error or false isometry and wrong sign error...... 185

Figure 3.10 – Schematic of the relationship of the true slope and the samples size to the

ability to categorized allometric trends...... 187

Chapter 4......

Figure 4.1 – Correlation of estimated and measured stratigraphic positions as well as

residuals ...... 239

Figure 4.2 – Histograms of estimation error for all quarries ...... 241

Figure 4.3 – Correlation of stratigraphic estimation accuracy with stratigraphic and

geographic position...... 243

Figure 4.4 – Graphical representation of stratigraphic estimation accuracy with

geography ...... 245

Figure 4.5 – Histograms of host sandstone rhythm thicknesses ...... 247

Figure 4.6 – Correlation between stratigraphic position of a quarry and the thickness of

the host rhythm ...... 249

Figure 4.7 – Comparison of the magnitudes of the stratigraphic precision and

stratigraphic accuracy ...... 251

Figure 4.8 – Hypothetical stratigraphic sections illustrating the relative effect of

accuracy of stratigraphic estimation and stratigraphic precision...... 253

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Figure 4.9 – Exemplars of evolutionary studies and the scale of the stratigraphic

resolution relative to the entire sample...... 255

Chapter 5......

Figure 5.1 – Hypothetical illustrations of the evolutionary modes of anagensis and

cladogenesis in horned dinosaurs ...... 357

Figure 5.2 – Illustration of the stratigraphic occurrence of centrosaurine bonebeds and a

selection of centrosaurine specimens within the Belly River Group,

Alberta ...... 359

Figure 5.3 – Hypothetical evolutionary patterns of a morphological trait in two

successive species through stratigraphy ...... 361

Figure 5.4 – Schematic representation of the skull of C. apertus showing the majority of

the morphometric measurements taken ...... 363

Figure 5.5 – Schematic representation of a hypothetical stratigraphic column illustrating

all methods of stratigraphic uncertainty simulation used ...... 365

Figure 5.6 – List of morphometric variables and their completeness, variation and

asymmetry...... 367

Figure 5.7 – Schematic representation of the skull of C. apertus showing areas most

consistently preserved in the articulated skulls ...... 369

Figure 5.8 – Schematic representation of the skull of C. apertus showing areas with the

highest variation across the entire sample of articulated skulls...... 371

Figure 5.9 – Schematic representation of the skull of C. apertus showing areas with the

highest variation across articulated skulls of C. apertus ...... 373

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Figure 5.10 – Density plots illustrating the distributions of the variation between skull

elements/regions, and between measurement types ...... 375

Figure 5.11 – Schematic representation of the skull of C. apertus showing areas with the

highest asymmetry across the entire sample of articulated skulls ...... 377

Figure 5.12 – Schematic representation of the skull of C. apertus showing areas with the

highest asymmetry across articulated skulls of C. apertus...... 379

Figure 5.13 – Plots illustrating the correlation of variation and asymmetry in the

measurements ...... 381

Figure 5.14 – Plots illustrating the correlation of variation and asymmetry in the cranial

elements/regions ...... 383

Figure 5.15 – Allometry of the nasal horncore in C. apertus ...... 385

Figure 5.16 – Plot of postorbital horncore height as a function of anteroposterior length

at base for all Belly River Group centrosaurine specimens...... 387

Figure 5.17 – Regression of postorbital horncore height on anteroposterior length at

base of the horncore for C. apertus specimens...... 389

Figure 5.18 – Plot of mean postorbital horncore height binned by intervals of basal

horncore length for all C. apertus specimens ...... 391

Figure 5.19 – Regression of epiparietal ossification height on their basal width for C.

apertus and S. albertensis...... 393

Figure 5.20 – PCA of nine linear measurements of the nasal showing PC1 vs. PC2 and

PC2 vs. PC3...... 395

Figure 5.21 – Proportion of variance explained by each axes, and loadings for the PCA

of the centrosaurine nasal ...... 397

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Figure 5.22 – PCA of nine linear measurements of the postorbital showing PC1 vs. PC2

and PC2 vs. PC3 ...... 399

Figure 5.23 – Proportion of variance explained by each axes, and loadings for the PCA

of the centrosaurine postorbital ...... 401

Figure 5.24 – PCA of 45 linear measurements of the parietal showing PC1 vs. PC2 and

PC2 vs. PC3...... 403

Figure 5.25 – Proportion of variance explained by each axes, and loadings for the PCA

of the centrosaurine parietal...... 405

Figure 5.26 – Stratigraphic position (and potential range of error) for of each

centrosaurine quarry in the area of DPP...... 407

Figure 5.27 – Putative Styracosaurus albertensis parietal fragment (TMP

1998.068.0033) from BB156 ...... 409

Figure 5.28 – Figures illustrating the effect of the two stratigraphic uncertainty methods

on both correlation of morphology and stratigraphy and time series

analysis of PC1 parietal scores ...... 411

Figure 5.29 – Significance tests for Pearson correlation between nasal PC scores and

stratigraphic position ...... 413

Figure 5.30 – Significance tests for Pearson correlation between postorbital PC scores

and stratigraphic position...... 415

Figure 5.31 – Significance tests for Pearson correlation between parietal PC scores and

stratigraphic position ...... 417

Figure 5.32 – Results of a time-series analysis of three evolutionary modes for the

multivariate results of the centrosaurine nasal...... 419

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Figure 5.33 – Results of a time-series analysis of three evolutionary modes for the

multivariate results of the centrosaurine postorbital...... 421

Figure 5.34 – Results of a time-series analysis of three evolutionary modes for the

multivariate results of the centrosaurine parietal...... 423

Figure 5.35 – Stratigraphic section of BB 156 (Quarry 124) from the Dinosaur Park

Formation...... 425

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List of Appendices

Chapter 1......

Appendix 1.A – Description of the 23 cranial measurements ...... 47

Appendix 1.B – Dataset of 226 specimens as well as their corresponding taxonomy and

cranial measurements...... 49

Chapter 2......

Appendix 2.A – Dinosaur faunal list for the DPF with data on date of

collection/description, literature reference, specimen references, regional

and overall completeness, and mass estimation ...... 126

Appendix 2.B – Database of body masses and measurements of coracoid widths for

both extant ornithurine birds and ornithurines from the DPF...... 130

Chapter 3......

Appendix 3.A – Dataset with the results of a literature survey investigating the sample

sizes used in intraspecies allometric studies...... 181

Appendix 3.B – Dataset of 108 specimens of Alligator mississippiensis, as well as their

cranial measurements...... 201

Appendix 3.C – Allometric power plots for the 22 variables, illustrating how the

conclusions of the best-fitting allometric trend change with increased

sampling ...... 205

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Chapter 4......

Appendix 4.A – Dataset of stratigraphic and geographic data for Centrosaurinae

quarries and bonebeds in the area of DPP ...... 257

Appendix 4.B – Stratigraphic sections for 18 of the surveyed quarries and bonebeds in

the area of DPP ...... 262

Chapter 5......

Appendix 5.A – Measurements for the articulated centrosaurine skulls examined in this

study...... 427

Appendix 5.B – Measurements for the isolated centrosaurine nasals examined in this

study...... 435

Appendix 5.C – Measurements for the isolated centrosaurine postorbitals examined in

this study...... 437

Appendix 5.D – Measurements for the isolated and articulated centrosaurine parietals

examined in this study ...... 440

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General Introduction

The deep time perspective offered to palaeobiologists by the fossil record is the largest difference between palaeobiology and other branches of biology. Although palaeobiologists cannot perform the manipulative tests of evolution the same way that neontologists can, the fossil record offers millions of years of data upon which quantitative analyses may be performed, which is the greatest strength palaeobiology brings to the study of evolutionary biology. However, limitations of the fossil record affect most palaeontological datasets which may be plagued by problems such as: small sample sizes (Hammer and Harper, 2006; Strauss and Atanassov, 2006), missing or incomplete data (Beale and Little, 1975; Kearney and Clark, 2003; Hammer and Harper, 2006;

Strauss and Atanassov, 2006), taphonomic and human-mediated sampling biases (Behrensmeyer et al., 1979; Signor and Lipps, 1982; Kidwell and Behrensmeyer, 1993), and time-averaging and stratigraphic uncertainty (Behrensmeyer, 1982; Kidwell and Behrensmeyer, 1993). These problems are common to many aspects of biology, but perhaps are most acutely seen when working with the vertebrate fossil record. Confoundingly, these problems are often not independent of each other, and the occurrence of one may limit our ability to deal effectively with the others. For instance, in extant/recent datasets issues of missing data can often be addressed by either recollecting the missing data, or deleting the problematic observations from the dataset (Beale and Little, 1975; Rubin, 1976; Legendre and Legendre, 1998), approaches not usually possible or practical in palaeontological datasets as the fossil record limits collection of additional data, and sample sizes are often too small with too high of a proportion of missing data to consider eliminating observations. Therefore, it is important to firstly understand both the extent and the effect of these issues in fossil datasets, and secondly determine best practices for dealing with them.

xxviii

Despite its limitations, the fossil record remains the only direct evidence of extinct life and is a prominent source of data for evolutionary studies (Norell and Novacek, 1992). In this manner, much of the work of palaeobiologists has been the documentation of evolutionary patterns revealed in the fossil record (Simpson, 1944; Eldredge and Gould, 1972; Gingerich,

1985; Sepkoski, 2009). Many recent analyses have objectively quantified intraspecific morphological change in biological lineages through well-sampled and often long periods of geologic time (Foote and Raup, 1995; Valentine et al., 2006; Hunt, 2008; Pachut and Anstey,

2009; Geary et al., 2010).

To date, evolutionary analyses of terrestrial vertebrates have largely concentrated on

Eocene mammals, due to large sample sizes, and have greatly added to our understanding of the modes and rate of evolution in these groups, as well as evolutionary theory (Gingerich, 1974,

1976; Gould, 1984; Gingerich, 1985; Gingerich, 1993; Gingerich and Gunnell, 1995). The increased sampling of the dinosaur fossil record, particularly from the Late Cretaceous of North

America, allows the possibility of testing modes of macroevolution within this group. The mode of dinosaur evolution, specifically the prevalence of anagenesis and cladogenesis, are debated

(Horner et al., 1992; Sampson, 1995; Scannella and Fowler, 2009), yet no analyses have attempted to test for the predictions of these modes using the available methods. One particular group, the horned Centrosaurinae from the Belly River Group of southern Alberta, make a particularly good model system for testing evolutionary modes. They are represented by large sample sizes, including population level variation represented in mass death assemblages (Currie and Dodson, 1984; Ryan et al., 2001; Eberth and Getty, 2005). They are diagnosed by taxonomically informative and taphonomically robust cranial ornamentation of the parietal

(Sampson et al., 1997; Sampson, 1999). Finally, they show distinct taxonomic turnovers within

xxix a well-sampled stratigraphic interval (Eberth and Getty, 2005; Ryan et al., 2007; Mallon et al.,

2013).

Despite this unparalleled fossil record, limitations in the data remain, including missing morphological data (making morphometric analyses difficult), small sample sizes for portions of the sample, unquantified taphonomic biases in the host geological formation, and potential stratigraphic mega-biases related to the alluvial host systems that confound detailed analysis of evolutionary modes, rate, and speciation. Here, I quantify and address these major problems utilizing novel approaches, and apply this to testing hypotheses of evolutionary modes in

Campanian centrosaurine dinosaurs, as a model for dinosaur evolution.

Thesis Design and Overview

The first four chapters of this thesis attempt to quantify the extent of the limitations in fossil datasets and/or test (using extant datasets) the best practices for overcoming them. Chapter 5 utilizes the unparalleled dataset of centrosaurine dinosaurs from the Belly River Group to test the predictions of evolutionary modes in this sample.

Chapter One

Chapter One investigates the issue of missing data in morphometric analyses. Missing data is particularly problematic in the context of multivariate explorative data analyses (e.g., PCA), as these tools often require complete datasets (Strauss et al., 2003). Analysis of palaeontological datasets therefore requires either estimating missing data based on the available data, or working around missing observation/variable pairs. Although the effects of missing data points in

xxx cladistic contexts have been widely explored (Wiens, 1998; Kearney and Clark, 2003; Wiens,

2003, 2006), their affect in morphometric analyses remains poorly studied (Strauss et al., 2003).

This analysis takes the novel approach of subjecting a complete dataset (crocodilian cranial measurements) to different simulations of missing data points, estimating these using multiple methods, and then comparing the results to the original using Procrustes superimposition. Previous studies have investigated missing data input randomly into the dataset

(Kramer and Konigsberg, 1999; Holt and Benfer, 2000; Strauss and Atanassov, 2006), and here the novel approach is taken of trying to simulate both taxonomic and anatomic biases in missing data distribution, and comparing these to random. This work attempts to more closely simulate the patterns of missing data seen in nature, which are not random but rather the result of multiple biological and physical factors. The results indicate that the pattern of distribution of missing data points (e.g., anatomically biased) is just as important as the amount of missing data, and this effect is magnified under some estimation methods. No clear mathematical threshold for the maximum amount of missing data that can, or should, be estimated was found, and this may have to be determined through consensus within the scientific community as future research employs this work.

In addition to the novel methods developed for testing the effect of missing data in morphometrics, this chapter is the first use of a Bayesian PCA missing data estimation method in the context of morphometrics (Brown et al., 2012), an approach that is now being incorporated more widely in morphometric analyses. The accurate handling of missing data in morphometrics has the potential to change the field of quantitative palaeobiology, by allowing multivariate analyses that could not be conducted previously. This research has spurred multiple other methods papers investigating this important topic, and these methods have been utilized in many

xxxi published morphometric analyses to date (Campione and Evans, 2011; Brink et al., 2012, Mallon and Anderson, 2013).

Chapter Two

Chapter Two investigates the role that taphonomic size biases play in our understanding of diversity, abundance, and palaeoecology in assemblages preserved in alluvial-paralic systems, by using the Dinosaur Park Formation of Alberta as a case study. Multiple recent papers have examined the distribution of dinosaur diversity and body size, and have suggested that strong negative skews (high diversity of large species and low diversity of small species) of total dinosaur diversity, as well as individual dinosaur faunas, are true biological signals and suggest that the life history strategies of dinosaurs were fundamentally different than other terrestrial groups (Codron et al., 2012; O'Gorman and Hone, 2013). An alternate explanation of these patterns is that the body size distribution of dinosaurs is heavily biased due the differential preservation potential of animals of differing body sizes (Dodson, 1971; Farlow et al., 1995).

This interpretation is consistent with the documentation of a taphonomic size bias in modern terrestrial systems (Behrensmeyer et al., 1979; Behrensmeyer and Dechant Boaz, 1980; Kidwell and Flessa, 1996).

Tests for the presence of a taphonomic size bias were performed on one of the most diverse, best-sampled, and well-understood dinosaur assemblages, Dinosaur Park Formation. A novel approach was taken by correlating estimated body mass of each dinosaur species to several independent preservational/taphonomic indices of that species. Strong correlations are found between all taphonomic indicators (e.g., skeletal completeness, time to discovery, taphonomic mode) and body mass. Although some of these correlations may be explained by other

xxxii mechanisms, only a strong taphonomic bias against small skeletons explains all of these results.

The presence of this taphonomic bias suggests the diversity and abundance of small species in the assemblage is greatly underestimated and that this may be true for other Mesozoic alluvial- paralic terrestrial systems. This work provides a stronger connection between studies of living and extinct taxa and communities, and may allow us to determine how much diversity in fossil assemblages is missing or unsampled. Since publication of this chapter (Brown et al., 2013), this research has not only been cited by many published papers to explain or quantify this pattern

(Longrich et al, 2012; Brown et al., 2013; Eberth et al., 2013; Evans et al., 2013; Longrich et al,

2013; Mallon and Anderson, 2013; Mallon et al., 2013; O’Gorman and Hone, 2013), but has spurred comments (Brown et al., 2013; Codron et al., 2013) and research incorporating similar methods into testing these questions in various other formations and depositional settings.

Chapter Three

Chapter Three investigates the effect of sample size on quantitative palaeobiological analyses in the context of bivariate allometry, or “relative growth”. Changes in shape through ontogeny are generally brought about by differential growth, and quantifying this can reveal patterns such as dimorphism or differentiation between closely related taxa (Gould, 1966; Bookstein, 1985).

Studies quantifying the pattern of relative growth in extinct species are common (Gould, 1973;

Dodson, 1975; Dodson, 1976; Evans, 2010; Campione and Evans, 2011), but may rely on less than optimal sample sizes. However, little work has quantified the optimal or even minimal sample size needed to perform allometric analyses in order to avoid potentially erroneous conclusions.

xxxiii

A literature review was conducted to determine the distribution of sample sizes that are generally used in studies of intraspecific allometry, and to identify any differences between fossil and extant taxa, or between invertebrates and vertebrates. Fossil datasets (both vertebrate and invertebrate) rely on significantly smaller samples than those of extant taxa, and the differences between vertebrates and invertebrates are much less than those seen between fossil and extant datasets. For the first time, we have an accurate idea of the distribution of sample sizes used in allometric analyses and how these samples differ between invertebrates and vertebrates, and more importantly between fossil and extant taxa.

The effect that small sample size in fossil datasets can have on analyses of allometry was tested using a large dataset of skull measurements for Alligator mississippiensis, spanning their entire growth trajectory. Each variable was regressed against the reference datum of skull length to determine if growth is isometric, or allometric (positive or negative) with respect to skull length. The dataset was then subsampled under multiple parameters to simulate analyses with smaller sample sizes, and the results compared to those of the larger sample. Because the null hypothesis is isometry, rate of Type II error (false isometry) is inversely correlated with sample size. Type I error (false allometry) shows no relationship to sample size. No clear threshold of minimum sample size for allometry studies is seen, but results of isometry should be regarded as likely due to Type II error in small samples. This research highlights the important role that both sample size, and effect size, have on the ability to detect patterns of differential growth. It also reveals the high rate of Type II error in fossil based analyses and suggests a nomenclatural change to take into account the false dominance of isometric growth in small datasets.

xxxiv

Chapter Four

Chapter Four encompasses a preliminary investigation into the scale of stratigraphic error, and the precision of stratigraphic positions in alluvial-paralic (river-dominated) systems. Precise stratigraphic position (of either specimens or sites) is important for establishing patterns such as species ranges, faunal turnovers, and evolutionary modes. Terrestrial systems, specifically alluvial-paralic systems, with alternating lithologies of palaeochannel sands and floodplain deposits, may result in large stratigraphic error due to the down-cutting effect of large palaeochannels from ancient river systems (Behrensmeyer, 1988). In this chapter a sample of centrosaurine quarries and bonebeds from the upper Belly River Group is used to document the observed scale of these palaeochannels and the absolute and relative stratigraphic offsets that result. Mean channel offsets in the Belly River Group represent a significant amount of error when placed in the context of the entire stratigraphic sample.

This analysis marks the first investigation attempting to quantify, and correct for, the down-cutting of palaeochannels in a stratigraphic context. These data provide a framework for either incorporating ranges of error, or simulations of stratigraphic error into palaeoecological or evolutionary analyses, which are explored in Chapter Five. The implications of these ideas regarding stratigraphic uncertainty go well beyond the Belly River Group, and are likely important for stratigraphic placement of specimens in any river-dominated depositional system.

xxxv

Chapter Five

Chapter Five represents a quantitative analysis of evolutionary mode in centrosaurine dinosaurs.

The contrasting suggestions that anagenesis (Horner et al., 1992) and cladogenesis (Sampson,

1995, 1999) are paramount in the evolutionary history of dinosaurs, allows for predictions regarding the patterns of morphological change though stratigraphy in centrosaurine dinosaurs from the Belly River Group. Monodominant bonebeds of Coronosaurus brinkmani,

Centrosaurus apertus, and Styracosaurus albertensis (Currie and Dodson, 1984; Ryan et al.,

2001; Eberth and Getty, 2005; Ryan and Russell, 2005; Ryan et al., 2007), combined with individual skeletons, provide a robust dataset for testing morphological change, at the population level, within a species and lineage of dinosaurs through stratigraphy. The effect of stratigraphic uncertainty is also taken into account through simulations of different patterns of stratigraphic error. Patterns of variation, asymmetry, allometry, and morphospace occupation are also investigated using C. apertus. Results of the evolutionary analyses are inconsistent with the predictions of anagenesis, and suggest the taxonomic turnover was due to either punctuated morphological change, or, perhaps more likely, ecological replacement of closely related taxa.

This analysis is the first to robustly quantify the patterns of morphological variation and asymmetry within a single species of dinosaur, with implications for interpretation of display structures. More importantly, this also represents the first study quantifying the variation within time successive populations of dinosaurs, and directly testing these patterns against models of evolution.

xxxvi

Contributions to Co-authored Chapters

Chapters 1 and 2 have been published in the peer-reviewed journals Systematic Biology and

Palaeogeography, Palaeoclimatology, Palaeoecology, and Chapters 3 and 4 will be submitted for publication as manuscripts similar to their current form. I am the primary author of all chapters, with additional contributions provided by co-authors as listed in each chapter. Chapter

5 will ultimately be represented by at least two individual papers with one focusing on variation and morphology, and the other on evolutionary analyses.

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xliv 1

Chapter One

Testing of the Effect of Missing Data Estimation and Distribution

in Morphometric Multivariate Data Analyses

Caleb Marshall Brown1,2, Jessica H. Arbour1, Donald A. Jackson1

1Department of Ecology & Evolutionary Biology, University of Toronto, 25 Willcocks Street,

Toronto, Ontario M5S 3B2, Canada

2Department of Natural History, Palaeobiology division, Royal Ontario Museum, 100 Queen's

Park, Toronto, Ontario M5S 2C6, Canada

Published as:

Brown, C., Arbour, J. and Jackson, D. 2012 Testing of the Effect of Missing Data Estimation and

Distribution in Morphometric Multivariate Data Analyses. Systematic Biology, 61(6): 941-954

Abstract

Missing data are an unavoidable problem in biological datasets and the performance of missing data deletion and estimation techniques in morphometric datasets are poorly understood. Here a novel method is used to measure the introduced error of multiple techniques on a representative sample. A large sample of extant crocodilian skulls was measured and analyzed with principal components analysis (PCA). Twenty-three different proportions of missing data were introduced into the dataset, estimated, analyzed, and compared to the original result using Procrustes superimposition. Previous work investigating the effects of missing data input missing values randomly, a non-biological phenomenon. Here, missing data were introduced into the dataset using three methodologies: purely at random, as a function of the Euclidean distance between respective measurements (simulating anatomical regions), and as a function of the portion of the sample occupied by each taxon (simulating unequal missing data in rare taxa). Gower’s distance was found to be the best performing non-estimation method, and Bayesian PCA the best performing estimation method. Specimens of the taxa with small sample sizes and those most morphologically disparate had the highest estimation error. Distribution of missing data had a significant effect on the estimation error for almost all methods and proportions. Taxonomically biased missing data tended to show similar trends to random, but with higher error rates.

Anatomically biased missing data showed a much greater deviation from random than the taxonomic bias, and with magnitudes dependent on the estimation method.

Introduction

Multivariate morphometric data analyses are becoming increasingly common in biological studies as a method of analyzing biological shape (e.g., Dodson, 1975a; Dodson, 1975b;

Pimentel, 1979; Bookstein, 1985; Strauss, 1985; Rohlf, 1990; Zelditch et al., 2003; Zelditch et al., 2004). Within biological datasets, missing values are a common and systemic problem

(Hadfield, 2008). This is highlighted by the fact that many methods to analyze multivariate morphometrics require complete datasets, and cannot function with missing data (Strauss et al.,

2003). Issues of missing data in very large datasets can often be addressed by deleting the incomplete observations or variables (list-wise deletion) from the dataset (Beale and Little, 1975;

Rubin, 1976; Legendre and Legendre, 1998). This approach has the unfortunate result of decreasing the sample size and, as a result, the statistical power of the analysis (Nakagawa and

Freckleton, 2008). This approach is also not always possible or practical with morphometric datasets, especially those with already small sample sizes. This situation often leaves morphologists in the dilemma of either estimating missing data based on limited present data, or working around missing observation/variable pairs. Several approaches to estimating missing values from partial datasets exist, from simple approaches such as using the mean of the known observations to complex multivariate methods (Beale and Little, 1975; Rubin, 1976; Legendre and Legendre, 1998; Strauss et al., 2003; Nakagawa and Freckleton, 2008). In addition to these, multiple procedures exist that down-weight or ignore missing data occurrences (Gower, 1966;

Gower, 1971; Reyment, 1991; Nakagawa and Freckleton, 2008).

Despite these limitations, multivariate explorative data analyses are widely used in biological analyses, often without a full discussion of the effects of missing data or estimated data (but see Dodson, 1975b; Dodson, 1979; Dodson, 1990; Hammer and Harper, 2006).

Additionally, the relative performance of different estimation methods have rarely been discussed, let alone tested. As a result their estimation accuracies as a function of the proportion of missing data and dataset size are unknown.

On those few occasions when the effect of missing data in morphological analyses have been tested, the tests have been undertaken with missing data input randomly into the dataset

(Kramer and Konigsberg, 1999; Strauss et al., 2003). This approach almost certainly stems from the simplicity of randomly assigning missing data within a dataset for multiple replicates, compared to the relative complexity of inputting missing values with a specific bias in a quantitative and repeatable fashion. The distribution of missing values within morphological datasets is not random, however, and is a result of multiple biological and sampling factors

(Hadfield, 2008; Nakagawa and Freckleton, 2008).

The most frequent cause of missing data in morphological datasets is the presence of incomplete, crushed or distorted specimens in the sample. Incomplete or damaged specimens do not affect all measurements in that specimen, only those measurements that relate to anatomical region of the specimen that is missing or damaged. Additionally, not all regions have the same probability of being damaged or lost. Extremities, delicate regions and elements subject to preferential taphonomic loss will have higher occurrence of missing data than central or robust regions. Missing data within a specimen is therefore not randomly distributed, but, rather, is often distributed in clusters based on the relative proximity of the measurements to each other, and the likelihood of those regions to be missing. It is unclear how missing data distributed regionally within specimens affects the analyses, or whether the error of estimation differs from the pattern seen in randomly distributed missing data.

In addition to the non-random distribution of missing data within a specimen, there is also a non-random distribution of missing data among specimens. The relative sample size of each taxon within a morphometric dataset is often negatively correlated with the relative amount of missing data within specimens of that taxon, with specimen completeness proportional to the sample size for that taxon. This tendency results from a principal factor. When a sample of one species is represented by abundant specimens the researcher can often be selective about which specimens to measure and can restrict inclusion to only complete specimens. Conversely, when a taxon is represented by a small number of specimens, there will be a desire to use all of the available specimens regardless of their completeness to increase sample size and ensure adequate taxonomic sampling. The lack of independence between the number of specimens for each taxon represented in the dataset, and the relative amount of missing data in that sample is problematic when we consider that it is often these rare/incomplete taxa that are the primary subject(s) of interest in morphometric studies. Estimation of values from randomly input missing data has a higher degree of error for taxa with low sample sizes relative to the whole sample, because smaller samples reduce the accuracy of estimated values for missing data values. Therefore, it is necessary to test whether higher rates of missing data in taxa represented by low specimen numbers compounds the problem seen in estimating randomly assigned missing data. This condition also represents the worst-case scenario in terms of relative distribution of missing data between taxa within the dataset.

We herein capitalize on an existing and complete morphometric dataset to test the effect that different proportions of missing data have on our ability to estimate the missing values. This approach allows for 1) a direct comparison of the accuracy of different methods of handling missing data, 2) a comparison of the relative accuracy of these methods across varying proportions of missing data, and 3) a direct comparison of the effect that the distribution of

6 missing data (random, anatomic and taxonomic) have given both the different proportions of missing data and the estimation method.

Institutional Abbreviations

CMN, Canadian Museum of Nature, Ottawa; FMNH, Field Museum of Natural History,

Chicago; ROM, Royal Ontario Museum, Toronto; RTMP, Royal Tyrrell Museum of

Palaeontology, Drumheller; UCMP, University of California Museum of Paleontology,

Berkeley; UCMVZ, University of California Museum of Vertebrate Zoology, Berkeley; UCMZ,

University of Calgary Museum of Zoology, Calgary; UM, University of Michigan Museum of

Zoology, Ann Arbour.

Materials

This study uses a dataset of crocodilian crania, complied from 226 specimens housed in eight museum collections. Crocodilians were chosen because they are relatively well-resolved, both taxonomically and phylogenetically, span a large range in body sizes to allow for testing the effect of body size and allometry, and large osteological collections exist in natural history museums to allow for ease of data collection (Romer, 1956; Norell, 1989; Densmore and White,

1991; Harshman et al., 2003).

This dataset contains 226 specimens representing 21 of the 23 extant taxa (Appendix B, also at http://datadryad.org/ — doi:10.5061/dryad.m01st7p0). All specimens reside within four

7 major groupings, two at the family level (Gavialidae and Crocodylidae) and two at the subfamily level within the family Alligatoridae (Alligatorinae and Caimaninae) (Romer, 1956; Norell,

1989; Densmore and White, 1991; Poe, 1996; Harshman et al., 2003). With a single exception, all skulls represent recent specimens, which are assigned taxonomically based on a combination of soft and hard tissue anatomy, and biogeography. One skull was measured from a fossil specimen - ROM 51011 from the Late Pleistocene (30,000 years old) of Florida, which is assigned to the extant taxon Alligator mississippiensis.

The measured sample includes specimens from hatchling-sized (or near hatchling) individuals to large, old adults in all of the four major taxonomic groups, with nearly compete size series seen in many species, particularly Alligator mississippiensis. As a result, the dataset shows a remarkable range of skull sizes. The skull length of the largest specimen (CMNAR

15792, Crocodylus porosus — 749mm) is more than twenty-five times larger (in linear dimensions) than that of the smallest (ROM R 7966, Alligator mississippiensis — 29mm). As such, it represents as great a scaling problem as will likely be encountered in any morphometric analyses.

Variables

Twenty-three cranial measurements were taken from each skull (See Fig. 1 and Appendix A, also at http://datadryad.org/ — doi:10.5061/dryad.m01st7p0). The measurements taken follow those of Dodson (1975a), with the measurements of the upper temporal fossa omitted (due to poor preservation and observation of this feature in many specimens). These osteological measurements represent functional complexes as opposed to dimensions of individual bones, and

8 have biomechanical and behavioral correlates (see Dodson, 1975a). Measurements were taken from the left side, unless this side was either incomplete or damaged, in which case the right side was used. Measurements of magnitude less than 150 mm were taken with digital calipers, between 150 and 300 mm with dial calipers, and above 300 mm were taken using a fiberglass measuring tape. All measurements were taken to the nearest millimeter.

Methods

All exploratory data analyses were carried out using the R software package (R Development

Core Team, 2011), using the packages “MASS” (Venables and Ripley, 2002), “vegan” (Oksanen et al., 2010), “e1071” (Dimitriadou et al., 2010), “pcaMethods” (Stacklies et al, 2010), “psych”

(Revelle, 2010), “cluster” (Maechler et al., 2010) and “LOST” (cited herein) available on the

CRAN (http://cran.r-project.org/) and/or Bioconductor (http://www.bioconductor.org/) websites.

Figures and plots were created using the R software package (R Development Core Team, 2011),

Microsoft Excel (v 12.3.0) and Adobe Illustrator (v 15.1.0).

Missing Data Input

The complete dataset of crocodilian measurements was subjected to artificial removal of data values to simulate the incompleteness of the fossil record, with differing proportions and configurations of missing data. The portion of missing data is represented as a percentage of the total. Twenty-three different proportions of missing data — 1% though 5% in intervals of 1, and

5% to 95% in intervals of 5 — were tested. Although we do systematically test the error

9 introduced by estimation of very high portions of missing data, this is done as a test of the performance of these methods, and should not be viewed as realistic amount of missing data.

Three functions were developed in the R statistical language to introduce a specified number of missing values into the dataset by substituting existing data values with “NA” following three criteria; random, anatomical bias, and taxonomic bias.

Random. Data values were removed from the dataset purely at random (henceforth the random missing data function or RMD), following the “missing completely at random” or MCAR model of Little (1988). “Random” data was generated by selecting values from the data matrix using the

“sample” function from the “base” package in R. “Sample” uses a fast pseudorandom number generator called the “Mersenne-Twister” (Matsumoto and Nishimura, 1998), which is commonly applied to Monte-Carlo simulations because of its long period and equidistribution property

(Matsumoto et al., 2006; Panneton et al., 2006). Although this random model does not represent the observed distribution of missing data in most morphometric datasets, it is the most basic simulation and, in most cases, would represent a best-case scenario.

Anatomic bias. To replicate the pattern of regional missing data caused by incomplete or damaged specimens, the probability of a measurement being removed from a specimen was calculated as being proportional to the inverse of the minimum Euclidean distances of its landmarks to the landmarks of all other measurements previously removed from that specimen

(LOST Package, Function ‘obliterator’). A single reference skull was used to establish the coordinates of each measurement landmark in three dimensions, and these were used to calculate the minimum absolute distances between measurements. While individual skull shape variation may have influenced the minimum distances between landmarks, it was not possible to generate

10 a set of coordinates for each skull and such variations would not have significantly altered the effect of anatomical bias from that produced using the ‘obliterator’ function.

To preserve the distribution of the number of missing data values across specimens (i.e., the number of specimens missing data and the number of measurements missing from each of those specimens), the function RMD was used to establish how many (and which) specimens would have measurements removed and how many measurements from each of those specimens would be removed. The first measurement removed from each “incomplete” skull (as well as which landmark of that measurement) was chosen at random. For subsequent removals, the minimum distance between each previously removed landmark and each remaining measurements’ landmarks were calculated. The probability of measurement A being removed after measurement B had been removed (P(A|B)) was proportional to the inverse of the minimum distance between the landmarks of measurements A and B, divided by the sum of inverses of the minimum distances of all other measurement landmarks to point B (Equation 1). Therefore, the likelihood of a measurement being removed was greater for those positioned close to missing measurements landmarks than those positioned far away, however the strength of this effect lessened with distance from the missing measurement landmark (i.e., measurements very far from a missing structure had similar, low likelihoods of being removed). For a given specimen, the probability of a measurement being removed was proportional to the product of the probability of that measurement being removed given each of the measurements already removed (Equation 2). This process was iterated for each specimen, after the first data point, until reaching the number of data values to be removed for a given specimen.

Eq. 1

Eq. 2

Where:

d = the minimum distance between a measurement (i or k) and a missing measurement (j)

i = a particular measurement

j = a missing measurement

k = any non-missing measurement

n = the number of measurements not missing

m = the number of measurements already missing

In the special case where measurements shared a terminus and the distance between the two would have been zero, the distance was instead set as one half of the minimum distance from the missing measurement to the next closest measurement. In reality if one such measurement was missing, both measurements would be missing; however this situation would dramatically

12 change the distribution of missing data values across specimens compared to RMD. By using a distance less than the minimum distance to the next nearest point, the overlapping measurement becomes the most likely to be removed next. This function “obliterator” is available on the

CRAN website under the package ‘LOST‘.

Taxonomic bias. To replicate the pattern of taxonomic bias (i.e., where taxa represented by only a few specimens tend to exhibit higher amounts of missing data), a function was developed

(LOST Package, Function ‘by.clade’) that weighted the probability of having the largest number of measurements removed from specimens belonging to taxa represented by few specimens. A sample matrix containing missing data was created using the RMD function. From this matrix the number of specimens missing data was calculated. A vector was produced containing the number of measurements missing from each incomplete specimen, and this vector (V) was sorted into descending order.

Specimens were sampled without replacement, with a probability for each specimen relative to the sum of the entire sample sizes divided by the number of specimens in that respective specimen’s taxon (Equation 3). Measurements were removed (randomly) from each sampled specimen based on the order in which they were sampled and corresponding number of measurements in V (i.e., the first specimen sampled had the largest number of measurements removed, the second specimen sampled had the second largest number of measurements removed, etc.). This meant that specimens being sampled from the least well-sampled taxon possessed the highest probability of being sampled first and thereby having the largest number of measurements removed. In this case, species was the taxonomic group on which this bias was based. This function “by.clade” is available on the CRAN website under the package ‘LOST‘.

Where:

i = a given specimen

j = the total number of specimens

N = the sample size of a taxon

n = the total number of taxa

Ni = the sample size for the taxon to which specimen i belongs

Missing Data Analysis

Diminished datasets were analyzed using non-estimation methods (Pairwise deletion and Gower distance matrix), or missing data was estimated using multiple procedures (Substitution of mean,

A priori size regression, Correlated variable regression, and Bayesian PCA missing value estimator).

Pairwise deletion. Pairwise deletion is a common, often-recommended approach for dealing with missing data (Dodson, 1975a), as no estimated values are inputted into the dataset. In this

14 method a distance matrix is produced base on pairwise dissimilarity of observations.

Observations missing from either variable used to calculate bivariate statistics are excluded from both variables. Here pairwise deletion was performed using the ‘principal’ function in the package ‘psych’ v. 1.0-92 (Revelle, 2010).

Gower distance matrix. Gower’s distance matrix handles missing data simply by weighting it as zero in the analysis, resulting in all the weight being associated with the present values

(Gower, 1966; Gower, 1971; Reyment, 1991). Rather than estimating the missing data and returning a complete matrix, this method produces a distance matrix based on pairwise similarity

(distance) between observations with missing values weighted as zero. Here a Gower similarity matrix was produced using the function ‘daisy’ in the package ‘cluster’ v. 1.32.3 (Maechler,

2011).

Substitution of mean. Mean substitution is a common estimation approach because the component axes of the ordination are centered on the grand mean and, as a result, substitution of the variable mean does not influence the axes. The mean of the observations for each variable was calculated from the existing dataset and substituted for all missing values for that variable.

In our analysis this was performed using the function ‘impute’ in the package ‘e1071’ 1.5-25

(Dimitriadau et al., 2011).

A priori size regression. When the sample is represented by a range of sizes, and variables are highly correlated with size, estimation of missing values may be based on their allometric equations relative to an a priori size variable (See discussion in Dodson, 1975a; Little, 1992;

Strauss et al., 2003). A function was developed (LOST Package, Function ‘est.reg’) to replicate the univariate estimation of missing data values based on a regression of all variables against an a priori determined variable that is a proxy for size. All variables (with the exception of variable

6 – skull length) were regressed independently against skull length (variable 6). Missing values for each variable were then estimated using the allometry of that variable relative to skull length.

Missing values for variable 6 (skull length) were estimated using the regression line of the most correlated variable for that specimen. This function “est.reg” is available on the CRAN website under the package ‘LOST‘.

Correlated variable regression. Rather than estimation based on the allometry of each variable against an a priori size variable, this method used the variable most highly correlated with the variable experiencing missing data. A second function was developed (LOST Package, Function

‘best.reg’) to estimate missing data values based on a regression of each variable against its most correlated variable, and not an a priori determined size variable. All variables were regressed individually against all other variables. The regression of the variable with the highest correlation

(the highest R2 value) with that of the variable experiencing missing data was used to estimate the missing data values. This function “best.reg” is available on the CRAN website under the package ‘LOST‘.

Bayesian PCA missing value estimator (BPCA). This method combines the expectation maximization approach for PCA (Strauss et al., 2003) with a Bayesian model (Oba et al., 2003) and has been used for missing data estimation in morphometrics (Campione and Evans, 2011). It is a highly complex method requiring multiple iterative matrix inversions and attempts to replace the missing value based via PCA by regressing the remaining values against the principal components for complete values (Strauss et al., 2003). This method was performed using the function ‘bpca’ in the package ‘pcaMethods’ (Stacklies et al., 2010).

Multivariate Explorative Data Analysis

Principal component analyses were performed on the original unaltered crocodilian dataset, as well as independently derived replicates of each of the 23 proportions of missing data for all of missing data handling techniques. Several methods were unable to effectively and consistently perform at relatively high proportions of missing data (with distribution of missing data values sometimes being a confounding problem), and as a result few replicates could be performed.

Mean substitution was robust even at small sample size so 1,000 replicates were performed for all proportions of missing data for random and taxonomic distribution. For the anatomic distribution of missing data, mean substitution is represented by 1,000 replicates for proportions of 65% or lower, 300 replicates for 70%, 100 replicates for 75% and 30 replicates for 80%.

Gower’s Coefficient is represented by 1,000 replicates for proportions less than 40%, 100 replicates at 40%, and 10 replicates at 45% for all distributions. BPCA is represented by 1,000 replicates for proportions of 65% or lower, 300 for 70%, 100 for 75% and 30 for 80%, for all distributions. Pairwise deletion is represented by 300 replicates for proportions lower the 10%

(all distributions), 200 replicates for 10-40% and 50 replicates for 45-75% (for random) and, 100 replicates for 10-35% and 20 replicates for 40-80% and 40-65 or 85% (for taxonomic and anatomic biases). The a priori size regression is represented by 1000 replicates for 1-60%

(taxonomic and anatomic) and 1-65% (random), 200 replicates for 65-70% (taxonomic) and 70%

(random), 50 replicates for 75-80% (taxonomic and random), and 30 replicates for 65%

(anatomic). The correlated variable regression is represented by 1,000 replicates for 1-60%, 300 replicates for 65-70%, 100 replicates for 75%, and 20 replicates for 80% for both taxonomic and random distributions, with anatomic distribution represented by 1,000 replicates for 1-55% and

100 replicates for 60%.

The original matrix and the estimated matrices were then analyzed using PCA.

Quantitative comparisons between the original and estimated PCA results were performed using

Procrustes superimposition to compare the relative position of specimens in multivariate PC space (Gower, 1975; Kendall, 1989; Bookstein, 1997). The overall set of differences (lack-of-fit) between the original and estimated scores, Procrustes sum of squares error, is due to error of estimation of data, and as such, the Procrustes sum of squares error is a metric of estimation accuracy of relative specimen positions in the multivariate ordination. Perfect matching in

Procrustes results in zero sum of squares error, and a complete mismatch results in a sum of squares error of one. As such, Procrustes sum of squares error provides a scale on which performance of missing-data estimation methods can be tested and compared in terms of the between-specimen distance in the ordination. It should be noted that this method only compares the change in the relative position of specimens in multivariate PC space, often the most important property for systematists, but does not consider other PCA properties such as change in loadings or variance in the vectors.

The sum of squares error for each replicate, as well as the means and standard deviations of the entire replicate series were recorded for all amounts of missing data. The means and standard deviations were then plotted against the relative proportion of missing data. Procrustes superimposition was chosen for comparison as it uniquely allows for quantitative comparison of the effect missing-data estimation has on the result of the PCA analysis, not the effect the estimation has on changing the dataset itself. Previous studies investigating the effect of missing- data estimation on morphometric analyses have concentrated on comparing the original and estimated datasets, not the result (Strauss et al., 2003). It is the result that is interpreted, and estimations that cause error here will have real and meaningful effects on the interpretation of the findings. Additionally, this approach allows for the direct comparison of both estimation and

18 non-estimation methods, a comparison not possible when analyzing the change in the dataset itself.

Results

Relative Performance

Figure 2 illustrates the estimation error (Procrustes sum of squares) incurred by the various estimation methods across all values of missing data when the missing data are A, randomly; B, taxonomically; and C, anatomically distributed. For all proportions of missing data (and amongst all missing data distributions) mean substitution introduced the highest amount of missing-data estimation error (Fig. 2). This method showed a linear increase in error with the proportion of missing data (Table 1). Although performing poorly in terms of estimating relative sample distances in the ordination, mean substitution is likely the estimation method with the least effect on the orientation of the PC axes.

Pairwise deletion of missing data resulted in less error than mean substitution, but more than the remaining methods and showed a non-linear relationship between error and increasing amount of missing data (Fig. 2). This relationship is best described by a power function (Table

1), which shows an increase in the error rate with a linear increase in proportion of missing data.

These trends, however, are based on a smaller number of replicates than those of the other methods. The relative performance of this method lies between mean substitution and the other methods tested.

The a priori size regression estimation method shows a linear relationship with an increasing proportion of missing data from 1-75% with random missing data, but a power function when then the data has taxonomic or anatomic biases (Fig. 2 and Table 1). However, above 75% for taxonomic and random and 55% for anatomic biased data, both the error rate and variation increase disproportionally. This increase in the mean error and variance resulted from a small number of results with magnitudes of up to an order of magnitude greater than those of the mean. Between 1% and 75% missing data the relative performance of this function lies between that of BPCA and pairwise deletion.

Unsurprisingly the correlated variable regression estimation method outperformed the a priori size regression method, but shows generally similar trends (Fig. 2). Unlike the a priori size method, the correlated variable shows a non-linear relationship with increasing missing data, best described as a power function, for all distributions of missing data (Table 1). As with the a priori size method, however, above 75% (and 55% for anatomic missing data) both the error rate and variation increased disproportionally. For random and taxonomic missing data, this method resulted in more estimation error than BPCA below 35%, and less above 35%. When the missing data were distributed with an anatomic bias, this method consistently introduced more error than

BCPA.

When the missing data were randomly or taxonomically distributed and in small proportions (<35%), BPCA introduced the lowest amount of error of any of the estimation methods (Fig. 2a, b). Above 35%, this method was out performed by the correlated variable regression. When the missing data were anatomically distributed, however, BPCA consistently introduced less error than all other estimation methods including correlated variable regression

(Fig. 2c), and all non estimation methods, including Gower’s above 15%. The increase in error

20 of estimation with this BPCA is not linear to the increase in proportion of missing data, but rather is represented by a power or exponential function (Table 1), which indicates a disproportional increase in error as missing data increases. The most dramatic increases in error rate and variation occurred at 35% and 75% for random and taxonomic missing data, and 55% for anatomic missing data (Fig. 2).

Gower’s distance matrix introduced the least error of all methods tests when the missing data were either randomly or taxonomically distributed (Fig. 2a, b). Anatomically biased missing data, however, caused Gower’s method to introduce the least error when the percentage was 15% or less, more error than BPCA when greater than 15%, and more error than the correlated variable regression and BPCA above 40% Fig. 2c). Although it performs well, Gower’s coefficient is limited in terms of the amount of missing data it is capable of handling. At values greater than ~30% missing viable results of the analysis are inconsistent, and at values above

45% nearly all datasets were unable to be estimated. This method showed a non-linear relationship between Procrustes sum of squares error and increasing amount of missing data

(Fig. 2). It is best described by a power function (Table 1), which shows an increase in the rate of error with a linear increase in proportion of missing data.

Effect of the Distribution of Missing Data

Missing data estimation did not affect all specimens equally, even when the missing data were randomly distributed throughout the dataset over multiple iterations. Specimens showing the largest movement (those with the largest introduced error) were those that were taxonomically underrepresented in the sample (here Gavialis, Osteolaemus, etc.) and those that lay on the

21 periphery of the occupied morphospace (here Gavialis). This contrasts with the relatively little movement that occurred in the abundantly sampled and tightly associated Alligator cluster.

Additionally, within taxonomic samples the outlying specimens were brought closer to the main cluster, and the tight cluster of central specimens spread out when the missing data were estimated.

The experiments contrasting the effect of both taxonomically and anatomically biased data with randomly distributed missing taxa allow for a quantitative test of the effect of random and non-randomly distributed missing data. The results of the introduced error for the various methods of estimation, at both different proportions and distributions of missing data, are summarized in Figure 3.

Taxonomic bias. Pairwise deletion of taxonomically biased missing data resulted in higher error than for random missing data across all proportions of missing data. These overall trends are similar and the absolute differences between to the two methods are small (Fig. 3a).

Although the overall trends were similar (Fig. 3b), Gower’s similarly matrix showed higher error in taxonomic biased distributions than random distributions for all proportions.

Similarity in the trends between random and taxonomic can be seen by comparing the equations describing their slope (Table 1).

For mean substitution, when the proportion of missing data was lower than 70%, missing data with a bias towards rare taxa systematically resulted in a lower rate of estimation error than missing data introduced at random (Fig. 3c). There is also a gradual decrease in the relative difference in error between the two distributions as the amount of missing data increases.

The regression estimation methods, both a priori size and correlated variable, showed similar response to the distribution of missing data. In both cases taxonomically biased missing data results in greater error than is encountered when the missing data is randomly distributed

(Fig. 3d, e). Both methods show a sharp increase in the error at 80% missing data, matching the spike at the same portion in the random sample (Fig. 3d, e). As with the random datasets, this increase in the mean error and variance seen at 80% resulted from a small number of results with magnitudes of up to an order of magnitude greater than those of the mean.

Taxonomic missing data estimated using BPCA showed a more complex relationship (Fig.

3f). As with the other methods, the patterns between random and the taxonomic biased samples were closer than either was to the regional bias, and taxonomic missing data incurred more estimation error than random across all proportions of missing data. Additionally, for reasonable amounts of missing data (<70%) the absolute differences between these two methods was quite small compared to the total error. For proportions of missing data above 70%, the rate of error for the taxonomically biased sample increased dramatically more than the same proportions in the random sample, resulting in a large difference in error values (Fig. 3f and Table 1).

Although the estimation error derived from taxonomically biased missing data were almost always higher than derived from random missing data, the overall trends were similar. In almost all cases, the distribution of the missing data, whether random or taxonomically biased, had less of an effect on the resultant estimation error than either the estimation method or the proportion of missing data.

Anatomic bias. Pairwise deletion of anatomically biased missing data resulted in a very large increase in error compared to the randomly introduced missing data (Fig. 3a). All error means

23 obtained, with the exception of at 10%, were greater than those obtained with random distribution.

As is the case with the majority of the methods, Gower’s showed a stronger deviation from the random distribution with anatomically biased missing data than with taxonomically biased

(Fig. 3b). The obtained result for the anatomic biased data showed higher error for all proportions with the exception of 10%.

Estimation error from mean substitution of missing data introduced to mimic regional loss of data is shown in Figure 3c. For small amounts of missing data (<10%), the error rate of random distributions was lower than that of the regional bias, with the relative difference between the two increasing with lower amounts of missing data. For portions of missing data above 10%, however, the regional bias systematically resulted in a lower rate of estimation error than missing data introduced at random (increasing from around 5 to 20%). The differences in the error rate were much greater than those seen between random and the taxonomically biased samples. In addition to having lower amounts of error at given proportions of missing data

(>10%), the rate of error increase with increasing proportion of missing data was less than those of both random and taxonomically biased missing data.

As with the taxonomically biased missing data, the two regression estimation methods again showed similar results to each other. For proportions of 50% or less anatomic biases showed a steady increase in error. Above 50% the error rate increased greatly, matching the pattern seen at 80% for both the taxonomic and random samples (Fig. 3d, e). For the a priori size method, between 5 and 55% missing data, the error for the anatomic samples was less than those findings seen for the random samples. For the correlated variable method, however, all proportions, except for 45%, had a higher error rate for the anatomically biased sample (Fig. 3d,

24 e). The dramatic increase in the mean error and variance seen above 50% for both the a priori size and correlated variable method was a result of a small number of results having differences up to an order of magnitude greater than those of the mean.

For BPCA, proportions of missing data between 10% and 60% resulted in anatomically biased samples experiencing less estimation error than those of a random distribution. Below

10%, although the difference is very small, results from regional data showed more error than random. Above 65% the anatomic error rate increased dramatically to exceed both those of the random and the taxonomic sample (Fig. 3f).

With the possible exception of the correlated variable regression method, the estimation errors resulting from anatomically biased missing data were consistently more disparate from those obtained at random than are the taxonomic results. The anatomic distribution tended to have lower error rates than random for small proportions of missing data, but in many methods

(BPCA, a priori size regression, correlated variable regression) this rate spiked once analyses were conducted above a certain threshold of missing data. Unlike taxonomically biased missing data, our simulation of anatomically biased missing data suggested that this regionalized distribution can have an equal or greater effect on the amount of introduced error than either the portioning of missing data or the method of estimation used.

Discussion

This analysis has attempted to answer questions pertaining to three main ideas. 1) Which of the many possible approaches to either working around missing data, or estimating them from the

25 existing data, introduces the least amount of error into the analysis? 2) How do the relative performances of these methods change as the amount of missing data increases? Is there a specific proportion or threshold of missing data estimation that should not be exceeded? 3) Does missing data having either taxonomic or anatomic biases result in differences in the results of the analysis than randomly allocated missing data? If the distribution of missing data does have an effect, do these biases result in a greater degree of estimation error?

Relative Performance of Missing Data Estimation Methods

Our study finds that approaches based on the substitution of the mean and pairwise deletion are likely to introduce the largest amount of error into the analysis, regardless of the proportion or distribution of the missing data. The use of these methods in estimation of missing data values is not recommended. The poor performance of mean substitution is neither surprising nor a new result. The literature testing its performance is well documented (Gornbein et al., 1992; Little,

1992; Strauss and Atanassov, 2006). Its inclusion is this study is mainly two-fold, firstly to provide a context in which the relative performance of the other methods can be measured and, secondly to add to the investigation of the effect of non-randomly distributed missing data – even in poorly performing methods.

Gower’s distance matrix consistently results in the lowest difference (error) between the result of the original dataset and the results of datasets with missing data. Between 1% and 30% missing data, the performance of BPCA is very similar, though slightly worse than that of

Gower’s. Although Gower’s consistently introduced the least amount of error, it possesses two significant drawbacks. Firstly, Gower’s does not return a dataset comparable to the original

26 dataset, rather it returns a distance matrix with the dimensions being equal to the number of observations. Results of the analysis therefore cannot illustrate the effect of the variables and can only illustrate the similarity between specimens in regards to all variables. This approach is useful for analyses interested in the relative clustering of specimens (irrespective of the variables), but may not be practical when the effect of individual variables is of interest.

Secondly, although it performs well at low amount of missing data, Gower’s cannot handle larger amounts of missing data. For this dataset, it performed consistently below 30%, inconsistently between 30% and 40%, and very rarely was able to handle 45% or more missing data. Preliminary results suggest this limited range of performance is related to dataset size and shape, and that its range of performance is more limited with smaller datasets.

If estimation of the missing data values is required, rather than generating a distance matrix, and the proportion of missing data is low, then BPCA is the most reliable method. At low proportions of random and taxonomically biased missing data (<35% missing, representative of most morphometric analyses), this study found that BPCA significantly outperforms all other estimation methods, and consistently results in the lowest amounts of error. Above 35% missing data BPCA shows an increase in error and is surpassed by correlated variable regression as the best performing estimation method. When the missing data is anatomically distributed, however,

BPCA is the best performing estimation method across all proportions of missing data.

Interestingly, even when the range of specimen sizes is very large, and the scaling patterns between the variables fairly constrained, the correlated variable regression consistently outperforms the a priori size regression. This outcome suggests that regression estimation approaches, when possible, should not depend on the regression against an a priori size value, but rather, should use the variable that is most correlated with each other variable.

Although for random and taxonomically biased distributions, the correlated variable regression method outperforms BPCA above 35%, the performance of this method (as well as the a priori size regression) is closely tied to the nature of the data, specifically the size range of the specimens and the isometric/allometric scaling patterns of the variables. The dataset tested here, with its large range of specimen sizes, and the constrained scaling patterns of the variables, likely represents the best conditions for regression estimation techniques. Datasets with more restricted ranges of specimen sizes or less constrained variable scaling would almost certainly return poorer results. Further testing of these methods with differing dataset types, sizes and shapes is required to fully understand the general performance of these methods.

Threshold of Missing Data Estimation

The second question addressed in our study related to how estimation error is related to the proportion of missing data, and if there is a significant increase in error beyond a certain proportion, indicating a threshold of missing data beyond which it should not be estimated.

Dodson (1975a) cited Pilbeam (1969) as suggesting that principal coordinates analysis can tolerate up to 1/3 missing data. Strauss et al. (2003) found that the expected maximization and

PC approach were reliable up to 50% missing data for very small number of characters. But, their reliability was strongly dependent on size of the dataset. It is unclear, however, on what they based their cut off for maximum estimation.

The majority of the methods and distributions tested herein showed either a linear or exponential/power correlation of increasing estimation error with increasing proportions of missing data. Simple methods, such as mean substitution, show a strong linear increase in

28 estimation error in response to a linear increase in percent missing data, with no distinct increase in error rate at any one portion of missing data. Because of this, no mathematically derived maximum threshold of missing data that can, or should, be estimated can be easily derived or justified.

For methods showing a disproportional increase in estimation error with increased missing data (Gower’s, pairwise deletion, BPCA, and the two regressions), the increase in error is not constant, and the rate of error per unit missing data increases as the percentage of missing data values increases. Visual inspection of the graphs permits identification of regions where certain methods experience dramatic increases in the error (e.g. 30-40% and 75-80% for BPCA and 75-

80% for the two regressions – Fig. 2). As illustrated in Figure 3, however, the distribution of missing data can dramatically affect these potential thresholds. Additionally, preliminary unpublished data (Brown, pers. obs.), as well as previous studies (Strauss et al. 2003), suggest that the size and the shape (i.e. number of variable vs. number of specimens) of the dataset can also have dramatic effect on the position of these potential thresholds. This finding suggests that there may not be a universal and distinct maximum amount of missing data that can or should be estimated in morphometric datasets. Rather, there is likely to be a combination of factors including, but potentially not limited to, the size and shape of the dataset, the distribution of the missing data, and the choice of estimation method. Researchers should be aware of the issues involved with missing data, as well as the degree of error of estimation that would be associated with their particular dataset. Further work is required to investigate this question.

Distribution of Missing Data

The disproportionate effect of the missing data estimation on underrepresented and morphological disparate specimens in the analysis, even when the missing data are randomly introduced, is an issue of concern. This is because it is often these very specimens (those of small sample size and those that are morphological disparate) that are 1) most likely to have missing data in the first place, and 2) are often of most interest to the scientist. The question of how a bias against poorly represented specimens in the distribution of missing data would confound this effect is, therefore, of great interest. The pattern seen when poorly represented specimens are subject to the highest amount of missing data is relatively constant across all estimation methods, with increased, but not dramatically, higher error rates. This finding indicates that estimates based on randomly input missing data will be reasonable proxies (will show similar trends) but will consistently underestimate the estimation error when there are systemic biases against completeness of rare taxa in datasets.

It is important to keep in mind that the measure of estimation error in these tests is the

Procrustes sum of squares, which measures the total difference (error) of all values between both results (original and estimated). This, in effect, averages the error seen across all values. If the error of estimation is concentrated on a few outlying values, the majority of the values will match closely and have little error. When the missing data (and resulting error) are biased against these rare specimens, allowing for the specimens of common taxa to be free of missing data, the result is disproportionate, with greater sum of squares error across the entire result. If the total error is greater, and the common specimens are complete, the relative movement of the rare specimens must be very large.

In addition to biases in the distribution of missing data among specimens, there are also systemic biases in the distribution of missing data within specimens. Interestingly, when missing data are biased towards anatomical regions the resulting estimation error deviates much more from the random pattern than does the taxonomically biased sample. Not only is the average deviation greater, but the anatomically biased samples often show trends that distinguish them from those seen commonly with the taxonomic or random samples. This outcome suggests that whether or not the missing data are distributed at random within the specimens has a greater effect on the result than whether it is distributed at random between the specimens.

Not only is the magnitude of the effect intriguing, but its direction is as well. For most estimation methods the anatomic distribution results in reduced estimation error – at least for small proportions of missing data. This is likely partly a factor of the metric used to compare the estimation results, Procrustes sum of squares, but may indicate a more general phenomenon. As discussed previously, Procrustes will compare all data values in the two outputs, create the best match for all the data, and then summarize the total difference with the sum of squares metric.

Regional biases in the distribution of missing data will result in certain anatomical regions having the majority of the missing values, whereas the remaining areas will encounter relatively little data loss. If the variables experiencing the loss are relatively consistent between specimens, as may be expected in morphological datasets with extremities and/or fragile areas, or as can be encountered in our simulation with measurements close together, systematic loss of information on the co-variance of these variables will occur. The unaffected variables, however, will likely perform with similar accuracy as seen in the original dataset. The result of this pattern may be similar to what would be expected if the variables experiencing data loss were removed (listwise deletion) from the dataset altogether.

As mentioned above, BPCA and Gower’s distance estimation perform very similarly when the missing data are randomly assigned to the dataset, with Gower’s showing slightly better performance. When the missing data are allocated with an anatomic bias, however, the gap in the performance of these two methods increases and BPCA shows less error than that seen in

Gower’s if more than 15% of the data are missing.

The dramatically higher rate of error seen when anatomical missing data are handled using pairwise deletion is of particular interest. This large deviation from the pattern seen with the random and increased rates of error, emphasizes the importance of both the distribution of missing data in addition to its amount, have on the resulting estimation error. Not only do the relative performance results suggest that pairwise deletion should be used with caution if used at all, but that its performance where the missing data show anatomic biases should be questioned.

It is almost certain that most biological datasets will have missing data with combined biases both among and within specimens, and that combining these scenarios may be a better proxy for biological data. Testing the effect of non-random distributions both among and within specimens in a single dataset was not performed here, however, due to complexity of the model involved. It is noted that taxonomic bias (between specimens) and anatomic bias (within specimens) tested independently, may not be representative of the combined effects that may result when co-occurring in the same dataset. Despite this possibility, we believe that the incorporation of testing of non-random distributions of missing data, within and among specimens, has increased our understanding of the effect these phenomena have on our analysis, and represents a marked improvement from testing of data missing at random.

Conclusion

BPCA and Gower’s distance measure introduce the least amount of error when handling missing data, and are herein recommended when dealing with missing values in datasets. Pairwise deletion of missing data, and mean substitution introduced the greatest amount of estimation error, and are not recommended in general. Future work systematically testing the effect of dataset size, and testing additional datasets is required to address this issue.

Mean substitution shows a linear relationship between the error of estimation and the proportion of missing data, does not show a disproportionate increase in error at a specific proportion of missing data, and therefore there is no obvious mathematical upper limit to their missing data estimation. Most methods do show a disproportionate increase in estimation error as the percentage of missing data increases and some show dramatic increases at specific amounts of missing data. These dramatic increases may superficially be seen as logical upper limits to missing data estimation (and they may indeed be for this dataset), but these sharp increases in error are a result of a combination of factors including dataset size and shape, which have not been tested systematically here. General conclusions regarding the upper limit of the different estimation techniques await systematic testing of multiple independent datasets and subsets, to quantify the effect of dataset size and shape.

Missing data input with biases both towards rare specimens and anatomical regions can dramatically affect both the amount of error at specific proportions of missing data, and the patterns of how estimation error responds to increases in missing data. In general, biases inputting more missing data in specimens that are represented by fewer numbers of specimens increase the amount of error seen in the overall results compared to randomly assigned missing

33 data. Although consistent, the absolute difference is relatively small and the pattern of response to increasing missing data is consistent in direction. Biases distributing missing data to mimic loss of whole anatomical regions result in much greater deviation from the pattern seen with random missing data. With smaller amounts of missing data, anatomical biases tend to show less error than with randomly assigned data, but as the proportion increases, this type of distribution experiences rapid increases sooner than is seen in the randomly generated samples. These results underscore the importance that the distribution pattern of the missing observations across the dataset, not merely the proportion, has on the estimation error and the interpretation of the result.

Funding

This work was supported by two National Science and Engineering Research Council of Canada

– Alexander Graham Bell Canada Graduate Scholarships, a Doris O. and Samuel P. Welles Fund

Travel Grant (UCMP), and a Dinosaur Research Institute Student Project Grant, and by the

University of Toronto.

Acknowledgements

We would like to thank multiple individuals at the University of Toronto and Royal Ontario

Museum for their contributions (both theoretical and practical) to the development and undertaking of this research project. D. Evans and R. Reisz provided guidance and support for the project. K. Brink, N. Campione, D. Larson, L. O’Brien, M. Ryan, R. Schott, J. Theodor, and

M. Vavrek provided assistance, and discussion for this project, as well as computer processing power. Access to specimens was facilitated by D. Evans and K. Seymour (ROM - Vertebrate

Palaeobiology), R. MacCullach (ROM - Herpetology), and K. Khidas and M. Steingerwald

(CMN), W. Fitch (UCMZ), B. Strilisky (RTMP), A. Resetar (FMNH), G. Schneider (UM), J.

McGuire (UCMVZ), P. Holroyd (UCMP). P. D. Polly and an anonymous reviewer, as well as N.

MacLeod and R. DeBry (Editors), provided useful revisions and suggestions that greatly improved this paper.

Supplementary Material

Supplementary material, including data files and/or online-only appendices, can be found in the

Dryad data repository (http://datadryad.org/ at doi:10.5061/dryad.m01st7p0).

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Table 1 – Summarizing of best-fit lines for the relationship between percent missing data and estimation error for each method and distribution of missing data.

Estimation Method Distribution Mode Equation R2 Sample Random Power 0.0047x0.8163 0.97402 1-80% Pair-wise Deletion Taxonomic Exponential 0.0192e0.0344x 0.94863 1-80% Anatomic Exponential 0.0138e0.0615x 0.95776 1-65% Random Power 8E-05x1.2838 0.98364 1-45% Gower Taxonomic Power 9E-05x1.2663 0.98205 1-45% Anatomic Exponential 0.0003e0.0999x 0.96455 1-40% Random Linear 0.0096x 0.99997 1-95% Mean Substitution Taxonomic Linear 0.0096x 0.99955 1-95% Anatomic Linear 0.0082x 0.99739 1-65% Random Linear 0.0011x 0.99980 1-75% A Priori Size Regression Taxonomic Power 0.0013x0.96 0.99982 1-75% Anatomic Power 0.0014x0.8603 0.99742 1-55% Random Power 0.0003x1.1401 0.99073 1-75% Correlated Variable Regression Taxonomic Power 0.0003x1.1202 0.98891 1-75% Anatomic Power 0.0004x0.9755 0.99073 1-55% Random Power 6E-05x1.5721 0.95518 1-80% Bayesian PCA Taxonomic Exponential 0.00040.0765x 0.94852 1-80% Anatomic Exponential 0.00050.0803x 0.94852 1-80%

Figure 1 – The 23 linear morphometric measurements used in this study illustrated on the skull of Alligator mississippiensis. For list of measurement see Appendix B. For biological explanation of measurements see Dodson (1975). Figure modified from Dodson (1975).

Figure 2 – Estimation errors obtained by the methods as a function of the proportion of missing data in the dataset introduced A) randomly, B) with a taxonomic bias and C) with an anatomic bias. Points represent the replicate mean with error bars of plus and minus one standard deviation. In all cases the horizontal axis is percent of missing data and the vertical axis is

Procrustes sum of squares error.

Figure 3 – Estimation error introduced by A) pairwise deletion, B) Gower distance matrix, C) mean substitution, D) a priori size regression, E) correlated variable regression, F) Bayesian

PCA as a function of both proportion of missing data and distribution of that missing data. Points represent the replicate mean with error bars of plus and minus one standard deviation. In all cases the horizontal axis is percent of missing data and the vertical axis is Procrustes sum of squares error.

Appendix A – Description of the 23 cranial measurements used in this study. See Figure 1 for reference diagram. Modified from Dodson (1975c).

Variable # Description of variable 1 skull width at posterior border of external nares 2 skull width at 4th maxillary tooth 3 skull width at anterior border of orbit 4 skull width at posterior border of quadratojugals 5 skull width across exoccipitals 6 skull length from tip of snout to quadrates 7 skull length from tip of snout to anterior border of orbits 8 skull length from posterior border of orbit to external condyle of quadrate 9 orbit length 10 orbit height 11 orbit separation 12 lateral temporal fenestra length 13 lateral temporal fenestra height 14 maxilla length 15 palatal fenestra width 16 distance from the tip of the posterolateral corner of the pterygoid to the medial condyle of the quadrate 17 height of skull from extremities of pterygoid process to dorsal surface of skull, perpendicular to long axis of snout 18 maximum depth of jaw 19 external mandibular fenestra length 20 external mandibular fenestra width 21 retroarticular process length, from crest of ridge posterior to articular cotyles to tip of process 22 palatal fenestra length 23 foramen magnum width

Appendix B – Dataset of 226 specimens as well as their corresponding taxonomy and cranial measurements.

Specimen Number Genus and Species 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 UCMP no data Alligator mississippiensis 8 10 12 17 14 29 11 10 10 8 2 3 4 13 3 8 13 4 5 2 3 6 4 ROM R 7966 Alligator mississippiensis 9 13 15 18 16 35 14 10 11 9 3 4 4 19 6 9 15 5 8 3 4 8 4 ROM R 7964 Alligator mississippiensis 9 12 13 18 15 35 13 10 10 8 2 3 3 17 5 8 13 5 8 3 4 7 4 UCMP 142041 Alligator mississippiensis 10 13 16 20 15 37 15 10 10 9 2 3 4 21 5 9 14 5 7 3 4 7 3 UCMP 142042 Alligator mississippiensis 10 15 17 21 15 38 16 11 11 10 2 3 4 20 5 10 14 6 8 3 3 9 4 ROM R 7965 Alligator mississippiensis 10 15 17 20 17 40 17 11 12 9 4 4 4 22 6 10 16 6 8 4 4 8 5 ROM R 6253 Alligator mississippiensis 11 16 20 22 17 44 19 12 13 10 3 4 4 23 5 12 17 7 7 3 4 9 4 ROM R 6251 Alligator mississippiensis 11 16 20 23 16 44 21 13 13 11 3 4 5 23 6 11 16 7 9 3 4 10 4 ROM R 6252 Alligator mississippiensis 11 16 19 23 17 45 20 12 13 11 3 4 4 23 6 11 16 7 9 3 4 9 4 UCMVZ 222424 Alligator mississippiensis 17 22 25 30 22 63 30 17 16 13 4 6 6 33 6 14 20 10 13 5 7 15 6 ROM R 0001 Alligator mississippiensis 18 25 28 33 23 67 34 18 18 14 4 6 6 37 6 17 22 10 13 5 7 15 5 CMNAR 35755 Alligator mississippiensis 19 25 29 34 25 70 34 20 18 14 5 7 6 37 8 18 24 11 15 5 8 17 6 UCMVZ 95933 Alligator mississippiensis 19 27 34 38 28 84 44 23 19 15 5 7 7 43 7 22 29 13 18 7 9 18 5 CMNAR 35753 Alligator mississippiensis 22 30 35 45 33 85 42 23 22 16 6 7 7 46 8 22 27 14 19 7 9 18 7 ROM R 0008 Alligator mississippiensis 25 35 41 47 32 97 51 26 25 16 7 8 9 53 11 23 32 16 18 7 10 20 8 ROM R 936 Alligator mississippiensis 26 37 44 51 36 102 55 29 24 17 7 9 9 56 10 26 33 17 23 8 14 22 7 ROM R 7852 Alligator mississippiensis 27 41 45 54 40 115 63 31 25 20 7 10 10 64 11 27 38 18 25 8 14 24 7 UCMVZ 65763 Alligator mississippiensis 30 43 49 57 39 116 65 33 26 19 10 10 11 66 11 30 40 19 24 9 14 24 8 CMNAR 35968 Alligator mississippiensis 32 45 57 77 54 128 64 42 28 23 10 18 16 67 16 36 50 24 20 10 19 36 10 ROM R 8039 Alligator mississippiensis 39 53 63 69 48 140 74 47 28 23 7 12 12 120 13 32 45 25 32 12 16 34 9 ROM R 7853 Alligator mississippiensis 36 52 61 74 50 146 80 40 32 26 9 14 14 77 12 38 50 23 31 12 17 32 9 UCMVZ 129985 Alligator mississippiensis 41 61 72 87 63 167 93 48 35 25 11 16 13 93 14 45 59 28 36 15 20 35 11 UCMP131080 Alligator mississippiensis 43 65 76 92 64 172 93 52 35 27 13 17 15 94 15 47 60 31 39 17 24 35 11 ROM Herp 46540 Alligator mississippiensis 43 62 71 82 57 182 101 50 39 26 14 17 16 97 13 41 53 28 42 14 22 34 10 UCMVZ 200608 Alligator mississippiensis 47 71 85 95 67 183 102 55 35 30 12 16 15 101 16 50 65 34 43 18 26 40 11 ROM R 4405 Alligator mississippiensis 54 68 75 91 62 188 106 55 37 26 12 19 16 101 15 44 55 28 39 15 24 39 12 UCMP 131688 Alligator mississippiensis 50 68 80 97 68 189 100 58 40 28 14 17 16 102 19 49 68 34 43 19 27 43 14 ROM 4414 Alligator mississippiensis 48 65 76 92 66 199 106 56 39 32 14 24 18 106 18 51 60 31 44 16 23 40 12 ROM R 1698 Alligator mississippiensis 52 70 81 93 62 205 115 52 44 27 15 16 16 115 17 48 66 33 43 16 23 43 12 ROM R 5855 Alligator mississippiensis 59 82 96 110 72 223 122 60 44 30 18 17 19 122 19 59 74 36 43 17 27 46 12 ROM R 4424 Alligator mississippiensis 56 83 110 136 86 224 121 62 45 29 19 20 20 108 20 66 84 44 46 20 31 51 15 ROM R 4418 Alligator mississippiensis 56 79 89 107 74 229 128 64 45 28 17 22 18 120 18 58 74 37 49 19 28 46 12 ROM R 4406 Alligator mississippiensis 67 83 97 118 80 241 137 77 45 32 17 22 18 130 19 65 78 37 52 20 29 51 14 ROM R 4419 Alligator mississippiensis 63 88 102 118 80 251 145 69 47 33 16 21 21 131 19 67 84 41 58 19 31 54 13 ROM R 4404 Alligator mississippiensis 62 88 108 120 80 252 142 67 49 32 17 21 24 133 24 62 94 42 52 22 29 56 16 UCMVZ 191313 Alligator mississippiensis 69 95 105 130 86 256 157 73 48 35 20 22 21 143 22 70 84 42 62 22 34 56 13 ROM R 4410 Alligator mississippiensis 64 92 111 122 74 257 147 73 48 34 15 26 21 137 22 68 87 43 45 19 30 54 15 UCMVZ 191314 Alligator mississippiensis 67 93 100 130 86 257 153 72 44 31 18 23 20 138 19 70 83 41 58 20 30 55 14 ROM R 4420 Alligator mississippiensis 69 94 106 124 81 267 144 74 51 35 18 23 23 141 22 69 88 45 56 24 30 57 14 UCMZ R 1975.229 Alligator mississippiensis 71 97 111 142 95 272 161 77 49 36 21 26 23 144 21 77 90 47 62 25 29 56 14 ROM R 4421 Alligator mississippiensis 77 107 121 148 102 287 156 79 51 32 24 24 23 156 23 82 98 50 58 27 37 59 15 ROM R 4407 Alligator mississippiensis 81 115 134 154 104 288 170 81 45 38 23 26 26 152 25 76 103 53 52 22 39 60 17 ROM R 4409 Alligator mississippiensis 80 115 129 158 102 298 180 82 54 40 21 26 27 160 21 81 104 54 61 25 37 69 16 TMP 1990.7.194 Alligator mississippiensis 85 115 130 168 108 301 174 83 54 37 20 22 25 158 24 89 99 55 64 25 43 59 16 ROM R 4413 Alligator mississippiensis 95 117 132 155 99 306 178 87 55 42 22 29 27 164 28 84 104 51 62 25 42 62 16 ROM R 388 Alligator mississippiensis 96 133 160 206 137 315 176 92 65 47 23 31 32 161 32 99 134 72 69 39 41 79 17 ROM herp 46539 Alligator mississippiensis 88 111 123 157 105 324 188 92 64 35 21 26 23 176 28 88 108 51 73 31 38 67 17 ROM R 4408 Alligator mississippiensis 115 131 151 183 115 324 190 100 55 42 24 29 29 169 29 95 127 59 64 24 49 69 18 ROM 4412 Alligator mississippiensis 79 112 131 154 95 325 172 86 56 38 22 28 24 169 24 83 102 52 59 25 39 70 17 ROM R 690 Alligator mississippiensis 85 111 126 159 108 331 186 58 64 36 22 27 23 174 29 88 109 52 73 30 39 67 16 ROM R 600 Alligator mississippiensis 107 142 173 208 143 334 181 89 61 44 21 26 28 151 20 93 135 77 66 34 44 75 19 ROM R 4416 Alligator mississippiensis 99 142 159 199 123 349 218 98 58 42 22 28 25 197 32 111 131 66 72 28 48 68 17 ROM R 4417 Alligator mississippiensis 99 136 150 186 119 352 205 111 59 43 24 33 30 189 27 107 130 69 79 32 44 66 16 ROM R 4422 Alligator mississippiensis 100 138 156 185 114 373 216 102 59 37 27 34 29 203 30 109 130 68 82 32 46 76 19 ROM R 4402 Alligator mississippiensis 131 149 172 218 133 398 231 115 70 52 26 29 32 214 32 128 152 75 92 34 56 91 19 ROM R 4401 Alligator mississippiensis 112 162 180 210 139 399 229 115 62 46 30 32 31 206 32 121 145 78 86 35 52 93 18 ROM R 4415 Alligator mississippiensis 124 163 179 232 155 432 265 122 67 50 26 35 36 228 31 125 155 80 86 35 57 79 19 ROM 4411 Alligator mississippiensis 135 190 210 259 166 479 275 139 74 49 33 39 33 241 32 160 180 86 101 35 57 89 21 ROM R 494 Alligator mississippiensis 140 196 212 282 182 497 302 148 71 45 41 20 39 262 34 153 191 95 99 47 66 99 23 ROM 51011 Alligator mississippiensis 203 274 289 371 243 689 441 195 84 57 51 53 47 371 62 181 236 132 141 64 60 139 32 FMNH 31302 Alligator sinensis 38 52 60 78 55 141 69 42 34 22 8 13 14 72 13 46 51 27 21 10 18 27 10 UCMVZ 67884 Alligator sinensis 42 59 69 88 59 153 76 46 34 24 9 16 15 80 14 52 58 32 22 10 22 28 12 FMNH 31303 Alligator sinensis 45 57 64 89 61 153 78 47 34 24 10 14 14 77 15 50 57 30 25 11 23 28 12 UM 61446 Alligator sinensis 59 80 92 122 81 212 114 67 42 31 13 22 19 105 37 72 83 42 36 16 29 37 15 UCMZ R 1975.209 Caiman crocodilus 9 12 14 19 14 35 13 11 10 10 3 4 4 19 4 9 12 6 6 3 4 8 3 51

Specimen Number Genus and Species 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ROM R1697 Caiman crocodilus 9 13 15 20 15 36 13 11 11 9 2 5 4 21 4 9 15 6 5 3 4 8 4 UCMVZ 67883 Caiman crocodilus 9 11 14 20 16 36 13 11 10 9 2 4 3 17 4 10 14 6 5 3 4 8 5 UCMVZ 68919 Caiman crocodilus 12 16 20 25 18 44 19 13 13 10 2 4 5 21 5 12 18 7 6 3 4 11 6 UCMP 123098 Caiman crocodilus 9 13 15 22 18 45 19 13 11 10 2 4 4 22 5 10 16 7 5 2 4 11 5 UCMP 138038 Caiman crocodilus 13 16 21 26 19 45 20 14 14 11 2 5 5 23 6 12 18 7 7 3 5 11 6 ROM Herp 46532 Caiman crocodilus 12 17 20 26 18 47 20 13 14 11 4 4 3 22 6 11 16 8 8 4 5 10 5 ROM Herp 46531 Caiman crocodilus 13 17 20 27 19 48 21 15 14 12 3 6 6 26 5 13 17 9 8 4 6 13 6 UCMP 123097 Caiman crocodilus 8 14 17 25 19 49 23 22 13 11 2 4 5 26 7 11 17 7 6 3 5 14 5 UCMVZ 95934 Caiman crocodilus 14 18 22 28 19 51 23 17 14 12 3 5 4 27 6 14 19 9 7 3 5 13 6 UCMVZ 207608 Caiman crocodilus 13 16 21 28 20 52 24 15 15 12 4 6 6 30 6 13 19 8 10 4 6 13 6 UCMP 119115 Caiman crocodilus 14 20 23 30 20 55 26 17 16 13 3 6 5 28 6 15 21 9 9 4 6 14 6 UCMP 132075 Caiman crocodilus 14 19 22 30 22 56 24 17 14 13 3 6 6 30 6 14 21 10 10 4 7 14 6 ROM R0015 Caiman crocodilus 16 23 27 35 26 64 30 22 16 14 5 8 5 37 7 17 25 13 9 5 8 15 5 UM 134962 Caiman crocodilus 22 30 34 44 31 77 35 24 21 16 5 8 9 41 11 22 30 13 12 6 10 18 8 UCMZ R 1987.3 Caiman crocodilus 21 31 37 48 35 87 42 25 23 18 4 13 11 47 10 23 33 15 13 7 11 22 8 UCMP 132076 Caiman crocodilus 25 34 40 53 37 89 42 28 24 20 4 11 10 48 11 24 34 16 13 6 12 24 8 CMNAR 35758 Caiman crocodilus 27 35 42 56 37 98 47 32 23 17 7 9 10 50 11 26 37 18 15 8 11 24 9 UCMVZ 199652 Caiman crocodilus 29 38 46 65 44 105 54 33 26 20 8 12 14 55 12 32 42 20 16 9 14 29 10 UCMP 123093 Caiman crocodilus 28 38 45 64 45 107 51 32 24 20 7 12 13 54 12 27 41 18 14 5 14 26 9 UM 130547 Caiman crocodilus 34 46 52 75 52 123 62 39 26 22 12 15 14 60 13 36 49 22 17 9 20 29 10 UCMVZ 58202 Caiman crocodilus 35 44 54 73 53 138 70 39 29 24 9 16 14 70 13 40 48 24 18 9 16 31 12 UCMP 123095 Caiman crocodilus 34 50 59 88 64 148 71 48 31 26 9 16 17 71 16 44 56 29 25 14 20 40 11 UCMZ R 1983.004 Caiman crocodilus 41 56 67 106 73 162 85 50 34 30 11 19 20 81 19 50 63 30 26 14 26 43 12 UM 53113 Caiman crocodilus 34 44 55 85 64 157 87 47 32 26 9 14 15 82 15 43 52 26 25 11 19 31 12 ROM R6587 Caiman crocodilus 40 56 67 102 73 174 92 52 36 31 16 17 19 91 18 45 63 32 30 13 24 39 12 ROM Herp 46525 Caiman crocodilus 47 62 82 120 90 177 95 59 36 30 15 19 19 89 25 54 74 36 28 12 31 43 13 ROM R6888 Caiman crocodilus 47 63 85 121 85 194 102 61 39 34 19 22 25 95 20 57 82 38 23 12 30 49 14 UM 53112 Caiman crocodilus 43 56 68 101 73 196 105 54 34 27 13 20 17 93 21 53 66 34 26 15 24 36 14 ROM R7714 Caiman crocodilus 47 65 85 121 83 196 107 60 38 29 15 23 19 105 18 55 76 40 36 18 29 55 16 ROM R7708 Caiman crocodilus 47 67 85 128 88 200 111 55 37 32 16 21 19 106 21 52 77 39 30 18 27 50 15 ROM R7719 Caiman crocodilus 45 64 83 122 86 206 109 63 39 29 19 25 22 106 23 53 71 36 34 18 33 52 15 ROM R6872 Caiman crocodilus 56 79 104 142 88 215 119 65 40 38 20 25 27 104 23 62 91 46 34 16 29 50 15 ROM R7706 Caiman crocodilus 50 66 90 132 92 216 122 67 40 34 16 23 24 114 22 63 80 42 37 18 34 56 16 ROM R7707 Caiman crocodilus 51 70 87 132 92 221 122 64 42 32 17 23 21 105 23 66 85 43 41 18 35 50 16 ROM R0275 Caiman crocodilus 55 73 88 124 84 228 126 66 32 42 16 23 22 113 21 65 80 43 37 18 32 20 15 UM 128024 Caiman crocodilus 61 78 101 145 98 237 131 74 43 40 18 24 27 123 22 73 86 48 36 22 38 46 18 UCMP 42842 C. c. apaporiensis 19 26 30 40 28 82 42 23 20 17 6 10 10 43 8 21 28 13 13 6 9 19 8 UCMP 42843 C. c. apaporiensis 38 48 63 104 73 217 127 59 40 27 15 23 18 109 17 54 67 35 35 16 25 45 15 UCMP 42844 C. c. apaporiensis 50 64 92 169 118 299 174 85 51 41 20 28 30 156 26 84 98 53 47 24 44 67 20 ROM Herp 46533 Caiman yacare 40 51 60 86 60 157 88 48 32 26 11 19 17 81 17 45 56 27 30 14 19 34 14 ROM Herp 46534 Caiman yacare 39 53 64 85 59 159 84 48 35 25 11 21 17 80 19 45 58 29 31 14 18 37 13 UM 155289 Caiman yacare 80 106 133 196 137 337 188 107 60 46 27 37 34 167 23 101 126 69 47 25 53 62 20 UM 155284 Caiman latirostris 24 34 38 47 31 81 39 26 21 18 6 9 10 39 11 23 29 15 16 7 9 21 8 UM 155285 Caiman latirostris 33 43 51 63 43 111 54 35 25 21 8 12 10 53 12 31 40 20 18 7 14 26 10 UM 155283 Caiman latirostris 43 62 70 86 54 148 75 49 31 24 14 19 15 74 15 46 53 28 20 9 17 34 12 UM 155288 Caiman latirostris 48 64 76 93 67 165 87 54 35 25 11 19 16 80 19 50 59 31 26 13 19 38 15 UM 155286 Caiman latirostris 51 68 81 94 65 173 92 56 32 26 14 21 17 85 21 49 62 31 28 13 22 36 14 UM 155287 Caiman latirostris 77 99 118 149 106 249 143 48 44 32 22 28 24 112 28 81 98 52 49 23 31 55 19 ROM R6692 Paleosuchus palpebrosus 14 20 24 34 23 61 26 18 17 15 6 6 6 33 7 16 23 11 9 4 7 16 6 CMNAR 35983 Paleosuchus palpebrosus 26 36 45 71 49 149 84 39 29 25 11 14 16 79 13 37 48 21 20 8 16 35 11 UCMVZ 71371 Paleosuchus palpebrosus 38 45 56 89 59 159 79 57 37 29 15 20 15 93 14 48 60 33 24 11 21 41 11 ROM Herp 42987 Paleosuchus palpebrosus 43 58 70 104 72 169 86 53 34 30 12 17 20 90 20 54 73 33 22 8 27 43 13 CMNAR 35747 Paleosuchus palpebrosus 41 53 62 99 70 222 129 60 41 34 14 21 21 118 19 56 71 32 24 10 25 49 14 TMP 1985.19.9 Paleosuchus trigonatus 45 59 75 121 80 192 107 59 37 36 16 20 22 102 23 61 79 40 18 9 20 42 12 ROM Herp 46535 Crocodylus niloticus 8 11 14 21 16 37 15 14 11 8 4 4 3 20 5 10 13 5 4 2 4 9 4 UCMP 140795 Crocodylus niloticus 24 37 48 72 51 147 84 41 27 22 9 11 11 80 14 35 47 21 20 8 19 36 10 UCMP 140796 Crocodylus niloticus 25 33 46 71 51 151 92 42 27 22 8 11 11 83 13 33 45 20 20 7 18 37 10 FMNH 19319 Crocodylus niloticus 38 65 83 120 85 261 168 70 39 26 19 18 15 147 62 62 83 41 27 14 29 63 15 UCMP 137149 Crocodylus niloticus 54 78 109 156 113 342 225 97 45 34 29 23 23 212 20 79 112 55 41 18 29 72 24 TMP 1985.19.10 Crocodylus niloticus 79 128 172 246 174 426 271 120 54 47 39 29 32 214 41 109 133 67 33 14 67 94 21 FMNH 217153 Crocodylus niloticus 128 206 263 349 233 609 372 203 69 49 77 37 39 327 59 184 253 111 61 29 95 120 28 ROM 6650 Crocodylus niloticus 124 203 274 384 254 659 402 185 79 62 77 43 47 345 74 163 195 109 64 23 94 141 32 ROM Herp 46537 Crocodylus porosus 8 11 13 20 15 40 17 11 11 10 3 3 3 18 4 9 13 5 5 3 5 8 4 ROM Herp 46538 Crocodylus porosus 7 11 13 20 16 41 18 13 12 9 3 3 3 20 5 9 12 5 5 2 3 11 4 ROM Herp 46536 Crocodylus porosus 8 12 16 23 17 49 22 15 14 11 3 4 3 28 6 12 15 7 8 4 4 12 4 52

Specimen Number Genus and Species 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ROM R 10 Crocodylus porosus 11 17 23 30 22 73 39 20 15 13 4 6 4 43 6 16 10 8 5 3 8 18 5 ROM R 6837 Crocodylus porosus 13 19 26 33 24 82 45 23 16 14 5 6 4 46 8 18 23 10 10 4 8 23 6 ROM R 6838 Crocodylus porosus 15 24 30 39 28 97 53 26 21 18 6 6 6 53 10 20 29 12 9 4 10 27 7 ROM R 15 Crocodylus porosus 14 23 30 42 29 103 56 27 20 15 5 5 5 57 7 24 27 13 12 5 9 23 7 FMNH 15222 Crocodylus porosus 17 26 31 41 30 103 59 25 21 16 5 8 6 59 10 21 25 12 11 4 10 26 7 FMNH 15223 Crocodylus porosus 16 25 31 43 32 109 65 26 21 17 5 7 7 61 11 23 28 14 12 5 11 27 8 UCMZ R ??? Crocodylus porosus 21 32 43 64 44 132 78 38 34 19 8 11 9 77 13 30 41 18 17 6 15 32 9 FMNH 15224 Crocodylus porosus 20 30 39 53 40 136 84 33 23 20 7 8 8 77 11 28 36 17 14 6 13 33 10 UCMP 123090 Crocodylus porosus 19 33 41 56 40 142 84 37 24 19 7 9 9 79 11 29 37 19 13 6 13 33 10 FMNH 15225 Crocodylus porosus 21 35 45 60 45 152 92 38 27 19 6 11 10 86 13 32 40 19 17 6 15 37 10 FMNH 14037 Crocodylus porosus 21 33 43 61 44 155 92 38 29 21 9 10 9 87 11 33 41 19 17 7 17 37 11 FMNH 15226 Crocodylus porosus 23 32 47 64 46 163 97 40 27 20 9 11 10 92 12 34 42 21 18 6 15 40 11 FMNH 14038 Crocodylus porosus 22 37 46 65 48 170 100 41 28 21 9 11 10 92 13 34 43 24 19 8 20 38 11 FMNH 21905 Crocodylus porosus 26 43 58 81 59 186 113 49 31 22 12 12 11 103 14 45 54 26 21 10 23 42 12 FMNH 11003 Crocodylus porosus 27 41 57 81 60 189 117 52 32 28 10 13 12 109 15 41 54 26 24 9 22 43 13 FMNH 14036 Crocodylus porosus 27 45 56 80 57 204 125 50 32 24 12 13 11 111 14 42 52 26 19 8 23 44 13 FMNH 15228 Crocodylus porosus 28 46 61 87 64 212 131 53 34 28 10 16 15 120 17 47 55 27 23 8 21 49 12 FMNH 15227 Crocodylus porosus 27 46 64 90 65 218 135 53 31 25 12 13 13 120 17 46 56 28 22 9 22 47 13 FMNH 21907 Crocodylus porosus 36 59 79 113 84 258 155 69 37 29 19 16 13 143 21 56 71 37 30 9 32 59 14 FMNH 15229 Crocodylus porosus 34 69 91 127 89 274 173 72 38 31 19 16 15 149 20 60 79 39 27 10 33 61 16 FMNH 14230 Crocodylus porosus 41 79 98 134 93 297 191 80 42 33 21 18 17 170 24 66 86 46 32 14 32 69 17 UCMVZ 226098 Crocodylus porosus 55 87 113 157 108 314 196 88 43 31 33 18 23 169 33 70 94 48 38 14 34 69 19 FMNH 28465 Crocodylus porosus 44 79 103 149 112 328 208 94 44 33 23 23 21 179 21 81 99 50 33 13 41 70 19 FMNH 28466 Crocodylus porosus 49 84 107 149 105 332 212 90 46 31 27 20 17 184 23 77 98 54 33 12 43 71 19 FMNH 15231 Crocodylus porosus 49 83 116 158 117 364 233 97 49 36 24 26 18 201 27 79 107 55 38 20 45 87 18 FMNH 28468 Crocodylus porosus 48 93 123 174 139 402 252 106 52 35 32 23 28 213 28 89 121 59 42 14 56 90 17 FMNH 15232 Crocodylus porosus 50 91 123 182 134 408 267 107 55 39 29 21 22 291 32 89 122 61 39 18 53 89 23 UM 167697 Crocodylus porosus 57 109 135 202 136 416 261 112 51 34 37 27 28 215 34 101 131 65 46 18 49 98 19 FMNH 10866 Crocodylus porosus 60 94 128 188 140 423 264 115 57 39 36 31 18 222 26 96 122 64 47 16 66 88 22 FMNH 28469 Crocodylus porosus 52 103 131 196 139 433 272 124 57 37 37 28 21 238 31 104 130 67 48 20 54 100 23 FMNH 13969 Crocodylus porosus 83 136 173 254 175 541 339 163 68 46 45 34 27 282 40 138 183 91 61 23 81 102 25 FMNH 10865 Crocodylus porosus 77 132 176 258 187 553 349 157 67 43 48 40 32 292 40 138 173 90 64 20 79 118 25 FMNH 14071 Crocodylus porosus 108 191 231 355 230 631 404 200 77 56 55 31 37 313 49 184 237 121 71 25 93 130 29 UCMVZ 81487 Crocodylus porosus 129 221 257 386 256 688 419 218 78 53 54 46 36 328 64 195 262 129 83 40 109 148 30 CMNAR 15792 Crocodylus porosus 132 199 279 408 259 749 462 241 90 56 68 56 49 388 68 203 264 127 86 44 109 163 29 FMNH 75658 Crocodylus intermedius 21 34 60 103 72 264 169 62 31 23 19 18 13 151 15 54 59 29 27 12 28 58 16 FNMH 75659 Crocodylus intermedius 33 51 92 157 113 391 259 96 42 27 32 27 19 214 23 81 92 47 37 15 44 88 19 FMNH 75661 Crocodylus intermedius 32 54 95 166 116 401 268 100 40 30 35 31 19 224 26 83 94 49 39 18 48 91 21 UCMP 123733 Crocodylus intermedius 46 70 121 191 146 462 300 118 51 34 41 24 19 258 30 92 119 54 32 13 57 90 23 FMNH 75600 Crocodylus intermedius 48 81 132 224 156 518 345 128 52 36 43 31 27 286 35 104 123 65 44 26 58 115 23 FMNH 75663 Crocodylus intermedius 53 87 126 237 164 529 352 114 51 36 45 31 25 286 40 111 132 65 48 22 58 115 24 FMNH 51691 Crocodylus palustris 64 95 139 200 132 291 165 87 39 39 29 26 29 146 46 84 106 50 33 20 43 82 21 FMNH 31537 Crocodylus palustris 117 197 239 308 201 544 333 132 62 36 51 36 37 276 57 160 221 103 50 25 82 140 25 FMNH 98937 Crocodylus acutus 15 23 27 38 26 84 46 23 16 13 5 8 6 48 7 19 24 11 9 4 9 21 7 FMNH 98938 Crocodylus acutus 16 25 31 43 31 98 57 27 18 15 7 8 7 56 9 22 27 13 12 5 9 25 7 UCMVZ unnumbered Crocodylus acutus 17 31 47 71 53 168 98 44 26 21 11 11 10 94 14 37 45 21 18 8 18 28 10 UCMVZ 222426 Crocodylus acutus 22 36 51 82 59 196 115 52 27 21 13 13 11 111 15 43 51 23 22 10 20 41 11 UCMVZ 39907 Crocodylus acutus 28 46 63 98 69 233 146 57 32 24 17 15 12 131 16 49 60 28 24 9 25 51 12 FMNH 5775 Crocodylus acutus 29 48 67 109 81 254 161 64 35 26 20 14 12 143 18 53 65 31 25 10 29 49 16 UCMVZ 39906 Crocodylus acutus 30 48 65 108 79 261 164 66 33 24 18 17 13 149 17 52 64 30 27 12 29 54 14 FMNH 5776 Crocodylus acutus 34 52 72 113 83 263 168 66 37 25 22 17 13 151 21 57 68 33 27 11 30 55 17 UCMP 81699 Crocodylus acutus 43 56 82 120 86 274 176 70 43 25 21 18 14 152 18 59 74 30 27 8 27 60 14 FMNH 74924 Crocodylus acutus 31 47 71 107 80 281 184 67 37 24 19 18 12 158 20 56 70 31 28 11 34 60 15 ROM 73??? Crocodylus acutus 35 62 94 139 104 354 223 88 45 32 23 22 20 199 24 69 85 41 34 13 44 71 19 ROM R 934 Crocodylus acutus 52 80 119 184 134 384 232 109 51 35 36 23 19 211 28 90 121 58 41 20 50 78 20 FMNH 11057 Crocodylus acutus 52 83 121 202 143 452 285 129 53 37 35 32 20 241 27 102 120 62 34 16 58 84 27 FMNH 34563 Crocodylus acutus 67 122 163 221 151 482 302 134 52 45 43 29 27 263 41 116 139 68 59 24 63 108 23 FMNH 28847 Crocodylus acutus 84 136 168 260 179 534 358 153 62 43 48 36 27 291 44 120 156 75 56 22 75 108 25 FMNH 69886 Crocodylus acutus 76 115 183 268 172 547 352 154 61 47 61 32 27 281 44 143 177 83 55 24 75 114 24 FMNH 73755 Crocodylus acutus 83 151 216 283 177 587 376 166 63 41 65 33 31 303 52 150 190 94 59 25 77 117 26 FMNH 59070 Crocodylus acutus 88 145 212 299 206 663 434 192 57 42 74 31 27 353 47 159 212 101 65 29 84 134 25 ROM Herp 23477 Crocodylus novaeguineae 10 15 22 32 23 72 40 19 13 16 4 5 4 41 8 15 20 9 8 3 7 19 6 FMNH 14053 Crocodylus novaeguineae 12 19 26 41 29 95 55 23 18 14 4 7 7 54 8 20 25 12 11 5 10 22 8 FMNH 34491 Crocodylus novaeguineae 18 32 38 55 41 140 82 33 24 18 7 9 8 78 10 27 35 16 12 5 13 33 10 UM 46780 Crocodylus novaeguineae 24 39 57 78 57 169 101 49 29 24 12 16 11 94 13 41 47 25 21 9 20 42 11 53

Specimen Number Genus and Species 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 FMNH 13966 Crocodylus novaeguineae 24 40 56 89 64 194 120 50 30 25 12 13 13 113 14 47 50 27 21 9 19 42 13 FMNH 15043 Crocodylus novaeguineae 50 88 128 172 130 383 234 108 49 37 29 32 22 212 37 99 118 57 32 17 45 81 20 FMNH 14044 Crocodylus novaeguineae 58 103 139 217 149 421 254 128 53 42 32 27 21 234 34 112 142 66 35 15 61 84 22 FMNH 4432 Crocodylus moreletii 18 32 37 55 40 120 68 31 22 17 8 10 10 65 11 26 34 16 13 6 11 31 8 UM 129718 Crocodylus siamensis 26 41 52 77 55 147 81 41 27 19 10 11 12 79 18 37 49 22 18 9 18 39 10 FMNH 21904 Crocodylus mindorensis 25 38 54 79 58 183 106 51 28 22 11 11 10 100 15 44 52 25 18 9 22 46 12 UCMP 81703 Crocodylus rhombifer 19 31 39 53 39 134 81 34 24 18 6 11 7 78 11 27 34 17 15 6 14 33 9 UM 155301 Crocodylus rhombifer 24 34 44 49 43 135 73 35 24 22 10 9 10 73 12 29 39 17 15 7 13 34 12 FMNH 44410 Osteolaemus tetraspis 12 20 23 34 24 59 27 18 16 13 6 4 6 32 6 16 24 11 6 3 7 18 7 FMNH 44442 Osteolaemus tetraspis 19 31 35 50 34 87 41 25 24 19 8 4 8 46 9 25 35 15 10 4 10 27 8 TMP 1985.19.8 Osteolaemus tetraspis 34 51 60 84 61 132 63 42 30 27 13 8 12 52 17 43 58 26 14 6 19 40 8 ROM Herp 46528 Osteolaemus tetraspis 33 50 60 89 63 141 71 44 31 25 14 8 16 77 11 41 64 28 16 8 21 42 12 ROM Herp 46530 Osteolaemus tetraspis 35 52 66 95 68 152 82 46 31 27 13 9 18 70 16 46 69 31 16 8 22 47 13 ROM R6533 Osteolaemus tetraspis 46 66 82 117 76 179 87 57 38 32 18 10 15 85 24 60 82 39 18 9 27 54 16 ROM Herp 46529 Osteolaemus tetraspis 41 70 82 117 64 182 88 59 43 32 16 13 21 90 23 62 85 41 16 9 28 56 12 ROM R19 Gavialis gangeticus 5 6 15 24 18 95 62 17 10 8 4 6 3 63 4 11 14 6 8 3 7 12 4 FMNH 98864 Gavialis gangeticus 13 11 36 63 48 234 170 44 21 21 15 20 10 152 9 32 34 17 15 5 24 33 10 FMNH 82681 Gavialis gangeticus 14 12 35 72 56 263 186 49 22 23 18 20 11 181 13 32 37 17 17 5 26 36 11 UM 155302 Gavialis gangeticus 33 36 103 197 146 548 396 119 41 44 52 45 28 370 34 85 96 50 40 15 66 71 22 TMP 1982.5.8 Gavialis gangeticus 40 46 151 231 181 711 512 150 53 58 67 55 35 502 46 110 120 65 46 20 82 86 26 FMNH 11084 Tomistoma schlegelii 7 9 22 42 33 138 88 28 20 16 4 8 6 85 9 22 26 14 15 5 13 8 8 UM 174416 Tomistoma schlegelii 10 10 27 55 42 176 115 36 26 16 5 11 9 100 10 27 32 16 15 5 18 33 8 UCMP 81702 Tomistoma schlegelii 12 15 36 72 52 218 147 45 27 22 6 14 10 136 10 37 43 24 19 8 21 37 11 UM 128552 Tomistoma schlegelii 14 16 40 81 60 222 151 47 31 24 8 15 9 131 14 38 47 24 25 10 26 42 12 UM 129397 Tomistoma schlegelii 16 20 46 90 66 239 161 52 32 26 8 17 11 145 16 43 51 27 22 9 27 45 13 CMNAR 43997 Tomistoma schlegelii 11 16 36 75 51 241 171 47 31 21 8 16 12 154 12 40 46 26 20 8 21 47 11 ROM Herp 46526 Tomistoma schlegelii 19 21 60 128 98 265 166 68 37 27 16 18 15 142 23 56 68 36 29 12 36 54 17 ROM Herp 46527 Tomistoma schlegelii 21 27 66 130 95 328 223 69 38 27 14 22 16 193 22 62 71 39 36 14 34 62 15 ROM R20 Tomistoma schlegelii 25 30 70 140 103 446 309 88 52 32 16 25 15 281 24 70 82 44 42 14 41 76 18 UCMVZ 77052 Tomistoma schlegelii 24 31 81 165 119 457 318 102 48 36 19 29 17 273 24 83 94 52 41 17 54 78 24 FMNH 206755 Tomistoma schlegelii 33 41 100 206 162 459 293 116 52 36 33 32 25 265 29 99 122 63 59 21 66 90 22 54

Chapter Two

Evidence for Taphonomic Size Bias in the Dinosaur Park

Formation (Campanian, Alberta), a Model Mesozoic Terrestrial

Alluvial-Paralic System

Caleb Marshall Brown1,2, David C. Evans1,2, Nicolás E. Campione1, Lorna J. O’Brien1, and

David A. Eberth3

1Department of Ecology & Evolutionary Biology, University of Toronto, 25 Willcocks Street,

Toronto, Ontario M5S 3B2, Canada

2Department of Natural History, Palaeobiology division, Royal Ontario Museum, 100 Queen's

Park, Toronto, Ontario M5S 2C6, Canada

3Royal Tyrrell Museum of Palaeontology, Drumheller, T0J 0Y0, Canada.

Published as:

Brown C. M., D. C. Evans, N. E. Campione, L. J. O’Brien and D. A. Eberth. 2013. Evidence for

Taphonomic size bias in a model Mesozoic terrestrial alluvial-paralic system. Palaeogeography,

Palaeoclimatology, Palaeoecology, 372: 108-122.

Abstract

A study of the distribution of dinosaurian body masses in the Dinosaur Park Formation (DPF;

Campanian; southern Alberta), reveals a prominent negative skew; a pattern distinct from those of modern terrestrial faunas. We find a direct and robust correlation between taxon size

(estimated live body mass) and known completeness. There is a clear dichotomy between large and small-bodied taxa at around 60 kg, in which taxa less than 60 kg are significantly less complete (mean completeness = 7.6%) than those with an estimated mass of 60 kg or greater

(mean = 78.2%). Along with completeness, there is also a strong association of body size and taphonomic mode, with small taxa known largely from isolated and occasionally associated remains, and large taxa known from articulated skeletons. In addition there is a significant correlation between taxon body mass and both date of discovery and of description, with taxa <

60 kg taking an average of 60.5 and 74.8 years to discover and describe, respectively, compared to 29.6 and 35.9 years for taxa > 60 kg. The rates of both cumulative discovery and description for large taxa are best described by a logarithmic curve nearing an asymptote, whereas small taxa show either a linear or power increase though time. This suggests our current knowledge of the large-bodied dinosaur assemblage is reasonably representative of the true biological fauna with few discoveries likely to be made in the future. However, small taxa are greatly underestimated in both their diversity and abundance, with many more potential discoveries to be made. Given that (1) the sedimentary deposits and fossil assemblages at DPF together represent one of the best studied examples of a Mesozoic alluvial-paralic (terrestrial) 'palaeoecosystem,' and (2) similar patterns have been suggested (but not documented) for other Mesozoic terrestrial ecosystems in the Western Interior of North America. We suggest this pattern of size bias may typify vertebrate fossil assemblages in terrestrial Mesozoic systems. If so, such biases must be

56 considered before patterns of diversity in dinosaur communities through time can be considered accurate, or used to compare and interpret Mesozoic palaeoecosystems.

Introduction

Non-avian dinosaurs represent a diverse group of extinct archosaurs that formed the dominant terrestrial megafauna from the Late Triassic to their extinction at the end of the Mesozoic.

During this time non-avian dinosaur populations and ecosystems experienced significant shifts in climate and palaeogeography (e.g., Smith et al., 2004; Miller et al., 2005), which are thought to have had a profound effect on patterns of distribution, evolution and diversity within this group.

As a result, non-avian dinosaurs are particularly useful for investigating terrestrial ecosystem evolution as well as identifying mechanisms that underlie major extinction events (Fastovsky et al., 2004; Sullivan, 2006; Wang and Dodson, 2006; Barrett et al., 2009; Butler et al., 2011;

Lloyd, 2012).

The study of evolution and diversity in deep time, however, is limited by gaps and biases in the fossil record including, among others: Lagerstätten effects, sediment availability, the ‘pull of the Recent’, and sampling/research intensity (Raup, 1972). Sampling intensity biases have received the most attention in studies of dinosaur diversity dynamics, and have been addressed using various methods (e.g., Fastovsky et al., 2004; Wang and Dodson, 2006; Barrett et al.,

2009; Lloyd, 2012). However, other biases, such as those related to taphonomy and preservation potential have received relatively little attention, particularly with respect to non-avian dinosaurs.

The vast majority of the non-avian dinosaur record is derived from terrestrial alluvial- paralic systems (e.g., the North American Hell Creek, Lance, Dinosaur Park, Oldman, and

Kaiparowits formations) (Currie and Koppelhus, 2005; Eberth, 2005; Rogers et al., 2007; Eberth et al., 2010; Horner et al., 2011) and a negative skew in the distribution of species body sizes is

57 common in non-avian dinosaur assemblages from these systems, where the majority of species in the assemblages are composed of larger-bodied taxa (Peczkis, 1994; Farlow et al., 1995).

However, this pattern of size distribution is unexpected when compared to modern ecosystems, which tend to have a much higher abundance of small-bodied taxa, and hence a positive skew, which is thought to reflect the ecological trade-offs between the selective advantages and risks of large body size (Elton, 1927; Colinvaux, 1978; Peters, 1983; Peters and Wassenberg, 1983;

Lawton, 1990; Griffiths, 1992; Blackburn and Gaston, 1994; Brown, 1995; Blackburn and

Gaston, 1996; Woodward et al., 2005; Clauset and Erwin, 2008). One possible explanation for this body size distribution is a taphonomic bias against small skeletons. This explanation has been invoked by several authors (Dodson, 1971; Farlow et al., 1995; Sereno, 1999; Longrich and

Currie, 2009a; Eberth et al., 2010; Carbone et al., 2011b) and is supported by actualistic taphonomic work on modern ecosystems (Behrensmeyer et al., 1979; Behrensmeyer and

Dechant Boaz, 1980; Behrensmeyer, 1988; Sept, 1994; Kidwell and Flessa, 1996; Arribas and

Palmqvist, 1998).

Although a taphonomic bias against small-bodied animals has been suggested to explain the non-analogue (i.e., not represented by extant systems – Williams and Jackson, 2007), negatively-skewed size distributions in many dinosaurian palaeoecosystems (Dodson, 1971;

Farlow et al., 1995; Sereno, 1999; Longrich and Currie, 2009a; Eberth et al., 2010; Carbone et al., 2011b), there have been few attempts to quantitatively test for this bias and establish its degree of influence. If such a bias could be rejected, then inconsistencies between modern and

Mesozoic assemblages could be more confidently attributed to intrinsic differences between the ecological structures of Mesozoic and modern terrestrial ecosystems, as was recently suggested by Codron et al. (2012). Alternately, if a bias is supported, it suggests that small-bodied taxa were more abundant and diverse in the Mesozoic than is indicated by preserved fossil

58 assemblages. Such conclusions have significant implications for our current understanding of both the structure of Mesozoic ecosystems, and diversity of small-bodied taxa within these ecosystems.

The Dinosaur Park Formation (DPF; Late Campanian at Dinosaur Provincial Park, southern Alberta, Canada) represents the best sampled and most diverse of any Mesozoic terrestrial vertebrate assemblage, and has been one of the most intensely studied (Dodson, 1971;

Dodson, 1983; Currie and Koppelhus, 2005). The DPF is an alluvial-paralic unit deposited during a transgressive phase of the Western Interior Seaway (Eberth, 2005). After more than a century of intense collecting, 50 valid dinosaurian taxa (Ryan and Russell, 2001; Currie, 2005;

Ryan and Evans, 2005; Longrich, 2009; Longrich and Currie, 2009a) have been described from the DPF. The dinosaur fauna of the DPF is unequivocally dominated by large-bodied animals, both in terms of the described taxa and abundance of fossils (Ryan and Russell, 2001; Currie and

Russell, 2005; Ryan and Evans, 2005). The dinosaur species represented in the DPF have a negatively skewed distribution in terms of their body sizes (Fig. 1) (skewness = -0.4079,

Shapiro-Wilk p-value = 0.000252) (Peczkis, 1994). It is therefore an excellent model system for quantifying modes of fossil preservation (e.g., Eberth and Currie, 2005), and testing the hypothesis that the body size distributions of dinosaurs is driven by taphonomic biases, where small-bodied animals are less likely to be preserved. Using the extensive dataset available for the

DPF, we quantitatively test for independence between body size and 1) the completeness of the skeletons from which a taxon is known; 2) the relative discovery time of the species; and 3) the date of species publication. Quantification of this bias in alluvial-paralic systems demonstrates that small dinosaurs were a more abundant and diverse component of Mesozoic communities than presently appreciated.

Institutional Abbreviations

AMNH, American Museum of Natural History, New York, New York, USA; CMN, Canadian

Museum of Nature, Ottawa, Canada; FMNH, Field Museum of Natural History, Chicago,

Illinois, USA; ROM, Royal Ontario Museum, Toronto, Ontario, Canada; TMP, Royal Tyrrell

Museum of Palaeontology, Drumheller, Alberta, Canada; UALVP, University of Alberta

Laboratory of Vertebrate Paleontology, Edmonton, Alberta, Canada; YPM, Yale Peabody

Museum, New Haven, Connecticut, USA.

Methods

Dinosaur Park Formation Faunal Data Set

An updated dinosaurian faunal list for the DPF was compiled from multiple sources (Ryan and

Russell, 2001; Ryan and Evans, 2005; Longrich, 2008; Longrich, 2009; Eberth and Evans, 2011;

Ryan et al., 2012). Attempts have been made to survey the literature as thoroughly as possible and to use the most current taxonomy. Literature was surveyed to establish dates of discovery/collection and dates of description for all DPF dinosaur taxa from its first systematic collections in 1897 until 2010. This sample is comprised of 50 dinosaur taxa that are here considered valid (Table 1).

Year of Discovery (YDi) - the year in which the first identified remains of a species were discovered or collected from the DPF, and Year of Description (YDe) - the year in which the first description of a species based on DPF material was published (often, but not necessarily the year the taxon was formally named), were determined for each species. Exact YDi for two taxa -

Ornithurine B and C (Longrich, 2009) were indeterminable. Dates for both YDi and YDe were taken as the first occurrence of that taxon, regardless of its completeness. For instance,

Stegoceras validum is known from a partial skeleton found in 1922 and described in 1924, but this taxon was first found in 1898 and documented in 1902 based on an isolated dome, and therefore the dates of 1898 and 1902 are used here (Lambe, 1902; Gilmore, 1924; Schott et al.,

2011).

Time to Discovery (TDi) – is the number of years it took from the first systematic prospecting of the DPF in 1897 (Lambe, 1902), until a taxon was discovered (the difference between the discovery date and 1897 plus one). Thus, for taxa discovered in 1897 the time to discovery is estimated as one year. Time to Description (TDe) – is the number of years it took to describe a taxon since the first publication of a DPF taxon description in 1902 (Lambe, 1902)

(the difference between the description date and 1902 plus one). For taxa described in 1902 the time to description is one year.

For analysis of familial diversity, all taxa in the DPF faunal list, with the exception of the unnamed ornthurines and Elmisaurus elegans, were assigned to family level (Table 1) following the taxonomy in the Dinosauria (Weishampel et al., 2004) and Hope (2002). Although

Elmisaurus elegans is likely an oviraptorosaur, it is unclear if it falls within the Caenagnathidae or Oviraptoridae, and as such we do not assign it to family.

Body Mass Estimation

Body mass, a measure of size, was estimated for all dinosaurian taxa at their presumed average adult size. The vast majority of estimates, with a few minor exceptions, were made using regression equations between skeletal measurements and body mass in extant taxa. The body masses of quadrupedal taxa, including ceratopsians, hadrosaurids, and ankylosaurids, were estimated using the combined minimum circumference of the humerus and femur input into the phylogenetically corrected prediction equation presented by Campione and Evans (2012). The

61 body masses of bipedal taxa, including all theropods (for which femora are known) and cf.

Orodromeus (TMP 1990.36.65), were determined based on the bipedal formula presented by

Anderson et al. (1985). In certain cases, a single body mass estimate was used for closely related taxa, which are of similar body size; for instance, all Lambeosaurus species were estimated based on the limbs of Lambeosaurus lambei (ROM 1218). Similarly, both Corythosaurus species were estimated based on a skeleton of C. intermedius (ROM 845). In a few cases when the humerus and/or femur were not preserved (Dyoplosaurus acutosquameus, Elmisaurus elegans) these measurements were estimated based on closely related taxa using a Bayesian PCA (Oba et al., 2003; Brown et al., 2012)

Estimates for taxa known only from teeth (e.g., Richardoestesia isosceles,

Dromaeosaurus morphotype A, and Paronychodon sp.), were based on estimates of closely related taxa that are known from more complete material, such as Richardoestesia gilmorei,

Dromaeosaurus albertensis, and Troodon inequalis, respectively (Carbone et al., 2011a).

Similarly, the mass estimates for taxa known only from jaws (Caenagnathus collinsi,

Caenagnathus sternbergi, and Unescoceratops koppelhusae) were based on the estimated masses of related taxa (Chirostenotes pergracilis, and Leptoceratops gracilis). Mass was not estimated for the putative indeterminate therizinosaur, cf. Erlikosaurus sp. (CMN 12355; Currie, 1987); it is based on a very small amount of material and its taxonomic status as a therizinosaur is equivocal (DCE, pers. obs.).

The ornithurine birds from the DPF (Longrich, 2009) are known primarily from coracoid bones, most of which preserve the width of the shaft, but are too incomplete to obtain their length. Therefore, body mass estimates were derived from a regression analysis between minimum dorsovental coracoid shaft width (taken along the lateral aspect of the coracoid) and body mass in a sample of 21 extant ornithurine birds. The regression equation,

log Body Mass = 2.2054 • log Coracoid Width + 1.3779 has a high coefficient of determination (R2 = 0.943) and serves as a rough estimate of body mass in these fragmentary taxa.

The database of measurements, of both taxa from the DPF, and the extant ornithurines can be found in the supplementary information (Appendix B).

Taxonomic Completeness Index

Each dinosaur taxon from the DPF was quantified in terms of its known completeness using a slightly modified version of the Skeletal Completeness Metric 2 (SCM2) of Mannion and

Upchurch (2010). This metric examines all skeletal material referable to each taxon and quantifies the percentage represented in each of ten anatomical regions (skull, cervical vertebrae and ribs, dorsal vertebrae and ribs, sacral vertebrae and ribs, caudal vertebrae and ribs, pectoral girdle, forelimbs, pelvic girdle, hindlimbs, and miscellaneous). Each region is expressed as a relative completeness; i.e. the number of bones that are preserved compared to the number that should be present for each region. All regions per taxon can then be average to obtain a mean taxon completeness. As this metric was developed primarily for sauropods, Mannion and

Upchurch (2010) weighted these regions by relative proportions (i.e. higher weight given to cervical vertebrae and ribs, dorsal vertebrae and ribs, and caudal vertebrae and ribs). Our sample encompasses a great diversity of variation in relative proportions of these anatomical regions, and as such, all regions were weighted equally in this study (equal weighting during the averaging).

Taphonomic Mode

Each dinosaur taxon from the DPF was assigned a taphonomic mode following the classification of Eberth and Currie (2005), modified from Dodson (1983), Koster et al. (1987) and Wood

(1988). Based on the criteria of Eberth and Currie (2005) the most complete or best preserved specimen from each taxon examined was assigned to a taphonomic mode that represents a disarticulation continuum; consisting of: Articulated (A) – complete (A1) or partial (A2),

Associated (B), or Isolated (C). Following Eberth and Currie (2005), an articulated skeleton is defined as having greater than 50 percent of its preserved elements in articulation, and a complete skeleton preserves all of its major body parts. Here mode C includes both elements that are regarded as truly isolated and elements derived from Microfossil Assemblages, which combines modes C and E of Eberth and Currie (2005).

Centrosaurus and Hadrosaurid Specimen Size Distribution

To test the pattern of distribution of specimen sizes within a taxonomic group, two of the most common taxonomic groups, one at species level (Centrosaurus) and one a family level

(Hadrosauridae), were evaluated in terms of their specimen size distribution. A dataset of

Centrosaurus occipital condyle diameter (transverse width) measurements, a good proxy for intraspecific body size (Anderson, 1999), was complied based on occipital condyles from both

Centrosaurus bonebeds, and individual articulated skulls. This dataset encompasses nearly the entire sample of known Centrosaurus occipital condyles (n=90). Because articulated skulls are essential for species-level taxonomic identification of hadrosaurid specimens from DPF, another dataset was compiled to quantify the size range of articulated hadrosaurid skulls from the park.

This dataset utilizes skull length (n = 56) and quadrate height (n = 55) as proxies for body size in the eight species recognised from the DPF.

Statistical Analyses

All analyses were conducted in Microsoft Excel (V 12.3.0) and R Statistical Language (V 1.4) (R

Core Development Team, 2009). In order to find a threshold where the difference in mean completeness is maximized across the continuous size ranked taxon list (and therefore the most natural size groupings), a sliding window approach was used. This was obtained by breaking the list into all possible combinations of two discrete clusters (i.e., all possible groupings of three or more while maintaining the ranked state) and measuring the difference in the mean of each group. For example, the mean completeness of the smallest three taxa is compared to the mean completeness of the largest 47 taxa, and then the mean completeness of the smallest four taxa was compared to the mean completeness of the remaining 46 taxa. This method was used to separate the taxa into small, intermediate, and large (or just large and small) size classes.

Results

Correlation of Body Mass and Completeness

A plot of taxonomic completeness (SCM2) as a function of log body mass for all DPF taxa shows a significant correlation between these two factors (Pearson’s correlation [r] = 0.81, p <<

0.001) (Fig. 2, Table 2). When the dinosaur taxa are divided into Ornithischia and Theropoda subsamples, significant correlations are still evident in both groups, with a higher correlation in

Ornithischia (r = 0.88 and 0.61, respectively, p < 0.001). This result holds true for all taxonomic groups for which a wide range of body masses are known, including Marginocephalia (r = 0.88),

Ornithopoda (0.95), and Ceratopsia (0.89) (Fig. 2, Table 2). This demonstrates that the

65 relationship between body mass and completeness is not restricted to, or driven by, patterns in certain taxonomic groups.

When the DPF taxa are positioned in rank order of increasing mass and plotted against completeness, a salient trend is evident where small-bodied taxa (< 60 kg) are relatively incomplete, large taxa (> 370 kg) are relatively complete, and the intermediate-sized taxa (60–

370 kg) show a great deal of variation in their completeness (Figs. 3 and 4). Using the sliding mean difference method (see Section 2.6), there were two areas where the highest difference in mean completeness occurred between two groups in the dataset, illustrated as peaks (Fig. 4). The first division occurs between the 25th and 26th taxa (Troodon inequalis or Paronychodon sp. and

Ornithomimus edmontonicus) at an estimated mass between 58 and 60 kg. This point marks the smallest taxon that exceeds 50% completeness, in this case 100% complete (Ornithomimus edmontonicus), and as such marks the point where the smallest nearly complete taxa are found.

The second division occurs at a less precise location between the 29th and 32nd taxa (inclusive of:

Unescoceratops koppelhusae (Ryan et al., 2012), the large unnamed ornithomimid (Longrich,

2008), Edmontonia rugosidens and Panoplosaurus mirus) at an estimated mass between 160 and

1600 kg. Although the sliding window mean difference method is ambiguous as to the precise location of this division, the middle of this range (between the large unnamed ornithomimid and

Edmontonia rugosidens) is characterised by both the most massive relatively incomplete taxon

(all larger taxa are more than 41% complete), and a distinct jump in size from 370 kg to 1400 kg.

For this reason, the second division is interpreted to occur between the large unnamed ornithomimid and Edmontonia rugosidens (the 30th and 31st taxa), at an estimated mass range of

370 to 1400 kg. Of these two divisions within the size range, the first (at ~ 60 kg) may be more significant, as it marks the smallest mass at which taxa are consistently found at relatively high completeness (Fig. 4). It is also a convenient division as half of the taxa in the faunal list occur

66 under this body mass, and half occur above it. These divisions allow us to conveniently divide the fauna into either small (<60 kg), intermediate (>60 and <360 kg), and large (>360 kg), or just small (<60 kg) and large (>60 kg) size classes for further comparisons of completeness and history of discovery.

All taxa smaller than 60 kg (n = 25) are less than 41% complete. Only five of these small taxa (Dromaeosaurus albertensis, Saurornitholestes langstoni, Stegoceras validum,

Chirostenotes pergracilis, and Troodon inequalis) are known from more than 10% of the skeleton, and the relatively high completeness of these five taxa can largely be attributed to a single specimen in each case (AMNH 5356, TMP 1974.10.5, UALVP 002, TMP 1979.20.01, and CMN 8539). The mean completeness for the small-bodied group is 7.6%, with both a median and mode of 1%. If the five taxa above are excluded, the mean completeness drops to

2.3%. Interestingly, all small-bodied taxa (< 60 kg, n = 25) are less complete than any of the large-bodied taxa (> 370 kg, n = 20), indicating no overlap in completeness between the two size classes (Figs. 3 and 4). The small size class is dominated by theropods (n = 20, relative to 5 ornithischians), specifically Aves (n = 8), Deinonychosauria (n = 8), and Oviraptorosauria (n =

4).

All taxa larger than 370 kg (n = 20) are more than 41% complete, with a mean completeness of 85.6%, a median of 95.5%, and a mode of 100% (Figs. 3 and 4). This size class is dominated by ornithischians (n = 18, relative to two theropods), specifically Hadrosauridae (n

= 7), Ceratopsidae (n = 6), and Ankylosauria (n = 4).

The five taxa in the intermediate size range, between 60 and 370 kg, exhibit a range of individual completeness scores between 1–100%, and an intermediate average value (mean =

50.2%, median = 45.5%). The intermediate size class consists of ornithomimids (n = 4) and the single leptoceratopsid (n = 1) (Figs. 3 and 4).

If the large and intermediate size ranges are pooled, small (< 60 kg) and a large (≥ 60 kg) size categories have equal numbers of species (n = 25). The pooled large size class (all taxa 60 kg or larger) has a mean completeness of 78.5%, median of 94.4%, and mode of 100% (Fig. 3 and 4). This pooled size class is used for much of the remaining analyses.

When ornithischian and theropod taxa are separated and positioned in rank order of increasing mass against completeness, the trends are the same as when the entire fauna is considered, but the threshold mass between small and large-bodied clusters – determined with sliding window mean difference method - is slightly different between the two groups (Fig. 5,

Table 2). Within the theropods every taxon with a mass of ≥ 60 kg (with the exception of the large unnamed ornithomimid) is more complete than every taxon < 60 kg. Similarly, for the ornithischians, every taxon ≥ 160 kg in mass is more complete than every taxon < 160 kg.

Although the proportion of taxa falling into the two size categories is substantially different (six small and 18 large for the ornithischians, and 20 small and six large for the theropods), the absolute body mass at which the distinction occurs (60 kg vs. 160 kg) is relatively consistent given the size ranges (0.10 kg – 4500 kg). This suggests that not only is this trend consistent in both Ornithischia and Theropoda, but a similar absolute body mass threshold is noted in both groups.

Correlation of Body Mass and Taphonomic Mode

In addition to completeness, there is also a significant correlation between body mass and taphonomic mode for the DPF taxa. The small-bodied taxa (< 60 kg, n = 25) show 19 taxa known from isolated remains, five from associated remains, and only one from an articulated skeleton (Fig. 3). The large taxa (> 60 kg, n = 25), however, are characterised by only two species from isolated, and one from associated remains, with the remaining 22 known from

68 articulated skeletons (Fig. 3). This dichotomy is highly significant (Chi-squared value = 39.3, p- value << 0.001), and shows an increase in the representation of taxa from articulated skeletons as body mass increases. Body mass distribution of the taxa assigned to each taphonomic mode illustrates clear size dichotomy between the taxa characterizing each mode (Fig. 6).

When using body mass as a continuous variable, and treating the taphonomic modes as a disarticulation continuum, from Articulated (3) to Associated (2) to Isolated (1), a tight and highly significant correlation exists (Spearman ranked [rs] mass = 0.81, Pearson [r] log mass =

0.82; all p-values << 0.001) (Table 3).

When the dinosaur taxa are segregated into their respective orders, both Ornithischia (n =

24, p-value << 0.001) and Saurischia (n = 26, p-value = 0.02) independently show a significant relationship between large and small size classes and the taphonomic modes that characterise the taxa (Fig. 5). If body mass is treated as a continuous variable Ornithischia (rs = 0.78, r = 0.89; all p-values << 0.001) and Theropoda (rs = 0.70, r = 0.65; all p-values << 0.001) both show significant correlations between body mass and dominant taphonomic mode (Table 3).

Segregation of the taxa into even less inclusive taxonomic groups (for which a range of body masses are known) continues to show very significant correlations between taphonomic mode and body size: Marginocephalia (r = 0.84, p-value < 0.001), Ornithopoda (r = 0.99, p-value <<

0.001), and Ceratopsia (r = 0.85, p-value = 0.02) (Table 3).

Correlation of Body Mass and Time to Description (TDe)/Discovery (TDi)

In addition to the correlation between the body size of a taxon and how much of its skeleton is represented in an assemblage, there is also a significant (r = -0.62, p < 0.01) correlation between body size and the length of time taken for a taxon to be described (TDe) (Fig. 7, Table 4). The description of new large-bodied dinosaur species in the DPF has slowed in the last 70 years.

Following a description plateau from 1940 to 1969, the only large (>160 kg) ornithischians named from the DPF are cf. Pachyrhinosaurus (Ryan et al., 2010) and Vagaceratops irvinensis

(Holmes et al., 2001). Similarly, the only large (≥ 60 kg) theropod named after 1933 is

Daspletosaurus sp. (Russell, 1970). However, since 1940 there have been four small ornithischians (<160 kg) and 16 small theropods (< 80 kg) described. If the TDe between the two size classes (< 60 kg and ≥ 60 kg) is compared, there is a significant (p < 0.01) dichotomy, with mean time to description of large taxa approximately half that of small taxa (35.9 years compared to 74.5 years) (Fig. 7, Table 5).

Importantly, four of the last five large-bodied taxa to be described from the DPF, the large unnamed ornithomimid (Longrich 2009), Vagaceratops irvinensis (Holmes et al., 2001),

Dromiceiomimus samueli (Russell, 1972) and Daspletosaurus sp. (Russell, 1970), were all found and collected much earlier than they were described (1967, 1958, 1926, and 1913, respectively) and were only recognised as distinct taxa when the jackets were prepared, or following taxonomic revisions. Based on this, a more appropriate metric to quantify size bias in the rate of discovery of new taxa, may be the date of discovery or collection of the taxa, rather than the date of description.

When body mass is plotted against time to discovery/collection (TDi) the relationship is similar (r = -0.59, p < 0.01) to that of body mass and TDe (Fig. 7, Table 4). The lack of recently discovered large-bodied dinosaur taxa is obvious. There has only been one new large-bodied

(>160 kg) ornithischian taxon discovered since 1958, cf. Pachyrhinosaurus (Ryan et al., 2010), and no new large bodied (≥ 60 kg) theropod taxa discovered since 1967 (1926 if the large unnamed ornithomimid is excluded). However, during these respective time periods there have been three small ornithischian and 13 small theropod taxa discovered. Comparison of TDi

70 between size classes shows that the mean time to discovery of large taxa is less than half that of small taxa (29.6 years compared to 60.5 years) (Tables 8, 6, Fig. 7).

We recognise that collected but heretofore unidentified taxa cannot be included in this type of analysis until they are recognised as distinct taxa. As such, and unlike the description curve, this approach may underestimate the number of recently discovered but as yet undescribed taxa.

Description and Discovery Rates of Small and Large Taxa

The cumulative discovery and description rates (5 year bins) are illustrated in Fig. 8. These curves are methodologically similar to those illustrating the pattern of cumulative discovery rates for dinosaurs as a whole calculated by Wang and Dodson (2006) and Benton (2008). The

Dinosaur Park Formation data show a high rate of both discovery and description in the initial 40 years (1897-1940), a distinct lag in discovery and description from 1940 to the late 1960’s, and then an increase in discovery and description from 1970 to 2010. The final increase from 1970 to

2010 has been steadier in taxonomic discovery, with taxonomic description showing a sharp increase after 2000 (Fig. 8). Differences in discovery and description dates suggest there are different accumulation rates between small and large size classes (Fig. 9). The rates of both discovery and description of large-bodied taxa (>60 kg) are initially high, and slow through time

(Fig. 9). These patterns of discovery and description (divided into 5 year bins) are best characterised by a logarithmic function (R2 = 0.91 and 0.96, respectively) (Fig. 9, Table 6). In contrast, the small taxa show a relatively constant or increasing pattern of discovery and description through time (Fig. 9), fitting a power function (R2 = 0.93 and 0.92) (Fig. 9, Table 6).

The same analysis using 10 year bins rather than five-year bins obtains similar results (Table 6).

Size Distributions of Specimens Within a Taxon

Within large-bodied species there is also a clear dichotomy between the abundance of specimens and their size. It has long been recognised that juvenile dinosaurs are rare in the DPF assemblage, even in the most common taxa, hadrosaurids and ceratopsids (Dodson, 1975;

Dodson and Currie, 1988; Sampson et al., 1997; Tanke and Brett-Surman, 2001; Evans, 2003;

Evans et al., 2005; Evans et al., 2007). In Centrosaurus, occipital condyle transverse diameter shows a size distribution of specimens around a mean adult size, with an absence of small, or juvenile, sized specimens (Fig. 10). This pattern is present whether or not all specimens, only articulated specimens, or individual bonebeds occurrences, are analysed (Fig. 10). Evidence from other elements suggests that these small-bodied juveniles are present both in bonebeds and as individual specimens (Dodson and Currie, 1988; Ryan, 1992; Sampson et al., 1997; Ryan et al.,

2001; Ryan, 2003; Brown et al., 2009), but are represented in low numbers. A virtually identical trend is present when articulated hadrosaurid skulls are analysed. Both length of skull (Fig. 11a) and quadrate (Fig. 11b) show a cluster of specimens from sub-adults to large adults, but with a nearly complete absence of articulated small-bodied juvenile skulls. As with the Centrosaurus data, juvenile material from the DPF is well documented on the basis of isolated elements, which demonstrates the presence of small juvenile hadrosaurs in the system despite their absence as articulated skeletons - a taphonomic mode common to adults (Dodson, 1975; Tanke and Brett-

Surman, 2001; Evans, 2003; Evans et al., 2005; Evans et al., 2007).

Correlation of Body Mass and Diversity

A plot of number of species per family as a function of mean body mass for each family reveals a significant (p-value = 0.0381) and positive (r = 0.579) correlation (Fig. 12). If the taxa based

72 purely on teeth are removed then the correlation becomes tighter (r = 0.623, p-value = 0.0228).

These results illustrate a relationship between diversity and body size in the DPF fauna where families with a higher abundance of species also have a higher mean body size.

Discussion

Taphonomic Bias and Compositional Fidelity of the Dinosaur Park Formation

Here we present the first comprehensive test of the correlations between body size and preservation to quantify patterns of size related taphonomic biases within a dinosaur assemblage.

Our results indicate that body size (estimated mass of the living dinosaurs) of a taxon is a significant factor influencing the completeness of its preservation in the DPF depositional system

(Fig. 2, Table 2). This is illustrated by strong correlations between body mass and both skeletal completeness and primary taphonomic mode. This result is not surprising given that the rarity of small forms, and prevalence of large-bodied taxa in the DPF, has been noted by many workers, with many also suggesting that taphonomic biases may play a role (Dodson, 1971; Dodson,

1983; Currie, 1985; Wood et al., 1988; Farlow et al., 1995; Sereno, 1999; Eberth and Currie,

2005; Longrich and Currie, 2009b; Brown et al., 2013). Indeed, Currie (1985) suggested a threshold body mass (100 kg), presumably based on the work of Behrensmeyer et al. (1979), under which skeletons would face strong biases against their preservation. However, the magnitude and pervasiveness of this correlation is remarkable. Approximately 80% (based on the r2; Tables 2 and 3) of the variation in skeletal completeness (i.e. the amount of the skeleton that is currently known) of dinosaur taxa can be explained by body size, suggesting that size is the most important factor in the overall pattern of dinosaur preservation in the DPF.

A strong taphonomic bias favouring the preservation of large skeletons has been recovered in live-dead, actualistic taphonomic experiments, of extant communities, in which

73 body size is an important factor influencing preservation frequency and species fidelity of death assemblages in relation to the living terrestrial vertebrate community (Behrensmeyer et al., 1979;

Behrensmeyer and Dechant Boaz, 1980; Behrensmeyer, 1988; Sept, 1994; Kidwell and Flessa,

1996; Arribas and Palmqvist, 1998). Studies of extinct Cenozoic terrestrial ecosystems corroborate the extant pattern of significant taphonomic biases against the preservation of small- bodied taxa in mammalian assemblages (Mendoza et al., 2005; Muñoz-Durán and Van

Valkenburgh, 2006). This bias is explained by the increased resistance of large skeletal remains to both physical and chemical weathering, and biological destruction (carnivory and scavenging) relative to that of small skeletal remains (Behrensmeyer et al., 1979; Behrensmeyer and Dechant

Boaz, 1980; Lyman, 1984; Kidwell and Flessa, 1996; Arribas and Palmqvist, 1998). Although derived from direct field observations in modern African mammal faunas, these processes are not expected to be unique to modern ecosystems, nor those dominated by mammals, and likely functioned in the same way in Mesozoic dinosaur dominated faunas (Behrensmeyer et al., 1979;

Kidwell and Flessa, 1996). Importantly, the size threshold at ~60 kg (or two thresholds at ~60 kg and 370 kg), which are derived from species-completeness data, are not drastically different than the threshold (~100 kg) observed in modern mammal death assemblages based on occurrence data (Behrensmeyer et al., 1979; Behrensmeyer and Dechant Boaz, 1980), given the magnitude of size ranges in both studies. This implies that a bias against preservation of small taxa has a universal effect on skeletons of certain absolute body sizes, irrespective of the fauna or taxa.

If a taphonomic bias against small skeletons is responsible for the disproportionate preservation of large-bodied animals in the DPF, then there are likely issues with both compositional fidelity and relative abundance. Therefore, interpretations of the palaeoecology and palaeobiology of this formation should be treated with caution. All large-bodied taxa, in their development, pass through a small-bodied stage comparable to those of the small-bodied taxa.

Differences in the physiology (Padian et al., 2001) and reproductive biology (Richmond 1965;

Janis and Carrano, 1992; Werner and Griebeler, 2011) between dinosaurs and mammals complicates the interpretation of the negatively skewed body size distribution in dinosaur assemblages as a strictly taphonomic artefact, as it is possible that these biological differences resulted in population and community structure that were fundamentally different from terrestrial ecosystems today (Codron et al., 2012). Suggestions that the disproportionate preservation of large-bodied animals is not the result of a taphonomic bias, but rather an accurate representation of an ecosystem dominated by large-bodied forms, would predict the absence of a strong correlation between body size and specimen abundance within a taxon. However, a taphonomic bias against small skeletons would predict a strong correlation, as juveniles of large taxa would experience similar taphonomic biases to the adults of small taxa. Within the most common dinosaur taxa preserved in the DPF, ceratopsians and hadrosaurids, the abundance of specimens is also heavily skewed towards large individuals in both bonebed and articulated skeleton taphonomic modes (Fig. 9 and 10; Evans, 2010; Sternberg, 1955; Dodson and Currie, 1988).

Juvenile remains that have been identified are generally isolated elements, and articulated skeletons are exceptionally rare, even for hadrosaurids (Tanke, 2001; Sternberg, 1955; Dodson and Currie, 1988). This bias against juveniles of large taxa mirrors that against the small taxa, and reciprocally indicates a strong bias against the preservation of skeletons of small absolute size, irrespective of whether this size is due to taxonomy or ontogeny (Currie, 2005).

The bias against small-bodied animals (both small-bodied taxa, and juveniles of larger taxa) is also illustrated in the significant positive correlation between mean family body size and the number of species that occur in each family (Fig. 12). Although the body size range of a clade is likely to increase with diversity as an artefact of the time that it has been in existence

(McClain and Boyer, 2009), average body size is generally considered to follow a positive skew,

75 which would result in a negative correlation in regards to taxonomic diversity in extant ecosystems (e.g., May, 1988; Gardezi and Da Silva, 1999). As such, the opposite, positive relationship recovered here for the DPF again emphasises the strong bias against small body size.

The robust correlation between the size and discovery of taxa further emphasises that animals of small size are severely underrepresented in the DPF assemblage, in terms of completeness, taphonomic mode, and abundance, and, almost certainly (but indeterminably) occurrence (fidelity). Our results demonstrate a systematic bias against the preservation of small taxa within this assemblage. The question now becomes; how many small-bodied taxa may have been part of the fauna but have yet to been found/identified in the assemblage? Given both the magnitude of the correlations, and the pattern of body size distribution in extant terrestrial ecosystems (Blackburn and Gaston, 1994; Woodward et al., 2005; Clauset and Erwin, 2008), this number may be very high. We may be missing large parts of the DPF ecosystem and niches of small-bodied dinosaur taxa, and are almost certainly missing a significant portion of their diversity. One example of an unrepresented niche in the DPF assemblage is the recent recognition of a microraptorine theropod from this unit (Longrich and Currie, 2009a), representing both a 45 million year range extension, and the first microraptorine known from

North America. One of the major implications of this work is that studies assuming (whether implicitly or explicitly) that the preserved faunal assemblage is an accurate representation of the true historical biological assemblage seriously underestimate the diversity, abundance, and ecological significance of the smaller-bodied taxa. As a result, accurate inferences based on the structure of the original biological fauna require a complete understanding of the taphonomic distortion of the preserved assemblage. These types of studies may include investigations of biogeography and provinciality (Lehman, 2001; Gates et al., 2010), community structure and implications for energetics (Bakker, 1972; Bakker, 1975; Farlow, 1976; Farlow, 1993; Baszio,

1997), or size distribution and niche partitioning (Foster et al., 2001; Farlow and Holtz, 2002;

Farlow and Pianka, 2002; Codron et al., 2012).

Historical Collection Bias

Although the correlations between size and, completeness, taphonomic mode and abundance demonstrate a strong taphonomic bias favouring the preservation of animals with large body size in the DPF, the significant correlation between size and discovery/description (Table 2) suggests the potential occurrence of a collection and/or discovery bias against small taxa, which may be responsible for compounding the described taphonomic pattern. Indeed, historical collection of specimens from the DPF concentrated on the collection of large articulated specimens suitable for museum displays. In the DPF there is likely a combination of preservational and collecting biases, however, given that there is an inherent difficulty in recovering the remains of smaller animals (Behrensmeyer et al., 1979; Behrensmeyer and Dechant Boaz, 1980; Western, 1980; Hill and Behrensmeyer, 1984), and that the true living assemblage can never be fully known, confidently determining the relative contribution of collector bias may not be possible. However, it is noteworthy that many recent prospecting/collecting expeditions of the DPF have been conducted systematically to reduce any of the intentional biases that may have been found in historical collections and indeed many workers have focused attention towards the identification and collection of smaller-bodied animals.

Rates of Discovery in the Dinosaur Park Formation

The overall patterns of cumulative discovery and description rates for the Dinosaur Park

Formation taxa (Fig. 9) show similarities, in periods of discovery and plateau, to that of the global dinosaur discovery rate (Benton, 2008), reflecting general patterns of the popularity of

77 dinosaur research. Beyond this, the cumulative rates (both discovery and description) of large- bodied taxa (≥ 60 kg) for the DPF show a logarithmic curve approaching, or at, a plateau, suggesting that we have, or are approaching, a robust and accurate understanding of the diversity of large-bodied dinosaur taxa (Fig. 9, Table 6). In contrast the accumulation rates of the small- bodied taxa show power (or possibly linear) relationship with no obvious signs of slowing, implying that our understanding of the diversity of small-bodied dinosaur taxa is lacking (Fig. 9,

Table 6). This also suggests that small-bodied taxa are under-sampled and, with future collections there should be more discoveries of new taxa, the majority of which will be small- bodied. In order to get a more complete understanding of the true DPF fauna there needs to be a concentration of worker effort towards the identification of small-bodied taxa (Longrich, 2008).

This task is understandably difficult since, as we have demonstrated here, these taxa are not likely to be represented by complete or articulated specimens.

Implications for Other Non-marine Systems

The DPF is proposed here as a model ancient alluvial-paralic system based on its well-sampled assemblage and the long history of multidisciplinary geologic and palaeontologic research associated with the locality. Given the strong taphonomic bias against the preservation of small- bodied taxa in the DPF, it is reasonable to question whether a similar bias is common in other alluvial-paralic systems or non-marine systems in general. Although not quantitatively tested, other studies suggest that these same biases occur in many alluvial-paralic formations. Dodson et al. (1983) suggested a taphonomic size bias as the cause for rarity of small to medium sized herbivores in the Morrison Formation (Jurassic). Carpenter (1982) discussed the possibility of a size bias against juveniles of large-bodied taxa in the Lance and Hell Creek (Maastrichtian) formations, and systematic surveys of the Hell Creek Formation document a dominance, in terms

78 of abundance of the large-bodied taxa, and/or a rarity of juveniles (White et al., 1998; Goodwin and Horner, 2010; Horner et al., 2011). Farlow et al. (1995) illustrated that the alluvial Nemegt

Formation of Mongolia is dominated by large hadrosaurids and tyrannosaurids. Additionally,

Benson et al (2012), noted low abundance of preservation, but high diversity, in small-bodied

Australian theropods, and the reverse for large bodied-taxa in the fluvial Strzelecki/Eumeralla formations.

This pattern, while appearing to be largely consistent with alluvial and paralic systems, may not hold for aeolian, paludal and lacustrine margin settings in smaller, craton-interior basins.

For example, Farlow et al. (1995) and Longrich (2010) demonstrated that the aeolian Djadokhta and alluvial-aeolian Baruungoyot formations are dominated by small to medium sized taxa, relative to the Nemegt Formation. Likewise, the lacustrine Jehol biota from northeastern China is dominated by small-bodied taxa (Zhou and Wang, 2005). Similarly, while Eberth et al (2010) described a bias against small theropod taxa in the Shishugou Formation of Xinjiang, China, and its effect on perceived small theropod diversity, the Shishugou fauna is associated with a mixed alluvial-paludal setting (Eberth et al., 2010) and is underrepresented in specimens of large- bodied dinosaurs but well-represented by small to medium size tritylodonts and ornithischian dinosaurs (Clark et al., 2006).

If our knowledge of the DPF, arguably the best sampled and understood terrestrial

Mesozoic assemblage, suffers from such strong taphonomic preservational biases likely affecting both species fidelity and relative abundances, then our understanding of other faunas preserved in similar alluvial-paralic systems is likely to be even less complete.

Implications for Global Dinosaur Diversity

One of the major goals in palaeobiology is the documentation of patterns of taxic diversity through time for the identification of diversification mechanisms, biotic revolutions, mass extinctions, and also to test what correlations these patterns have to major global events, such as climate change, tectonism, and palaeogeographic evolution (Valentine, 1970; Raup, 1972; Raup,

1975; Bambach, 1977; Sepkoski et al., 1981; Benton, 1985; Raup and Boyajian, 1988; Benton,

1995; 2006; Alroy et al., 2008; Alroy, 2010). Dinosaurs have been of particular interest to palaeobiologists in this regard (e.g., Fastovsky et al., 2004; Sullivan, 2006; Wang and Dodson,

2006; Barrett et al., 2009; Butler et al., 2011; Lloyd, 2012).

The different preservation potential of different sized taxa in different depositional settings influences our understanding of patterns of diversity through time. Lagerstätten assemblages, which typically preserve a high diversity of small-bodied taxa, will affect diversity curves differently than large-bodied assemblages that characterise Mesozoic alluvial-paralic systems.

The non-uniform distribution of Lagerstätten in space and time will alter and potentially confound attempts to recover true, holistic diversity patterns (Butler et al., 2009; Benson et al.,

2010; Butler et al., 2013; Evans et al., 2013). Two approaches can be used to ensure more equal sampling in this regard. Firstly, the Lagerstätten may be ignored or down weighted from an analysis, decreasing the number of sites, but maintaining the diversity within each site. Secondly, all sites can be retained, but the taxa limited to the size classes that are consistently taphonomically favoured, and more likely to reflect the true species fidelity of those size classes across all sites. This second approach allows for the number of sites to remain constant, but the diversity will be decreased based on the size-dependent taphonomic properties of the assemblages. The first of these methods has been used in analysis of global diversity patterns through time (Benson et al., 2011; Butler et al., 2011). The second of these techniques has been

80 more widely used for compositional comparison in ecological studies (Mendoza et al., 2005;

Campione et al. 2011). A better understanding and appreciation for taphonomic biases, such as those related to size as quantified here, will contribute to a more comprehensive picture of diversity dynamics through Earth history, particularly for non-analogue ecosystems during the

Mesozoic.

Acknowledgements

We thank Victoria Arbour, Roger Benson, Don Brinkman, Michael Burns, Phil Currie, Peter

Dodson, Jason Head, Derek Larson, Nick Longrich, Tetsuto Miyashita, Anthony Russell,

Michael Ryan, Brandon Strilisky, Darren Tanke, Matthew Vavrek, for discussions and contributions to this project. Peter Dodson and Michael Benton (reviewers), and Roger Benson

(editor) provided feedback that improved the manuscript. Of course, none of the analysis would have been possible without the generations of scientist who have come before us, and who have spent countless hours collecting the data that we build upon.

Access to specimens was facilitated by Kevin Seymour (ROM), Brandon Strilisky

(RTMP), Carl Mehling (AMNH), Kieran Shepherd and Margaret Currie (CMN).

Funding for this project was provided by an NSERC-CGS Alexander Graham Bell Canada

Scholarship, and Dinosaur Research Institute Student Project Grant (to CMB), a National

Sciences and Engineering Research Council Discovery Grant (to DCE), a Queen Elizabeth II

Graduate Scholarship in Science and Technology (to NEC), and a University of Toronto

Fellowship (to LOB).

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Table 1 – Dinosaur Park Formation dinosaur faunal list, ordered by increasing mass.

Discovery Description Mass Taph. Genus and Species Family Completeness (YDi) (YDe) (kg) Mode Ornithurine B 2009 1% 0.098 C Ornithurine F 1993 2002 1% 0.110 C Ornithurine C 2009 1% 0.123 C Ornithurine E 1993 2002 1% 0.180 C Ornithurine D 1988 2009 1% 0.480 C Palintropus species B Quercymegapodiidae 1983 2002 8% 1.481 C Ornithurine A 1986 2009 1% 1.636 C Hesperonychus elizabethae Dromaeosauridae 1982 2009 9% 2.000 C Palintropus species A Quercymegapodiidae 1986 2002 3% 2.540 C Cf. Orodromeous Thescelosauridae' 2008 2009 8% 13 C Saurornitholestes langstoni Dromaeosauridae 1974 1978 33% 13 B Elmisaurus elegans 1926 1933 3% 13 C Dromaeosaurus morphotype A Dromaeosauridae 1987 2002 1% 16 C Dromaeosaurus albertensis Dromaeosauridae 1914 1922 12% 16 B "Stegoceras" breve Pachycephalosauridae 1918 1918 1% 16 C New unnamed Pachycephalosaur Pachycephalosauridae 2002 2005 1% 16 C Stegoceras validum Pachycephalosauridae 1898 1902 37% 16 B Richardoestesia gilmorei Dromaeosauridae 1917 1990 1% 20 B Richardoestesia isosceles Dromaeosauridae 1984 2001 1% 20 C Caenagnathus sternbergi Caenagnathidae 1936 1971 1% 30 C Caenagnathus collinsi Caenagnathidae 1936 1940 1% 30 C Chirostenotes pergracilis Caenagnathidae 1914 1924 41% 30 B Hanssuesia sternbergi Pachycephalosauridae 1928 1943 1% 40 C Troodon inequalis Troodontidae 1898 1932 20% 58 C Paronychodon sp. Troodontidae 1979 1990 1% 58 B Ornithomimus edmontonicus Ornithomimidae 1933 1933 100% 60 A1 "Dromiceiomimus" samueli Ornithomimidae 1926 1972 46% 90 A1 Struthiomimus altus Ornithomimidae 1901 1902 100% 110 A1 Unescoceratops koppelhusae Leptoceratopsidae 1995 1998 1% 160 C Large unnamed Ornithomimid Ornithomimidae 1967 2008 4% 370 C Edmontonia rugosidens Nodosauridae 1915 1930 58% 1400 A2 Panoplosaurus mirus Nodosauridae 1917 1919 63% 1600 A2 Gorgosaurus libratus Tyrannosauridae 1913 1914 100% 1950 A1 Dyoplosaurus acutosquameus Ankylosauridae 1919 1924 42% 2000 A2 Daspletosaurus sp. Tyrannosauridae 1913 1970 76% 2200 A1 Centrosaurus apertus Ceratopsidae 1901 1902 100% 2800 A1 Euoplocephalus tutus Ankylosauridae 1897 1902 97% 2800 A2 Prosaurolophus maximus Hadrosauridae 1915 1916 99% 2900 A1 Lambeosaurus magnicristatus Hadrosauridae 1919 1935 81% 3100 A1 Lambeosaurus lambei Hadrosauridae 1902 1923 100% 3100 A1 Lambeosaurus clavinitialis Hadrosauridae 1935 1935 100% 3100 A1 Corythosaurus intermedius Hadrosauridae 1919 1923 100% 3200 A1 Corythosaurus casuarius Hadrosauridae 1912 1914 100% 3200 A1 Chasmosaurus russelli Ceratopsidae 1938 1940 68% 3300 A2 Styracosaurus albertensis Ceratopsidae 1913 1913 100% 3900 A1 Vagaceratops irvinesis Ceratopsidae 1958 2001 89% 4000 A1 Gryposaurus notabilis Hadrosauridae 1913 1914 100% 4000 A1 Chasmosaurus belli Ceratopsidae 1898 1902 94% 4100 A1 Cf. Pachyrhinosaurus Ceratopsidae 2001 2010 63% 4200 B Parasaurolophus walkeri Hadrosauridae 1920 1922 82% 4500 A1

Table 2 – Correlation coefficients for estimated body mass (log kg) and skeletal completeness

(SCM2). Pearson’s product-moment correlation is significant in all taxonomic groups (p < 0.01), and highly significant in most (p << 0.001).

Dinosauria Ornithischia Theropoda Ornithopoda Marginocephalia Ceratopsia Sample (n) 50 24 26 9 11 7 Range (kg) 0.098 - 4500 13 - 4500 0.098 - 3000 13 - 4500 16 - 4200 160 - 4200

Pearson's 0.81 0.88 0.61 0.95 0.88 0.89 p value p << 0.001 p << 0.001 p < 0.001 p << 0.001 p < 0.001 p < 0.01 Significance *** *** *** *** *** **

Table 3 – Correlation coefficients and significance values for body mass (log kg and rank mass) and ranked taphonomic mode. Pearson product-moment correlation is significant in all taxonomic groups, and Spearman rank correlation is significant in taxonomic groups with a large sample.

Dinosauria Ornithischia Theropoda Ornithopoda Marginocephalia Ceratopsia Sample (n) 50 24 26 9 11 7 Range (kg) 0.098 - 4500 13 - 4500 0.098 - 3000 13 - 4500 16 - 4200 160 - 4200

Log mass 0.82 0.89 0.65 0.99 0.84 0.85 p value p<< 0.001 p << 0.001 p << 0.001 p << 0.001 p = 0.001 p = 0.02 Significance *** *** *** *** ** *

Ranked mass 0.81 0.78 0.7 0.94 0.79 0.59 p value p << 0.001 p << 0.001 p << 0.001 p << 0.001 p = 0.004 P = 0.16 Significance *** *** *** *** ** -

Table 4 – Correlation coefficients for estimated body mass (log kg) and both discovery and description dates. All correlations are significant (p < 0.01) for both dated of discovery and dates of description.

Discovery Description Sample (n) 48 50

Pearson -0.568 -0.6 p value p << 0.001 p << 0.001 Significance *** ***

100

Table 5 – Average years to discovery and description of the small and large-bodied dinosaur taxa in the Dinosaur Park Formation. All differences between large and small are significant (p <

0.01).

<60 kg >60 kg Sample (n) 23 and 25 25

Years to Discovery mean 65.9 33.6 median 88 25

Years to Description mean 75.6 34.1 median 100 22

101

Table 6 – Goodness of fit between accumulated discovery and description rate of DPF dinosaur taxa and linear, logarithmic and power functions describing the curve (measured in r2 and AIC).

Calculations for rates binned at both 10 and 5 years are included. Functions with the highest r2, and/or lowest AIC, and therefore the best fit to the observed rate, are in bold.

Discovery Bin Size Class Linear Logarithmic Power Large (> 60 kg) 10 year r2 0.676 0.89 0.807 AIC 73.78 60.85 68.74 ∆AIC 12.93 - 7.89 Small (< 60 kg) 10 year r2 0.926 0.769 0.928 AIC 56.34 69.93 53.33 ∆AIC 3.01 16.6 -

Large (> 60 kg) 5 year r2 0.693 0.915 0.843 AIC 130.75 101.37 116.95 ∆AIC 29.38 - 15.58 Small (< 60 kg) 5 year r2 0.925 0.731 0.934 AIC 100.93 130.44 95.56 ∆AIC 5.37 34.88 -

Description Large (> 60 kg) 10 year r2 0.827 0.969 0.884 AIC 60.99 40.39 52.43 ∆AIC 20.6 - 12.04 Small (< 60 kg) 10 year r2 0.806 0.628 0.945 AIC 68.49 76.34 64.74 ∆AIC 3.75 11.6 -

Large (> 60 kg) 5 year r2 0.822 0.956 0.907 AIC 112.06 79.7 93.47 ∆AIC 32.36 - 13.77 Small (< 60 kg) 5 year r2 0.801 0.611 0.912 AIC 119.11 134.59 115.16 ∆AIC 3.95 19.43 -

102

Figure 1 – Histogram illustrating the distribution of estimated body masses of the dinosaur fauna of the Dinosaur Park Formation. Large-bodied taxa greatly outnumber the smaller taxa.

103

104

Figure 2 – Plots of skeletal completeness (SCM2) as a function of estimated body mass for A, all Dinosaur Park Formation dinosaur taxa; B, theropods; and C, ornithischians, all showing an increase in skeletal completeness as body mass increases. Boxplots on the right indicated significant dichotomy of completeness when the samples are divided into large and small size classes.

105

106

Figure 3 – All Dinosaur Park Formation dinosaur taxa ranked by estimated body mass showing an increase in skeletal completeness (above), and taphonomic mode assignment (below), as body mass increases. Large taxonomic groups are indicated by colour.

107

108

Figure 4 – All Dinosaur Park Formation dinosaur taxa ranked by estimated body mass showing an increase in skeletal completeness as body mass increases. Vertical dotted lines indicates peaks of statistical deference (sliding window group mean difference) between group completeness means, dividing series into small (<60 kg, black circles), large (>370 kg, white circles), and intermediate (>60 kg <370 kg, grey circles). Dashed line indicates moving averages of completeness values for 10 taxa. Grey areas illustrated areas with no overlap in completeness between the small-bodied taxa and large-bodied taxa.

109

110

Figure 5 – Dinosaur Park Formation dinosaur species segregated into A) ornithischian and B) theropod taxa ranked by estimated body mass showing an increase in skeletal completeness

(above) and taphonomic mode assignment (below) as body mass increases. Large taxonomic groups are indicated by colour. Vertical dotted lines indicate peaks of statistical deference

(sliding window group mean difference, above) between group completeness means, indicating group division (~160 kg for ornithischians and ~60 kg for theropods). Dashed lines indicate moving averages of completeness values for 10 taxa.

111

112

Figure 6 – Size distributions (log mass) of DPF dinosaur species for the three taphonomic modes: Mode A – Articulated (black), Mode B – Associated (grey), Mode C – Articulated

(white).

113

114

Figure 7 – Discovery year (A) and description year (B) of Dinosaur Park Formation dinosaur taxa plotted against estimated body mass (log kg). Boxplots on top indicated significant dichotomy in both discovery and description times when the samples are divided into large and small size classes.

115

116

Figure 8 – Cumulative taxonomic diversity of Dinosaur Park Formation dinosaur taxa through time based on time of discovery (A) and time of description (B). Description year (C) and discovery year (D) of Dinosaur Park Formation dinosaur taxa segregated into large-bodied

(black) and small-bodied (grey) taxa.

117

118

Figure 9 – Discovery (A) and description (B) curves of rates of cumulative taxonomic diversity of Dinosaur Park Formation dinosaur taxa segregated into large-bodied (black squares) and small-bodied (grey diamonds) classes with five year bins. Lines of best fit and 95% confidence intervals are also illustrated (see Table 5).

119

120

Figure 10 – Histograms illustrating the abundance of size classes of Centrosaurus specimens based on occipital condyle diameter (log mm) from A, all Centrosaurus specimens; B, articulated Centrosaurus skulls; C, All Centrosaurus bonebeds; D, Bone Bed 43; E, Bone Bed

91.

121

122

Figure 11 – Histograms illustrating the abundance of size classes of articulated hadrosaur skulls based on A, skull length (log mm); and B, quadrate height (log mm).

123

124

Figure 12 – Correlation between mean estimated body mass of each dinosaur family and number of species per family. Correlation is both positive, increasing diversity with increasing body mass, and significant (r = 0.579, and p-value = 0.0381).

125

126

Appendix A – Dinosaur faunal list for the Dinosaur Park Formation, with corresponding data on date of collection/description, literature reference, specimen references, regional and overall completeness, and method of body mass estimation for each species. See methods for details on the completeness metric and on body mass estimation.

127 00 3 2 , n 00 2 v a i ., 2 l 99 0 99 0 Sul l 1 1 00 9 00 9 00 9 00 9 , , n n 2 2 2 2 00 9 922 , , , , 2 943 ; 1 99 8 , 00 5 key et a 1 ric h ric h ric h ric h S lo a S lo a 1 n 2 , d d rrie , l 00 1 00 2 ke r n n u s , e i S a n 01 0 a C l s 93 5 93 3 93 5 93 2 94 0 Lon g Lon g Lon g Lon g ., 2 ., 2 94 0 n Brow l l d u 00 8 00 9 00 9 00 9 00 9 d 1 1 1 1 1 97 2 93 0 n 1 ., 2 92 4 , , , , , 97 2 97 0 n l gb y a gb y a Ev a R 90 2 90 2 90 2 91 9 91 4 90 2 91 8 90 2 90 2 91 3 91 4 001 ; 2 2 2 2 2 00 9 91 6 91 4 i i 1 1 , , , a , , Sc h d d 92 4 92 2 93 3 98 7 1 , r g r g r g r g r g rr , , 002 ; 002 ; 002 ; 002 ; 1 1 a a 97 8 1 1 1 1 1 1 1 1 1 1 1 e , 2 1 1 n n t R R e e e e e e , , , , , , , , , , , f t c s et a , , , & 1 1 1 192 3 ll , ll , 2 2 2 2 r e 1 e e e e e e e e e e e b b b b b b a r e e , , , , a a e ric h ric h ric h ric h ric h ric h e e o e n b b b b b b b b b b b o e e e e key et a key 2 s , n n n he w c r r n r n r n r n r n r n m t p p p p l e r rrie , s m rrie 1 s rrie , n n rks rks , rks , rks , m m m m m m m m m m m e e e e e e l m a o o o o o y a y a u u y a u u u r a a a a a e f R R Lon g H S t G i L a S a R H H C S t R L a n Brow L a H S t Lon g Lon g P Lon g G i S t L a L a L a S t R C Lon g Lon g M L a L a S a S t L a P R n Brow n Brow L a H L a P C n Brow Su e P C ed b i 200 5 200 8 200 2 193 5 193 0 190 2 200 2 199 8 200 2 200 2 199 0 193 3 192 8 190 2 201 1 192 3 200 2 193 5 200 9 200 9 192 3 200 9 192 4 193 2 191 9 191 4 190 2 194 0 201 0 198 7 200 9 200 9 192 2 191 8 190 2 200 1 194 0 190 2 192 4 197 0 191 6 191 4 191 3 200 1 191 4 192 2 199 0 194 3 197 8 193 3 197 1 es c D e a t D ed t ? ? ll e c 197 2 196 7 198 6 192 8 191 5 189 7 198 7 199 5 199 3 199 3 197 9 191 6 192 6 189 8 200 8 190 2 198 3 191 9 191 9 198 2 191 4 189 8 191 7 191 3 190 1 193 8 200 1 196 8 198 8 198 6 191 4 189 8 189 8 198 4 193 6 190 1 191 9 191 3 191 5 191 2 191 3 195 8 191 3 192 0 191 7 192 8 197 4 192 6 193 6 C o / un d F o e a t D r cies cies u e e p p A d s a i e o S S l m i a y p s t h m li s m ne d ne d o p u i i a o c u 5 i t i eu s i s h e s t li s t a h n u i a u t a i m i n s i . r m r m o i r p s s t h s s yc e n a u i i i o r t u t ri d e e e o r g cie s s t t s u en sis q s e e

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t a h n . a p u t i m n s i o i r p li s s i s t h s o yc e ue l a A B i i l o u t ri . d a e o r g . s t t i u en sis r e e o e e s p r n c r t m i i e m . s n ci l g c h i i s k o rius r t en s b s si ll i e en sis l s b s q s m s p o n n n a a x i u s p e m a i s r g r a e ili s u a r u r m cie s cie s b O r t t u g a s b e u v i r u C r n t m e b ll i si d u m il m r t s p l e e e s i u lli n d P s u li z s ne sis s" s a a t r g m a n sos ce l b l a s l a e ve d o u e l a b li s p o u e u r a e t d g i a l c ut o e s s t g a p s a s o s e r n r e o b s e g s a t s m s p s p m li d u s e B F ( C s l a s c s m E D A s r s b s t u s in t s c eu s m n e i u r u l hu s w e si a si a s p s a s a r u r u u s irv i e l e u r u r u s a hu s m o a r a m u l s l i e e e r u p s n r u r u r u r u r u m r u r u u u p m a s t u t s" b r u hu s st e hu s c p h d r u m i o s s u m u u u u u r u u u i s t s t s v a t t o o s e s a i r u c e yc h ne q r u u a m r o u o s a s a i p n h t u u yrhinos a nn a r a s a u a a e e i o u t o n s a u s a s a s a s a s a r a s a s a m o o d o i m p p r ol o r u n t o n n o o nn a s a n r a t en o c e c e o o o o e e t o o s a t r ol o o o o u c h c e i l o u u s a r o p li k g g i s a hu rin e hu rin e hu rin e hu rin e hu rin e hu rin e o o ue si a r o n yc h c e o r n a a r o r o e e e a u h h r d r d s t o e l u h t t t t t t e p o cos a c e t t h o r o d o n o m i i i i i i s a t s a e n b b b a a P E r O t s g s m s m a p i i o m m r o p L e

s a r o o w g r g e s p r g r a i ry t a ry t s pl e n en a en a m o g n r a r o li n li n r o u m m m e r u yr a r n r n r n r n r n r n c h c h e f. f . f . f . r n r o r o y op l a a e a o o h a h en t h i i a a a r o a a r o S t D C u R C S t R " N D " S t L a C L a T E d P G o L a C O C P O O El m D C C H P C L a D D C E u S t V a Gr y C C C P O O S a C O O P P H 130

Appendix B – Database of body masses and measurements of coracoid widths for both a sample of extant ornithurine birds and ornithurines taxa from the Dinosaur Park Formation. Regression of the mass on coracoid width on extant birds allows for the estimation of body mass in the extinct sample.

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Extant Ornithurines width of Speices Specimen # mass (g) log mass (g) log width (mm) coracoid (mm) Coccyzus erythropthalmus ROM R8593 51.4 1.71 1.5 0.18 Aegolius acadicus ROM R8568 90.4 1.96 2.4 0.38 Melopsittacus undulatus ROM R8566 44.1 1.64 1.5 0.18 Scolopax mino ROM R8596 180 2.26 2.9 0.46 Junco hyemalis ROM R8628 20.4 1.31 0.95 -0.02 Pipilo erythrophthalums ROM R8390 45.5 1.66 1.2 0.08 Zonothrichia leucophrys ROM R8401 35.6 1.55 1 0.00 Coturnix japonica ROM R8683 136 2.13 2.8 0.45 Bonasa umbellus ROM R8308 602 2.78 4.6 0.66 Grus canadensis ROM R6559 3960 3.60 12.9 1.11 Columba livia ROM R7686 403 2.61 3.9 0.59 Somateria spectabilis ROM R6827 1900 3.28 6.3 0.80 Lophodytes cucullatus ROM R6410 565 2.75 4.7 0.67 Aechmorphorus occidentalis ROM R5760 965 2.98 4.6 0.66 Mergus merganser ROM R5797 1126 3.05 7 0.85 Pandion haliaetus ROM R2644 1288 3.11 5.8 0.76 Strix nebulosa ROM R5791 1700 3.23 7 0.85 Chen caerulesens ROM R7319 2288 3.36 6.6 0.82 Larus marinus ROM R2345 2536 3.40 6.2 0.79 Corvus caurinus ROM R5761 233 2.37 2.5 0.40 Gallus gallus ROM R5544 1548 3.19 4.7 0.67

Extinct Ornithurines Est. log mass width of Speices Specimen # Est. mass (g) log width (mm) (g) coracoid (mm) Palintropus species A TMP 1986.36.126 2540.01 3.40 8.30 0.92 Palintropus species B TMP 1986.112.6 1481.49 3.17 6.50 0.81 Ornithurine A TMP 1988.116.0001 1636.50 3.21 6.80 0.83 Ornithurine D TMP 1988.087.0027 480.21 2.68 3.90 0.59 Ornithurine E TMP 1993.019.0001 180.10 2.26 2.50 0.40 Ornithurine C UALVP 47942 122.61 2.09 2.10 0.32 Ornithurine F TMP 1993.116.0001 110.10 2.04 2.00 0.30 Ornithurine B UALVP 47943 98.32 1.99 1.90 0.28

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Chapter Three

Testing the Limits of Small Sample Sizes in Allometric Analyses;

Implications for Palaeobiology

Caleb Marshall Brown1, Matthew J. Vavrek2 and David Evans1,2

1Department of Ecology & Evolutionary Biology, University of Toronto, 25 Willcocks Street,

Toronto, Ontario M5S 3B2, Canada

2Department of Natural History, Palaeobiology division, Royal Ontario Museum, 100 Queen's

Park, Toronto, Ontario M5S 2C6, Canada

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Abstract

Quantitative morphometric analyses, particularly allometric analyses, are common methods used in quantifying shape in both extinct and extant organisms. Due to incompleteness and the potential for restricted sample sizes in the fossil record, palaeobiological analyses of allometry may encounter higher rates of error. Differences in sample size between fossil and extant studies, and any resulting effects on allometric trends have not been thoroughly investigated, and a logical lower threshold to sample size is not clear. Here we show that studies based of fossil dataset show smaller sample sizes than those based on extant animals. Interestingly the similar pattern between vertebrates and invertebrates indicates this is not a problem unique to either group, but common to both. We investigate the relationship between sample size, allometric relationship and statistical power using a large dataset of skull measurements of Alligator mississippiensis. Across a variety of subsampling techniques, used to simulate different taphonomic or sampling effects, smaller sample sizes gave less reliable and more variable results, often with the result that allometric relationships will go undetected due to Type II error

(failure to reject the null hypothesis). This may result in a false impression of fewer instances of allometric growth in fossil compared to living organisms. No mathematically derived minimum sample size for allometric studies is found; rather results of isometry (but not necessarily allomery) should not be seen with confidence at small sample sizes.

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Introduction

Morphometric analyses are becoming increasingly common in biological studies to quantify and investigate biological shape (Bookstein, 1985; Rohlf, 1990; Bookstein, 1997; Zelditch et al.,

2003; Zelditch et al., 2004). One of the key uses of morphometric methods in both neontology and palaeobiology is as a more objective and repeatable means to quantify patterns of growth and allometry within organisms. Allometry, often defined as ‘relative growth’ or ‘differential growth’, is most strictly defined as the “the study of size and its consequences” (Gould, 1966) or

“the study of the consequences of size for shape” (Bookstein, 1985), and has been an integral aspect of the study of growth in the context of ontogeny and evolution since Julian Huxley and

George Simpson (Huxley, 1932; Huxley and Teissier, 1936; Simpson, 1944; Gould, 1966;

Blackstone, 1987; Strauss, 1987). Given the nature of the data available, and the evolutionary questions of interest, much of the theoretical underpinning of allometry has been of particular interest to paleobiologists, particularly during the ‘paleobiological revolution’ of the 1970s and early 1980s (Gould, 1966; Gould, 1977; Alberch et al., 1979; Sepkoski and Ruse, 2008).

Neontological studies often use allometry to elucidate relative growth of anatomical structures, patterns of dimorphism, and to differentiate between closely related taxa (Dodson,

1975c, b, 1978, 1979; Zelditch and Fink, 1995; Zelditch et al., 2003; Zelditch et al., 2004).

Similar questions are asked by palaeobiologists, but researchers are limited to a small subset of the data available to neontologists, most often consisting of hard tissue anatomy (Dodson, 1975a;

Chapman et al., 1981; Chapman, 1990; Chapman and Brett-Surman, 1990; Dodson, 1990;

Heathcote, 2004; Marugan-Lobon and Buscalioni, 2004; Stayton and Ruta, 2006; Evans, 2007).

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Of particular interest among many biologists is the pattern of intraspecific ontogenetic allometry, the ‘heterauxesis’ of Simpson (1953), the allometry of elements or structures relative to the total size of an organism (Simpson, 1953; Alberch et al., 1979; Strauss, 1987; Rice, 1997). This can, in theory but rarely in practice, be derived directly from multiple measurements of a single individual through its lifespan (e.g., Jungers and Fleagle, 1980), rather than the less desirable but more practical indirect method of estimating patterns of individual growth from samples of multiple individuals at various stages of ontogeny (see Gould, 1966 and citations therein;

Alberch et al., 1979). Palaeobiologists cannot however, due to the nature of the fossil record, measure a single individual at differing times in its life span, and must rely on bulk sampling of a population or taxon to indirectly infer ontogenetic allometry. These ontogenetic trajectories illuminate the developmental dynamic of an organisms life history. Anatomical elements showing strong positive allometric growth in extant species, such as horns, antlers and crest, may often be taxonomically diagnostic display structures and thought to be under sexual selection and, likewise, secondary sexual characters are often positively allometric (Huxley, 1931; Geist,

1966, 1968; Brown and Bartalon, 1986; Alatalo et al., 1988; Green, 1992; Petrie, 1992; Simmons and Tomkins, 1996; Pomfret and Knell, 2006). Based on this, features which show strong positive allometric growth within extant clades may be taxonomically diagnostic, potential display structures, and under sexual selection (Gould, 1973, 1974; Dodson, 1975a; Sampson et al., 1997; Goodwin and Horner, 2004; Goodwin et al., 2006; Brown et al., 2009; Horner and

Goodwin, 2009; Evans, 2010; Tomkins et al., 2010; Knell and Sampson, 2011; Padian and

Horner, 2011; Schott et al., 2011; Hone et al., 2011). Determining the pattern of relative growth of these structures is therefore often important for interpretation of their paleobiological significance.

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As ontogenetic allometry is of interest to palaeobiologists, and can usually only be inferred based on multiple individuals preserved at various stages of ontogeny, there may be systemic methodological problems in the determination of ontogenetic trajectories (see Gould,

1966 and citations therein). Further compounding this problem, palaeobiologists are often limited to the small sample sizes that are associated with fossil taxa. Small sample size is arguably the most limiting factor in most palaeobiological studies, particularly those of vertebrates, and this is most evident in quantitative analyses such as morphometrics. The effect of small sample size in morphometric studies include reducing the number and type of analyses that can be performed, reducing the statistical and resolving power of those analyses, and increasing the probability of Type II error. This last point is of particular interest when a goal is to categorize each variable as positively allometric, negatively allometric, or (essentially) isometric. Because isometry is usually treated as the null, and small sample sizes will have reduced power, there will be large amounts of type II error (incorrect conclusions of isometry, when it should be allometry) when small sample sizes are used.

The implications and limitations of small sample sizes have be empirically tested and discussed in other aspects of paleobiology, namely paleoecology (Wolff, 1975; Grayson, 1978;

Koch, 1987; Forcino, 2012) and diversity studies (Raup, 1975; de Caprariis et al., 1976;

Grayson, 1981; Signor and Lipps, 1982; Miller and Foote, 1996), but this effect in morphometrics, particularly allometry, is less well documented (but see Cobb and O'Higgins,

2004; Strauss and Atanassov, 2006; Cardini and Elton, 2007).

Here we provide an empirical investigation into the practical limits that small sample size has on allometric analyses on extinct vertebrates, using an extensive ontogenetic series of a well- understood extant taxon, Alligator mississippiensis. We also perform a literature survey to

137 quantify differences in sample size between neontological work and palaeontological work, and between vertebrate and invertebrate taxa.

Institutional Abbreviations

AM, Australian Museum, Sydney; AMNH, America Museum of Natural History, New York;

CMN, Canadian Museum of Nature, Ottawa; FMNH, Field Museum of Natural History,

Chicago; TMM, Texas Memorial Museum, Austin; ROM, Royal Ontario Museum, Toronto;

RTMP, Royal Tyrrell Museum of Palaeontology, Drumheller; UCMP, University of California

Museum of Paleontology, Berkeley; UCMVZ, University of California Museum of Vertebrate

Zoology, Berkeley; UCMZ, University of Calgary Museum of Zoology, Calgary; UF, Florida

Museum of Natural History, Gainesville; UM, University of Michigan Museum of Zoology, Ann

Arbour; USMN; National Museum of Natural History (Smithsonian), Washington.

Materials

Literature Survey

In order to understand the relative effects of sample sizes on allometric studies, and to determine the range and distribution of sample sizes that have been used in previous studies, a survey of studies performing intraspecies allometric analyses was conducted. Studies were retrieved using the search term “allometry” in Google Scholar, and those investigating interspecies allometry were disregarded. 542 samples (intraspecific ontogenies), were recorded from 102 studies. Many studies, specifically those comparing intraspecific allometry between species, contained samples pertaining to more than one species. For all samples, the author and year, genus and species,

138 sample size, and whether the data pertained to invertebrates or vertebrates, and extinct or extant taxa were recorded (Appendix A). This allowed for direct comparisons of the distributions of sample sizes between extinct and extant taxa, and between vertebrate and invertebrate taxa.

Empirical Dataset

The dataset analyzed consists of 23 skull measurements of Alligator mississippiensis. A. mississippiensis was chosen for a number of reasons. This taxon is well understood and grows to large body sizes allowing for investigations into the effect of shape as a result of size to make used of an extensive body size axis (Romer, 1956; Norell, 1989; Densmore and White, 1991), large osteological collections exist in North American natural history museums that allow for ease of data collection, and modern A. mississippiensis has experianced an extensive history as modern analogue for testing ideas of extinct archosaur palaeobiology (Dodson, 1975c; Brochu,

1996).

The cranial dataset contains 108 specimens of A. mississippiensis (Appendix B). The measured sample includes specimens from hatchling (or near hatchling) sized individuals to large adults, and as such represents a complete size series for A. mississippiensis. As a result, the dataset encompasses a remarkable range in skull sizes. The skull length of the largest specimen

(ROM 51011 — 689mm) is more than twenty times larger (in linear dimensions) than that of the smallest (ROM R 7966 — 29mm). As such, it represents as great a scaling as will likely be encountered in any palaeontological analysis.

Twenty-three cranial measurements were taken from each skull (See Fig. 1 and Table 1).

The measurements taken following Brown et al. (2012), which were modified from Dodson

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(1975c). These osteological measurements represent functional complexes as opposed to dimensions of individual bones, and have biomechanical and behavioral correlates (see Dodson,

1975c). Measurements were taken from the left side, unless this side was either incomplete or damaged, in which case the right side was used. Measurements smaller than 150 mm were taken with digital calipers, those between 150 and 300 mm with dial calipers, and those above 300 mm with a fiberglass measuring tape. All measurements were taken to the nearest millimeter.

Methods

Regression

All data analyses were carried out using the R software package (R Development Core Team,

2009), using the packages SMATR (Warton et al., 2011) available on the CRAN website.

The complete dataset of A. mississippiensis was analyzed for the allometric trajectories of

23 linear cranial measurements. Each variable (with the exception of the reference datum) was plotted against the reference datum, skull length, for the entire sample size. Basal skull length has been cited a good reference datum for allometry studies (Gould, 1974; and references therein). Both Ordinary Least Squares (OLS) and Reduced (Standardized) Major Axis (RMA) regressions were used to determine the slopes, 95% confidence intervals of the slopes, and correlation coefficients for each variable relative to skull length. In addition to the continuous variables of slope and confidence intervals, each variable was also assigned to a categorical variable of positively allometric (95% confidence interval of slope is greater than 1), negatively allometric (95% confidence interval of slope is less than 1), or isometric (95% confidence

140 interval of slope includes 1). The values for the slope, 95% confidence intervals of the slope, correlation coefficient, and allometric category (all based on the entire ontogenetic series) were recorded as the ‘true’ regression parameters for each variable (with 108 specimens), to which the subsamples were then compared.

Subsampling

To test the effect that smaller sample sizes have on the ability to reliably obtain similar slopes and allometric trends, the complete size series was systematically subsampled (1,000 replicates) using four distinct Monte Carlo subsample methods (Random, Population Binned, Length

Binned, and Adults Plus One), regressed, and the results compared to the ‘true’ results.

Random Subsample (Without Replacement). This was the simplest form of Monte Carlo subsampling performed, and consisted of randomly selecting (without replacement) from the entire size series the number of specimens corresponding to the desired sample size, n. The relative position of specimens within the ontogenetic series had no influence on their probability of being selected, and other than the lack of replacement, the choice of the subsequent specimens was not affected by the choice of the proceeding specimens (Fig. 2A).

Binned Subsamples. Two basic methods of binned subsampling were used, occupancy-based and length-based. Both of these divided the ontogenetic series into n bins, with n being the number of samples in the replicate.

Even Length Binned Subsample. This method divided the size series into n bins, with the size of the bins determined by equal divisions of the reference variable (basal skull length) –

141 that is, each bin represents an equal amount of the magnitude of the reference measurement (Fig.

2B). One specimen is then selected at random from each bin. This subsample method represents the best-case scenario as it both maximizes the size range (for the reference variable) of the sampled specimens, and distributes them relatively evenly across that range.

Occupancy Binned Subsample. This method divided the size series into n bins, with the size of the bins determined by equal occupancy of the bins – that is, each bin has the same number of specimens within it (Fig. 2C). One specimen is then selected at random from each bin.

This acts to maximize the range, and even out the distribution, of subsamples within the complete sample, but is dependant on the relative distribution of the sampling intensity.

Adult Bias Subsamples. Ontogenetic datasets derived from existing samples (as opposed to captive breeding), such as museum specimens or fossil data, rarely preserve an even number of samples across the ontogenetic series and often show distinct biases towards sampling of large/adult specimens, and against sampling of small/juvenile specimen. To replicate this, a method of subsampling was developed to bias the distribution of specimens towards having a greater number of large/adult specimens and few small/juvenile specimens.

Adults Plus One. A method was developed to segregate all specimens into two groups, those with a length less then half that of the largest reference variable (less than ‘adults’), and those with a length greater than half that of the largest specimen (‘adults’). For a given samples size (n), the group of larger specimens was subsampled randomly for n-1 specimens, while the group of smaller specimens was subsampled for one specimen (Fig. 2D). This method simulated a sample composed largely of adults, but with one juvenile specimen (from the smaller size class).

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Subsampling Intervals. The Random Subsample was performed over a range from three to 100 specimens, increasing with an interval of one specimen. ‘Occupancy binned’, and ‘adults plus one’ subsample methods were performed over a range from 3 to 20 specimens, increasing with an interval of one specimen. The ‘equal length binned’ method was performed over a range from

3 to 10 specimens, as the unequal distribution of specimens across the axis of the reference specimen would not allow for smaller bin size. All subsample methods were performed for

1,000 independent replicates.

Comparison of Subsamples to Whole Sample

Any difference between the ‘true’ results (i.e., those for all 108 specimens in the entire sample) and those of the subsample replicates is interpreted as error due to small sample size.

Comparisons of differences in results between each subsample level, and for each variable, and the ‘true’ results allowed for determination of the sample sizes for which there are differences in the categorical allometric trends (i.e., positive allometry, isometry, negative allometry) between

‘true’ and subsample results. For the categorical allometric trends, this was determined as the sample size at which 95% for the replicates result in the same allometric trend as that of the

‘true’ (entire) trend.

This analysis was performed for each subsample method, and the resulting minimum sample size compared between subsample methods. Minimum sample sizes were then correlated against ‘true’ slope and/or ‘true’ allometric trend for each variable to test for correlations between the allometric nature of the original variable and the sample needed to reliably recover the slope or trend. This allows for a direct comparison of rates of type I and II errors as a result

143 of sample size, and known slope. Additionally, comparison of the results for the two regression methods (OLS and RMA) was also performed, to highlight any systematic effects that small sample sizes may have on these methods.

Results

Distributions of Sample Size in Allometry

A survey of the literature reveals a wide range of sample sizes (n = 542, range = 3-1449) used for quantifying intraspecific allometry. When these studies are segregated based on their taxa of interest (i.e. vertebrate vs. invertebrate) and ages (i.e. extinct vs. extant) distinct patterns are clear

(Fig. 3, Table 2). Samples from extant invertebrates and vertebrates illustrate very similar distributions, which are not significantly different from each other (KS test, p-value =0.4381)

(Fig. 3, Table 3). In contrast, those studies examining extinct taxa use systematically smaller sample sizes than those of extant taxa, a pattern that is consistent for both vertebrates and invertebrates (p-values << 0.001) (Fig. 3, Table 3). Although not nearly as distinct and the pattern between extinct and extant (for either group), the difference between extinct vertebrates and extinct invertebrates is marginally significant (p-value = 0.0375). The mean, median, minimum, and maximum of the extant samples (both vertebrate and invertebrate) are systematically larger than those of the extinct samples. The systematic use of smaller datasets for extinct taxa can be illustrated in that only 5.6% (invertebrate) and 3.0% (vertebrate) of the extant samples are based on 10 specimens or fewer, while 20.1% (invertebrate) and 34.7% (vertebrate) of the extinct samples are of this size.

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Minimum Sample Size for Determination of Allometry

The minimum sample size for which 95% of the replicates result in the same allometric trend as that of the entire sample are listed in Table 4. This includes the results for all subsampling methods, and for both OLS and RMA. The minimum sample required is only listed for results of allometric trends (positive or negative), and not listed for results of isometry, as it is the null hypothesis.

For the random subsampling method, the sample size at which 95% of the replicates result in the same allometric trend as that of the entire sample ranged from 6 to 95 specimens for

OLS and from 7 to 80 specimens in RMA (Table 4). For both OLS and RMA, five of the variables were found to be isometric at 100 specimens, and these may require samples of greater than 100 specimens to determine an allometric trend, or may truly represent isometry.

Due to the low range of sample sizes over which subsampling could be performed for the

‘occupancy binned’, ‘adults plus one’ and ‘equal length binned’ methods, few variables were able to be identified as allometric (Table 4). For the variables that were found to be allometric under the alternative subsampling methods, those that spread out the selected specimens maximally across the size range reduced the sample size needed for a correct conclusion of allometry. The reduction of the number of specimens needed ranges from 1 to 3 (73 to 88% of the original sample) for ‘occupancy binned’, and from 0 to 10 (44 to 100% of the original sample) for ‘equal length binned’ for both OLS and RMA.

Conversely, those methods which attempted to simulate more realistic sampling (i.e., disproportionate sampling of certain size classes – in this case adult or large size), resulted in a

145 systematic increase in the sample size needed for a correct collusion of allometry. This increase in required sample size ranges from 9 to 12 (130% to 200% of the original sample).

Slope of Allometric Relationship Versus Sample Size. The effect of sample size on the conclusions of allometric trends can be visualized through the use of allometric power plots (Fig.

5 and Appendix C), which plot the proportion of subsample replicates resulting in the categorical results (i.e., positive allometry, isometry, and negative allometry) against sample size. For all variables, subsampling methods, and regression types, the trends at low sample sizes were dominantly isometry, and as the sample size increased the percentage of replicates that were isometric either decreased (for ‘true’ allometry) or increased (for ‘true’ isometry). This is due to large confidence intervals of small samples (low power), making the null hypothesis difficult to reject, and decreasing confidence intervals as sampling increases. The sample size at which 95% of the replicates had the same result as the ‘true’ trend (i.e., that of the entire sample), for which the true trend was allometric was easily determined and recorded. This represents the sample size required for 95% confidence in a conclusion of allometry for that particular variable. When the ‘true’ trend was isometric, however, the level at which 95% of the replicates resulted in the correct trend was more difficult to determine, as the smallest samples usually resulted in the correct conclusion, but due to low statistical power.

Correlation of Slope and Minimum Number of Specimens. There is a strong correlation between the slope of the relationship between two variables and the minimum number of specimens needed to determine an allometric trend with 95% confidence. The further the slope deviates from one (either positively or negatively) the fewer specimens are needed to conclude allometry (Table 4, Fig. 6). Conversely, as the slope becomes closer to one, the number of specimens increases exponentially.

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The relationship between slope and minimum sample size is hyperbolic, with vertical asymptotes at a slope of just below and just above one, and horizontal asymptotes at a minimum sample size for extreme slopes (Fig. 6). The basic equation describing the relationship is shown by Eq. 1.1, where ‘m’ describes the shape of the curve and ‘b’ describes the position of the vertical axis (Fig. 6A). An additional term ‘c’ can be added to this equation (see Eq. 1.2), which allows for additional error in the y-axis (Fig. 6B)

Eq. 1.1

Eq 1.2

Appropriateness of these two equations given the empirical data is compared using

Akaike Information Criterion (AIC) to evaluate goodness of fit given model complexity. The results are shown in Table 5. In all cases Eq. 1.1 is preferred to Eq. 1.2, but this difference is marginal.

Correlation of Variation and Minimum Number of Specimens. There is marginally insignificant correlation between the variation of two variables (measured here as the r2) and the minimum number of specimens needed to conclude an allometric trend with 95% confidence.

This relationship is linear (r2 = 0.347), and negative, in that the higher the variation (the lower the r2) the fewer specimens are required on average to correctly conclude allometry. This negative correlation is counter-intuitive, as one would expect highly correlated relationships (low variance: narrower 95% confidence interval) to result in increased rejection of the null hypotheses of isometry. The reason for this negative relationship is not that a low variance is

147 negatively correlated with the minimum sample size, but rather that the variance is strongly correlated with the deviation of the slope from one (R2 = -0.717, p-value << 0.001) (Fig. 7).

Slopes that deviate greatly from one (strong allometry) tend to also have higher variation (lower r2) values, and this relationship is strong enough to cause r2 and minimum sample size to be negatively correlated. To fully discern the relationship between variance, independent of slope, and minimum sample size a dataset with independence between slope and variance is required.

Comparison of Minimum Sample Required for OLS and RMA. Comparing the minimum number of specimens required for 95% of replicates to correctly conclude allometry between

OLS and RMA reveals an interesting pattern. For variables that are positively allometric RMA consistently requires fewer specimens than does OLS to correctly conclude allometry, and the difference between the two increases as the slope approaches 1 (Fig. 8). Conversely, for variables that are negatively allometric OLS requires fewer specimens than does RMA, and the difference also increases as the slope approaches 1 (Fig. 8). This represents a minimum sample size advantage of OLS for negative allometry and RMA for positive allometry. The relationship between the differences between OLS and RMA minimum sample sizes and slope is best described by the Equation 3, where the sample size advantage of OLS relative to RMA is inversely proportional to the deviation of the slope from 1.

Eq. 1.3

148

Error Rate as a Function of Sample Size

There are three potential errors that can be made when concluding which allometric trend best describes the relative growth of one variable relative to another. Firstly, the two variables do grow at the same rate (i.e. their slope is not different from one) and are isometric, but one concludes that they are growing at different rates (their slope is not one). This represents an incorrect rejection of a true null hypothesis (isometry), and is Type I error or ‘false allometry’.

Secondly, the two variables do grow at different rates (i.e. their slope is not one), and are thus allometric, but one incorrectly concludes they are isometric (their slope is not different from one). This represents an incorrect failure to reject a false null hypothesis (isometry), and is Type

II error or ‘false isometry’. Finally, the two variables do grow at different rates (i.e. their slope is not one), and are thus allometric, and one correctly concludes they are allometric, but the sign of the allometry is wrong (e.g., they are negatively allometric, but are found to be positively allometric. This is referred to here as ‘sign error’, and is distinct from both Type I and II errors.

The relative rates of these different errors changes drastically as a function of the sample size. Figure 9 illustrates the relative dominance of these errors as the allometric sample size increases, for both OLS (A) and RMA (B) in the random subsample. In both cases, Type II error is consistently very high (mean = >50% when n < 12) for small samples, and decreases as the sample size increases. In contrast, Type I error and Sign error are low, and very low respectively, (mean =~10% when n < 12, and mean <3% when n > 12), with Type I error changing little in response to increased sample size, and ‘sign error’ not being a factor at sample sizes great than twenty.

149

Discussion

Extinct and Extant Sample Sizes

The results presented here illustrate that studies of intraspecfic allometry, investigating patterns of relative growth and ontogeny within a species, based on extinct animals consistently use smaller sample sizes than those based on living animals (Fig. 1). Commonality in the difference between samples of extinct and extant taxa in both vertebrates and invertebrates, and the similarity between samples of both extant vertebrates and invertebrates and, to a lesser extent, extinct vertebrates and invertebrates suggest these small sample sizes are a result of work on extinct taxa, irrespective of whether they are vertebrate or invertebrate.

This distinction between sample sizes of extant and extinct taxa is driven by the nature of the specimens available, namely that investigations into allometry in extinct taxa require fossil datasets. In this regard, palaeobiologist are restricted to using those data preserved in fossil record, which greatly restricts the number of specimens and types of specimens available.

In addition to their rarity, the collection of fossil specimens is often more difficult than collection of their extant counterparts. Fossil samples often require great investments of time and money for prospecting, excavation, and preparation exceeding most collection of neontological specimens. This also acts to limit the number of specimens available. When large numbers of specimens from one species can be obtained, these samples often see two distinct and negative effects of taphonomic processes. Firstly, many of the specimens may suffer from either incompleteness or distortion making them difficult or impossible to use in allometric studies

(Strauss and Atanassov, 2006; Brown et al., 2012). Secondly, taphonomic biases often act on the absolute size of the organism, and as a result may skew the relative abundance of samples across

150 the ontogenetic trajectory (e.g. biases reducing the abundance of small-bodied specimens)

(Behrensmeyer et al., 1979; Kidwell and Flessa, 1996; Brown et al., 2013). Occasionally large samples for vertebrate fossils, suitable for allometry studies, can be recovered, but in many these cases they are often restricted to small portions of the anatomy that are both taphonomically resistant and taxonomically informative (e.g., teeth, pachycephalosaur domes) (e.g., Evans et al.,

2013).

Despite these limitations, the fossil record offers unique data not available in neontological datasets, specifically for studies of evolutionary biology requiring deep time data.

Relative Performance of OLS and RMA at Small Sample Size

In general, RMA is more effective at discerning positive allometry while OLS is more effective at discerning negative allometry, when sample sizes are small. This result, is not necessarily surprising in itself, as due to the way the best fit line is calculated (RMA reducing the squared area between the points and the line, and OLS reducing the squared vertical distance between the points and the lines), may result in RMA given consistently higher slopes than OLS. This may indicate that one method may be more practical than the other depending on the nature of the structure analyzed. However, unless there is a strong a priori assumption regarding the nature of the allometric relationship, neither can be recommended over the other based on this. In these cases, it is suggested that both methods be used, and having agreement (or disagreement) between methods can be a useful tool for indicating the confidence one might have for the given conclusion of allometry.

151

The choice of which allometric model (OLS vs. RMA) is preferred should not be determined by their relative ability to discern allometric relationship. Rather, each model makes assumptions regarding the nature of the data and scaling relationship, particularly the symmetry of the data, as well as the location of error (Seim and Saether, 1983; Smith, 2009). Consistency or violation of these assumptions, are the parameters upon which allometric model choice should be based.

Type I and Type II Error and Sample Size

Given the results from the empirical A. mississippiensis data, it is likely impractical to delimit a distinct minimum sample size that is recommend for allometry studies. When the sample size is low, the statistical power is reduced, and the confidence intervals become wider, making the null hypothesis more difficult to reject (high Type II error rate at low sample size) (Fig. 9). This trend is consistent across all variables, regression types, and subsampling methods. Results of allometry are relatively robust regardless of sample size. In contrast, results of isometry, are affected by high Type II at low sample sizes, and may therefore indicate less about the relative growth of a structure, and more about the statistical power of the analysis (Fig. 9).

The sample size at which high Type II error (false isometry) occurs are dependant upon the slope of the variable, but the range of sample size for the 22 A. mississippiensis cranial characters is illustrated in Figure 9. The mean of Type II error is consistently higher than the mean of Type I error for all sample sizes smaller than ~70 specimens for both OLS and RMA.

Above ~70 specimens the rates of Type I and II errors are relatively similar. Importantly, the average sample for both fossil invertebrates (39.7) and vertebrates (21.4) are well below this

152 level, while those of extant invertebrates (72.3) and vertebrates (75.7) are at or above this region of equal Type I and II errors. This suggests that the majority of analysis based on fossil data suffer from significant amounts of Type II error, resulting in disproportionate levels of isometry.

This is not the case for the majority of studies based on extant data. This could lead to the misleading result that isometry is, on average, more common in extinct taxa, due to smaller sample sizes. Further investigations on other taxa will help to determine how representative the empirical data for A. missippiensis are for other taxa and other systems.

Allometric Nomenclature and the False Dominance of Isometry

For ‘true’ isometry the rate of change of one variable relative to another is exactly one. This is only one of the infinite number of possible slopes between the two variables, with all other possible slopes being allometric (either positive or negative). Given this, a slope of exactly one should be rare in biological datasets, but in many cases it is seen as default. As isometry in the null hypothesis, failing to reject a test for isometry does not require a slope of one, just that the slope is not significantly different than one, which will occur with small sample sizes.

We suggest a modification to the nomenclature of isometry to clarify this potential imprecise terminology. The term ‘true isometry’ is suggested for the case of the slope being equal to exactly 1. This is largely a theoretical concept in the context of biology, and would be impossible to prove with empirical data. The term ‘hard isometry’ is suggested for the case in which the slope is not statistically different from 1, and continued sampling will not change this result (i.e., the result is not due to low sample size or low power) (Fig. 10). Conversely, the term

‘soft isometry’ is suggested for the case in which the slope is not statistically different from 1,

153 but this is due to low sample size and may become statistically different from 1 with further sampling (Fig. 10). Allometry, not being prone to high error related to sample size, does not require further subdivision.

The use of ‘hard’ and ‘soft’ to indicate the level of confidence in the isometric relationship is borrowed from the similar usage in phylogenetics (Maddison, 1989). A ‘soft’ polytomy is the situation of unresolved branching pattern due to insufficient or conflicting phylogenetic resolution. Likewise, ‘soft’ isometry is used to indicate the uncertainty in the scaling trend due to low statistical power. Conversely, a ‘hard’ polytomy is used to describe the interpretation of having multiple, simultaneous speciation events associated with a single common ancestor (interpretation of a biological phenomenon). Likewise, ‘hard’ isometry is used to indicate interpretation of equal scaling of two variables, within statistical error.

As with hard and soft polytomies, distinguishing between hard and soft isometry may be not be easy, but the distinction between them is important as they lead to different biological interpretations. Isometric results based on small samples should be interpreted as soft isometery, and due to low statistical power. Subsampling of large datasets, can reveal how the isometeric result changes with sample size and can allow for interpretation of hard isometry.

Acknowledgments

We would like to thank multiple individuals for their contributions (both theoretical and practical) to the development and undertaking of this research project including: J. Arbour, K.

Brink, N. Campione, P. Dodson, D. Evans, D. Jackson, L. O’Brien, and R. Reisz. Access to specimens was facilitated by K. Seymour (ROM- Vertebrate Palaeobiology), R. MacCullach

154

(ROM - Herpetology), and K. Khidas and M. Steingerwald (CMN - Amphibians and Reptiles),

W. Fitch (UCMZ), B. Strilisky (RTMP), A. Resetar (FMNH - Amphibians and Reptiles), G.

Schneider (UM), J. McGuire (UCMVZ), P. Holroyd (UCMP), R. Sadlier (AM). P. Dodson was kind enough to share his Alligator measurement dataset. Special thanks is reserved for Grant

Hurlburt and Kevin Seymour for their work in acquiring the ROM Alligator collection, upon which most of this research is based. Funding for this project was provided by an NSERC-CGS scholarship, the University of Toronto, and the Dinosaur Research Institute.

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Table 1 – Description of the 23 cranial measurements used in this study. See Figure 1 for reference diagram. Modified from Dodson (1975c).

# Description of variable 1 skull width at posterior border of external nares 2 skull width at 4th maxillary tooth 3 skull width at anterior border of orbit 4 skull width at posterior border of quadratojugals 5 skull width across exoccipitals 6 skull length from tip of snout to quadrates – Reference Datum 7 skull length from tip of snout to anterior border of orbits 8 skull length from posterior border of orbit to external condyle of quadrate 9 orbit length 10 orbit width 11 orbit separation 12 lateral temporal fenestra length 13 lateral temporal fenestra height 14 maxilla length 15 palatal fenestra width 16 distance from the posterolateral corner of the pterygoid to the medial condyle of quadrate 17 height of skull from pterygoid process to dorsal surface of skull, perpendicular to long axis 18 maximum depth of jaw 19 external mandibular fenestra length 20 external mandibular fenestra width 21 retroarticular process length, from crest of ridge posterior to articular cotyles to tip of process 22 palatal fenestra length 23 foramen magnum width

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Table 2 – Summary statistics for the sample of published sample sizes in studies examining interspecies allometry or ontogenetic trajectories in both invertebrates and vertebrates, and in both extant and extinct taxa.

N Min Max Mean Median n ≤ 10 Invertebrate Extant 178 6 1449 72.3 32.5 5.6% Vertebrate Extant 169 6 984 75.7 40.0 3.0% Invertebrate Extinct 119 3 733 39.7 19.0 20.1% Vertebrate Extinct 76 4 110 21.4 14.0 34.7%

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Table 3 – The results of a Kolmogorov-Smirnov test of the allometric study sample sizes, testing the hypothesis that the two samples were drawn from the same distribution.

p value Significance Invertebrate: Extant vs. Extinct 1.74E-10 *** Vertebrate: Extant vs. Extinct 2.89E-10 *** Extant: Vertebrate vs. Invertebrate 0.4381 Extinct: Vertebrate vs. Invertebrate 0.03757 *

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Table 4 – Results of allometric analysis for OLS and RMA regression of the 22 cranial skull variables in A. mississippiensis. For each variables, the r2, intercept, slope, lower and upper 95% confidence intervals for the slope, allometric trend, significance of allometric trend, and the minimum sample size required for 95% of replicates to achieve allometry were recorded. Green

= positive allomerty, red = negative allometry, black = isometry.

167

OLS Minimun Sample for 95% Allometry Var. r2 Intercept Slope lCI uCI Trend Sig. Rand. Occ. Leng. Out. 23 0.928 -0.408 0.651 0.616 0.685 - **** 7 6 7 16 10 0.969 -0.073 0.666 0.643 0.689 - **** 6 5 4 18 9 0.981 -0.103 0.737 0.717 0.757 - **** 6 5 4 15 15 0.951 -0.541 0.784 0.749 0.818 - **** 15 12 >10 >20 13 0.974 -0.860 0.916 0.887 0.944 - **** 43 >20 >10 >20 5 0.990 -0.335 0.946 0.922 0.969 - *** 61 >20 >10 >20 22 0.977 -0.571 0.959 0.931 0.988 - ** 98 >20 >10 >20 12 0.976 -0.986 0.969 0.940 0.998 - * >100 >20 >10 >20 14 0.993 -0.244 0.991 0.974 1.007 h. iso 0.2266 >100 >20 >10 >20 17 0.988 -0.445 0.996 0.975 1.017 h. iso 0.7146 >100 >20 >10 >20 3 0.993 -0.379 1.003 0.987 1.020 h. iso 0.6929 >100 >20 >10 >20 8 0.994 -0.567 1.010 0.994 1.025 s. iso 0.1199 >100 >20 >10 >20 4 0.990 -0.327 1.017 0.997 1.038 s. iso 0.0898 >100 >20 >10 >20 2 0.994 -0.491 1.025 1.010 1.040 + ** 77 >20 >10 >20 19 0.986 -0.746 1.033 1.009 1.057 + ** 95 >20 >10 >20 11 0.977 -1.247 1.043 1.012 1.073 + ** 95 >20 >10 >20 1 0.991 -0.687 1.052 1.032 1.072 + **** 36 >20 >10 >20 16 0.981 -0.702 1.054 1.026 1.082 + *** 64 >20 >10 >20 20 0.982 -1.218 1.066 1.038 1.093 + **** 53 >20 >10 >20 18 0.991 -0.970 1.089 1.070 1.109 + **** 18 15 8 >20 21 0.984 -1.133 1.100 1.073 1.127 + **** 35 >20 >10 >20 7 0.995 -0.562 1.132 1.117 1.147 + **** 12 9 7 >20 RMA Minimun Sample for 95% Allometry Var. r2 Intercept Slope lCI uCI Trend SIG Rand. Occ. Leng. Out. 23 0.928 -0.465 0.675 0.641 0.711 - **** 8 7 7 >20 10 0.969 -0.098 0.677 0.654 0.700 - **** 7 6 4 >20 9 0.981 -0.120 0.744 0.724 0.765 - **** 8 6 4 >20 15 0.951 -0.587 0.804 0.770 0.839 - **** 18 15 >10 >20 13 0.974 -0.888 0.928 0.899 0.957 - **** 54 >20 >10 >20 5 0.990 -0.354 0.954 0.930 0.978 - *** 76 >20 >10 >20 22 0.977 -0.597 0.971 0.943 1.000 - * >100 >20 >10 >20 12 0.976 -1.013 0.981 0.952 1.010 h. iso 0.1918 >100 >20 >10 >20 14 0.993 -0.253 0.994 0.978 1.011 h. iso 0.4874 >100 >20 >10 >20 17 0.988 -0.459 1.002 0.981 1.023 h. iso 0.8428 >100 >20 >10 >20 3 0.993 -0.387 1.007 0.991 1.023 h. iso 0.4070 >100 >20 >10 >20 8 0.994 -0.574 1.013 0.998 1.029 s. iso 0.0983 >100 >20 >10 >20 4 0.990 -0.340 1.023 1.003 1.043 + * >100 >20 >10 >20 2 0.994 -0.498 1.028 1.013 1.043 + *** 67 >20 >10 >20 19 0.986 -0.763 1.041 1.017 1.065 + *** 80 >20 >10 >20 11 0.977 -1.275 1.055 1.025 1.086 + *** 75 >20 >10 >20 1 0.991 -0.698 1.057 1.038 1.077 + **** 31 >20 >10 >20 16 0.981 -0.725 1.064 1.036 1.092 + **** 48 >20 >10 >20 20 0.982 -1.240 1.075 1.048 1.103 + **** 42 >20 >10 >20 18 0.991 -0.981 1.094 1.074 1.114 + **** 15 13 8 >20 21 0.984 -1.154 1.109 1.083 1.136 + **** 29 >20 >10 >20 7 0.995 -0.568 1.134 1.120 1.149 + **** 11 8 7 >20

168

Table 5 – Results of model fitting of equations 1.1 and 1.2 to the empirical data for the minimum number of specimens required for determination for allometry (95% confidence) in the crocodilian skull variables. Results include those for both combined and separate OLS and RMA regressions.

Combined RMA/OLS RMA OLS Eq. 1.1 Eq. 1.2 Eq. 1.1 Eq. 1.2 Eq. 1.1 Eq. 1.2 Residual SS 4925 4889 2773 2740 2077 2075 AIC 251.087 252.855 133.885 135.698 122.531 124.512 delta AIC 0.000 1.767 0.000 1.183 0.000 1.981 Akaike Weight 0.708 0.292 0.712 0.288 0.729 0.271 m 3.14 3.25 3.21 3.35 3.04 3.09 95% CI of m (2.83-3.45) (2.66-3.84) (2.74-3.68) (2.46-4.24) (2.53-3.54) (2.10-4.07) b 0.992 0.992 0.993 0.993 0.992 0.992 95% CI of b (0.988-0.997) (0.988-0.996) (0.987-0.999) (0.987-0.999) (0.985-0.999) (0.984-1.000) c NA -2.07 NA -2.68 NA -0.81 95% CI of c NA (-11.29-7.16) NA (-17.51-12.15) NA (-15.08-13.48)

169

Figure 1 – The 23 linear morphometric variables of Alligator mississippiensis skulls used in this study. For description of the measurements taken see Table 1. Figure modified from Dodson

(1975c).

170

171

Figure 2 – Diagram illustrating the five subsampling techniques utilized here: Random

Subsample (A); Even Length Binned Subsample (B); Occupancy Binned Subsample (C); Adults

Only (D); Adults Plus One (E).

172

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Figure 3 – Distribution of 542 sample sizes in published studies examining intraspecies allometry, or ontogenetic trajectories, in invertebrates (A) and vertebrates (B). Solid vertical lines indicate the median, and dotted vertical lines indicate the mean.

174

175

Figure 4 – Comparison of allometric sample size distributions between extant and extinct taxa

(A) and between invertebrates and vertebrates (B). Extinct taxa show a systematically smaller sample size in both invertebrates and vertebrates. Conversely, the sample sizes between invertebrates and vertebrates, for both extinct and extant taxa, are similar. “*” indicates significance results of Kolmogorov-Smirnov tests for differences in distributions.

176

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Figure 5 –Allometric power plots illustrating the effect of sample size on the allometric trend of three representative variables: A) Variable 1, positively allometric; B) Variable 3, isometric; C)

Variable 10, strongly negatively allometric. In all cases, white indicates isometry, green indicates positive allometry, red represents negative allometry, and grey indicated disagreement between

OLS and RMA. The bars at the top represent the minimum sample size needed to achieve the same allometric trend as the entire dataset.

178

179

Figure 6 – Minimum sample size required for correct identification of allometric trend (in 95% of replicates) as a function of slope, for each of the 22 variables that were found to be allometric for both OLS (black) and RMA (grey). Minimum required sample sizes are small away from

1.00 (i.e., strong allometry) and increase exponentially to vertical asymptotes as the slope approaches one (i.e., isometry). The relationship is best described by a hyperbolic function (solid line), with 95% confidence intervals indicated in grey. The fitted model includes both the RMA and OLS data. A) Simpler model (Eq. 1.1) with two parameters. B) More complex model (Eq

1.2) with third term, allowing for error in y-axis. The vertical dashed line indicates a slope of

1.00.

180

181

Figure 7 – Correlation between the absolute deviation of the slope from 1.00, and the correlation coefficient (r2). As the slope deviates from one, the correlation coefficient decreases (the variance increases).

182

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Figure 8 – Relationship between the slope of the variable and the difference in the minimum sample size for allometry (95% or replicates) between OLS and RMA. No difference (0) indicates that RMA and OLS require similar sized samples to conclude allometry, the upper half of the graph shows OLS needing larger sample sizes to determine allometry, while the lower half shows RMA needing larger sample sizes to determine allometry.

184

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Figure 9 – The effect of sample size on the frequency of Type I error or false allometry (green),

Type II error or false isometry (red) and wrong sign error (blue) in the subsample replicates for

A) OLS and B) RMA. Solid lines represent the means and dotted lines represent one standard deviation of replicates. Mean sample sizes for fossil and extant, and vertebrae and invertebrate, allometric studies are indicated with the vertical bars.

186

187

Figure 10 – Schematic of the relationship of the true slope and the sample size to the ability to categorized allometric trends.

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189

Appendix A – Dataset with the results of a literature survey investigating the sample sizes used in intraspecies allometric studies. Each sample includes the literature reference, genus and species, whether it is invertebrate or vertebrate and extinct or extant, and the sample size (N).

190

Referance Genus and Species Group Extinct? N Emlen, 1997 Onthophagus acuminatus Invertebrate Extant 600 Kawano, 2000 Cyclommatus bicolor Invertebrate Extant 27 Kawano, 2000 Cyclommatus canaliculatus Invertebrate Extant 46 Kawano, 2000 Cyclommatus cupreonitens Invertebrate Extant 22 Kawano, 2000 Cyclommatus dehaani Invertebrate Extant 26 Kawano, 2000 Cyclommatus elaphus Invertebrate Extant 29 Kawano, 2000 Cyclommatus lunifer Invertebrate Extant 32 Kawano, 2000 Cyclommatus metalifer Invertebrate Extant 33 Kawano, 2000 Cyclommatus montanellus Invertebrate Extant 22 Kawano, 2000 Cyclommatus pahanensis Invertebrate Extant 32 Kawano, 2000 Cyclommatus simaluensis Invertebrate Extant 21 Kawano, 2000 Cyclommatus zuberi Invertebrate Extant 50 Kawano, 2000 Neolucanus baladeva Invertebrate Extant 20 Kawano, 2000 Neolucanus brevis Invertebrate Extant 28 Kawano, 2000 Neolucanus cingulatus Invertebrate Extant 26 Kawano, 2000 Neolucanus giganteus Invertebrate Extant 25 Kawano, 2000 Neolucanus laticollis Invertebrate Extant 20 Kawano, 2000 Neolucanus maximus Invertebrate Extant 22 Kawano, 2000 Neolucanus nitidus Invertebrate Extant 34 Kawano, 2000 Neolucanus parryi Invertebrate Extant 35 Kawano, 2000 Neolucanus perarmatus Invertebrate Extant 25 Kawano, 2000 Neolucanus sinicus Invertebrate Extant 27 Kawano, 2000 Odontolabis alces Invertebrate Extant 114 Kawano, 2000 Odontolabis belicosas Invertebrate Extant 33 Kawano, 2000 Odontolabis brookeanus Invertebrate Extant 26 Kawano, 2000 Odontolabis castelnaudi Invertebrate Extant 65 Kawano, 2000 Odontolabis cuvera Invertebrate Extant 130 Kawano, 2000 Odontolabis cypri Invertebrate Extant 28 Kawano, 2000 Odontolabis femoralis Invertebrate Extant 50 Kawano, 2000 Odontolabis gazella Invertebrate Extant 38 Kawano, 2000 Odontolabis imperialis Invertebrate Extant 21 Kawano, 2000 Odontolabis intermedius Invertebrate Extant 68 Kawano, 2000 Odontolabis kazuhisai Invertebrate Extant 22 Kawano, 2000 Odontolabis lacordairei Invertebrate Extant 57 Kawano, 2000 Odontolabis latipennis Invertebrate Extant 38 Kawano, 2000 Odontolabis leuthneri Invertebrate Extant 30 Kawano, 2000 Odontolabis ludekingi Invertebrate Extant 48 Kawano, 2000 Odontolabis micros Invertebrate Extant 23 Kawano, 2000 Odontolabis mouhoti Invertebrate Extant 46 Kawano, 2000 Odontolabis platynota Invertebrate Extant 48 Kawano, 2000 Odontolabis siva Invertebrate Extant 40 Kawano, 2000 Odontolabis sommeri Invertebrate Extant 73 Kawano, 2000 Odontolabis spectabilis Invertebrate Extant 22 Kawano, 2000 Odontolabis stevensi Invertebrate Extant 35 Kawano, 2000 Odontolabis vollenhoveni Invertebrate Extant 21 Kawano, 2000 Odontolabis wallastoni Invertebrate Extant 48

191

Referance Genus and Species Group Extinct? N Klingenberg and Zimmermann, 1992 Gerris argentatus Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 Gerris argentatus Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. costae Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. costae Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. gibbifer Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. gibbifer Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. lacustris Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. lacustris Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. lateralis Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. lateralis Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. odontogaster Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. odontogaster Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. thoracicus Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 G. thoracicus Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 Aquarius najas Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 Aquarius najas Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 A. paludum Invertebrate Extant 20 Klingenberg and Zimmermann, 1992 A. paludum Invertebrate Extant 20 Clayton and Snowden, 1991 Ilyoplax stevensi Invertebrate Extant 109 Clayton and Snowden, 1991 Ilyoplax stevensi Invertebrate Extant 95 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 101 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 122 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 54 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 79 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 38 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 44 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 59 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 58 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 102 Tzeng and Yeh, 2001 Penaeus japonicus Invertebrate Extant 66 Clayton 1990 Ceistostoma kuwaitense Invertebrate Extant 134 Hall et al., 2006 Portunus pelagicus Invertebrate Extant 1287 Hall et al., 2006 Hypothalassia acerba Invertebrate Extant 571 Hall et al., 2006 Chaceon bicolor Invertebrate Extant 392 Hall et al., 2006 Farfantepenaeus duorarum Invertebrate Extant 927 Kapiris, 2007 Aristaeomorpha foliacea Invertebrate Extant 1449 Kemp and Bertness, 1984 Littorina littorea Invertebrate Extant 36 Kemp and Bertness, 1984 Littorina littorea Invertebrate Extant 22 Kemp and Bertness, 1984 Littorina littorea Invertebrate Extant 80 Kemp and Bertness, 1984 Littorina littorea Invertebrate Extant 54 Kemp and Bertness, 1984 Littorina littorea Invertebrate Extant 200 Chaparro, 2008 Ostrea chilensis Invertebrate Extant 180 Baker and Wilkinson, 2001 Chaetodiopsis meigenii Invertebrate Extant 40 Baker and Wilkinson, 2001 Cyrtodiopsis dalmanni Invertebrate Extant 93 Baker and Wilkinson, 2001 Cyrtodiopsis quinqueguttata Invertebrate Extant 102 Baker and Wilkinson, 2001 Cyrtodiopsis whitei Invertebrate Extant 86 Baker and Wilkinson, 2001 Diasemopsis aethiopica Invertebrate Extant 104 Baker and Wilkinson, 2001 Diasemopsis albifacies Invertebrate Extant 63

192

Referance Genus and Species Group Extinct? N Baker and Wilkinson, 2001 Diasemopsis conjuncta Invertebrate Extant 65 Baker and Wilkinson, 2001 Diasemopsis dubia Invertebrate Extant 128 Baker and Wilkinson, 2001 Diasemopsis elongata Invertebrate Extant 30 Baker and Wilkinson, 2001 Diasemopsis fasciata Invertebrate Extant 63 Baker and Wilkinson, 2001 Diasemopsis hirsuta Invertebrate Extant 67 Baker and Wilkinson, 2001 Diasemopsis longipedunculata Invertebrate Extant 74 Baker and Wilkinson, 2001 Diasemopsis nebulosa Invertebrate Extant 66 Baker and Wilkinson, 2001 Diasemopsis obstans Invertebrate Extant 124 Baker and Wilkinson, 2001 Diasemopsis signata Invertebrate Extant 65 Baker and Wilkinson, 2001 Diasemopsis ilvatica Invertebrate Extant 82 Baker and Wilkinson, 2001 Diasemopsis sp. Invertebrate Extant 21 Baker and Wilkinson, 2001 Diopsis apicalis Invertebrate Extant 31 Baker and Wilkinson, 2001 Diopsis fumipennis Invertebrate Extant 25 Baker and Wilkinson, 2001 Diopsis gnu Invertebrate Extant 7 Baker and Wilkinson, 2001 Eurydiopsis argentifera Invertebrate Extant 6 Baker and Wilkinson, 2001 Sphyracephala beccarii Invertebrate Extant 99 Baker and Wilkinson, 2001 Sphyracephala brevicornis Invertebrate Extant 32 Baker and Wilkinson, 2001 Sphyracephala bipunctipennis Invertebrate Extant 23 Baker and Wilkinson, 2001 Sphyracephala munroi Invertebrate Extant 45 Baker and Wilkinson, 2001 Teleopsis breviscopium Invertebrate Extant 60 Baker and Wilkinson, 2001 Teleopsis quadriguttata Invertebrate Extant 29 Baker and Wilkinson, 2001 Teleopsis rubicunda Invertebrate Extant 89 Baker and Wilkinson, 2001 Teloglabrus entabenensis Invertebrate Extant 18 Baker and Wilkinson, 2001 Trichodiopsis minuta Invertebrate Extant 41 Baker and Wilkinson, 2001 Chaetodiopsis meigenii Invertebrate Extant 27 Baker and Wilkinson, 2001 Cyrtodiopsis dalmanni Invertebrate Extant 91 Baker and Wilkinson, 2001 Cyrtodiopsis quinqueguttata Invertebrate Extant 66 Baker and Wilkinson, 2001 Cyrtodiopsis whitei Invertebrate Extant 86 Baker and Wilkinson, 2001 Diasemopsis aethiopica Invertebrate Extant 93 Baker and Wilkinson, 2001 Diasemopsis albifacies Invertebrate Extant 67 Baker and Wilkinson, 2001 Diasemopsis conjuncta Invertebrate Extant 65 Baker and Wilkinson, 2001 Diasemopsis dubia Invertebrate Extant 107 Baker and Wilkinson, 2001 Diasemopsis elongata Invertebrate Extant 29 Baker and Wilkinson, 2001 Diasemopsis fasciata Invertebrate Extant 63 Baker and Wilkinson, 2001 Diasemopsis hirsuta Invertebrate Extant 67 Baker and Wilkinson, 2001 Diasemopsis longipedunculata Invertebrate Extant 50 Baker and Wilkinson, 2001 Diasemopsis nebulosa Invertebrate Extant 66 Baker and Wilkinson, 2001 Diasemopsis obstans Invertebrate Extant 107 Baker and Wilkinson, 2001 Diasemopsis signata Invertebrate Extant 63 Baker and Wilkinson, 2001 Diasemopsis ilvatica Invertebrate Extant 112 Baker and Wilkinson, 2001 Diasemopsis sp. Invertebrate Extant 22 Baker and Wilkinson, 2001 Diopsis apicalis Invertebrate Extant 31 Baker and Wilkinson, 2001 Diopsis fumipennis Invertebrate Extant 21 Baker and Wilkinson, 2001 Diopsis gnu Invertebrate Extant 6 Baker and Wilkinson, 2001 Eurydiopsis argentifera Invertebrate Extant 9 Baker and Wilkinson, 2001 Sphyracephala beccarii Invertebrate Extant 110 Baker and Wilkinson, 2001 Sphyracephala brevicornis Invertebrate Extant 33 Baker and Wilkinson, 2001 Sphyracephala bipunctipennis Invertebrate Extant 22

193

Referance Genus and Species Group Extinct? N Baker and Wilkinson, 2001 Sphyracephala munroi Invertebrate Extant 14 Baker and Wilkinson, 2001 Teleopsis breviscopium Invertebrate Extant 30 Baker and Wilkinson, 2001 Teleopsis quadriguttata Invertebrate Extant 18 Baker and Wilkinson, 2001 Teleopsis rubicunda Invertebrate Extant 52 Baker and Wilkinson, 2001 Teloglabrus entabenensis Invertebrate Extant 12 Baker and Wilkinson, 2001 Trichodiopsis minuta Invertebrate Extant 34 Hines, 1982 Cancer magister Invertebrate Extant 10 Hines, 1982 Geryon quinquidens Invertebrate Extant 17 Hines, 1982 Hemigraphsus nudus Invertebrate Extant 20 Hines, 1982 Hemigraphsus oregonensis Invertebrate Extant 21 Hines, 1982 Pachygrapsus crassipes Invertebrate Extant 19 Hines, 1982 Sesarma reticulatum Invertebrate Extant 6 Hines, 1982 Libinia emarginata Invertebrate Extant 8 Hines, 1982 Loxorhynchus Invertebrate Extant 12 Hines, 1982 Mimulus foliatus Invertebrate Extant 11 Hines, 1982 Pugettia producta Invertebrate Extant 23 Hines, 1982 Pugettia richii Invertebrate Extant 28 Hines, 1982 Scyra acutifrons Invertebrate Extant 18 Hines, 1982 Uca minax Invertebrate Extant 6 Hines, 1982 Uca pugnax Invertebrate Extant 11 Hines, 1982 Callinectes sapidus Invertebrate Extant 12 Hines, 1982 Ovalipes ocellatus Invertebrate Extant 6 Hines, 1982 Eurypoanopeus depressus Invertebrate Extant 13 Hines, 1982 Lophopanopeus leucomanus Invertebrate Extant 6 Hines, 1982 Panopeus herbstii Invertebrate Extant 20 Hines, 1982 Rhithropanopeus harrisii Invertebrate Extant 22 Lagergren et al., 2007 Bosmin gibbera Invertebrate Extant 50 Lagergren et al., 2007 Daphnia cucullata Invertebrate Extant 50 Lagergren et al., 2007 Bosmina retro-extensa Invertebrate Extant 30 Lagergren et al., 2007 Leptodora kindtii Invertebrate Extant 150 Helluy et al., 1995 Homarus americanus Invertebrate Extant 16 Mooi and David, 1993 Urechinus mortenseni Invertebrate Extant 23 Mooi and David, 1993 Urechinus mortenseni Invertebrate Extant 16 Hale et al., 2004 Aporrectodes Invertebrate Extant 241 Hale et al., 2004 Dendrobaena octaedra Invertebrate Extant 157 Hale et al., 2004 Octolasion tyrlaeum Invertebrate Extant 57 Hale et al., 2004 Lumbricus Invertebrate Extant 88 Dodson, 1975 Aligator mississippiensis Vertebrate Extant 52 Yezerinac et al., 1992 Passerculus sandwichensis Vertebrate Extant 20 Yezerinac et al., 1992 Passer domesticus Vertebrate Extant 25 Yezerinac et al., 1992 Acridotheres tristis Vertebrate Extant 25 Yezerinac et al., 1992 Empidonax ainorum Vertebrate Extant 19 Yezerinac et al., 1992 Molothrus ater Vertebrate Extant 25 Yezerinac et al., 1992 Euphagus cyanocephalus Vertebrate Extant 23 Yezerinac et al., 1992 Zonotrichia capensis Vertebrate Extant 21 Leamy and Bradley Mus musculus Vertebrate Extant 383 Piacentino and y Barrera Oro Trematomus newnesi Vertebrate Extant 180 Kowalewski et al., 1996 Glottidia albida Vertebrate Extant 11

194

Referance Genus and Species Group Extinct? N Kowalewski et al., 1996 Glottidia audebarti Vertebrate Extant 27 Kowalewski et al., 1996 Glottidia palmeri Vertebrate Extant 83 Kowalewski et al., 1996 Glottidia pyramidata Vertebrate Extant 30 Cassemiro et al., 2007 Satanoperca pappaterra Vertebrate Extant 258 Larson, 2003 Bufo americanus Vertebrate Extant 105 Ismen et al., 2006 Rhinobatos rhinobatos Vertebrate Extant 225 âoban et al., 2009 Sparus aurata Vertebrate Extant 784 Dodson, 1975 Casuarius casuarius Vertebrate Extant 43 Dodson, 1975 Sceloporus olivaceous Vertebrate Extant 80 Dodson, 1975 Sceloporus undulatus Vertebrate Extant 69 Pounds et al., 1983 Dipsosaurus dorslis Vertebrate Extant 132 Pounds et al., 1983 Uta stansburiana Vertebrate Extant 124 Pounds et al., 1983 Callisaurus draconoides Vertebrate Extant 116 Pounds et al., 1983 Cophosaurus taxanus Vertebrate Extant 128 Pounds et al., 1983 Sceloporus woodi Vertebrate Extant 159 Garland, 1985 Amphibolurus nuchalis Vertebrate Extant 68 Wotherspoon and Burgin, 2010 Pogona barata Vertebrate Extant 197 Written, 1985 Tympanocryptis cephalus Vertebrate Extant 12 Written, 1985 Tympanocryptis intirna Vertebrate Extant 12 Written, 1985 Tympanocryptis tetraporophora Vertebrate Extant 17 Written, 1985 Tympanocryptis adelaidensis Vertebrate Extant 16 Written, 1985 Tympanocryptis diemensis Vertebrate Extant 49 Written, 1985 Arnphibolurus rnuricafus Vertebrate Extant 65 Written, 1985 Arnphibolurus nobbi Vertebrate Extant 147 Written, 1985 Cairnanops amphiboluroides Vertebrate Extant 22 Written, 1985 Chlamydosaurus kingii Vertebrate Extant 14 Written, 1985 Diporiphora albilabris Vertebrate Extant 14 Written, 1985 Diporiphora australis Vertebrate Extant 15 Written, 1985 Diporiphora bennettii Vertebrate Extant 14 Written, 1985 Diporiphora bilineata Vertebrate Extant 29 Written, 1985 Diporiphora lalliae Vertebrate Extant 10 Written, 1985 Diporiphora magna Vertebrate Extant 14 Written, 1985 Diporiphora winneckei Vertebrate Extant 13 Written, 1985 Lophognathus gilberti gilberti Vertebrate Extant 14 Written, 1985 Lophognathus g. centralis Vertebrate Extant 61 Written, 1985 Lophognathus longirostris Vertebrate Extant 12 Written, 1985 Lophognathus ternporalis Vertebrate Extant 12 Written, 1985 Ctenophorus decresii Vertebrate Extant 14 Written, 1985 Ctenophorus fionni Vertebrate Extant 19 Written, 1985 Ctenophorus ornatus Vertebrate Extant 10 Written, 1985 Ctenophorus pictus Vertebrate Extant 23 Written, 1985 Ctenophorus fordi Vertebrate Extant 31 Written, 1985 Ctenophorus isolepis Vertebrate Extant 62 Written, 1985 Ctenophorus cristatus Vertebrate Extant 18 Written, 1985 Ctenophorus caudicinctus Vertebrate Extant 31 Written, 1985 Ctenophorus reticulatus Vertebrate Extant 20 Written, 1985 Ctenophorus gibba Vertebrate Extant 17 Written, 1985 Ctenophorus maculosus Vertebrate Extant 28

195

Referance Genus and Species Group Extinct? N Written, 1985 Ctenophorus nuchalis Vertebrate Extant 52 Written, 1985 Pogona barbata Vertebrate Extant 21 Written, 1985 Pogona vitticeps Vertebrate Extant 16 Written, 1985 Moloch horridus Vertebrate Extant 12 Written, 1985 Phjaignathus lesueurii Vertebrate Extant 14 Written, 1985 Chelosania brunnea Vertebrate Extant 6 Luque et al., 2007 Arctocephalus gazilla Vertebrate Extant 153 Luque et al., 2007 Arctocephalus tropicalis Vertebrate Extant 124 Livezey, 1989 Pelucunoides urinutor Vertebrate Extant 60 Livezey, 1989 Aptenodvtes forsteri Vertebrate Extant 49 Livezey, 1989 Pygoscelis udeliae Vertebrate Extant 39 Livezey, 1989 Eudyptus chrysocome Vertebrate Extant 63 Livezey, 1989 Megadyptes antipodes Vertebrate Extant 45 Livezey, 1989 Eudyptula minor Vertebrate Extant 42 Livezey, 1989 Spheniscus mugellanicus Vertebrate Extant 63 Devillers et al., 1993 Equus caballus prjewalski Vertebrate Extant 13 Devillers et al., 1993 Equus caballas (domestic) Vertebrate Extant 44 Devillers et al., 1993 Ponies Vertebrate Extant 7 Devillers et al., 1993 Equus quagga Vertebrate Extant 31 Devillers et al., 1993 Equus hemionus Vertebrate Extant 35 Devillers et al., 1993 Equus zebra Vertebrate Extant 17 Hills et al., 1983 Homo sapiens Vertebrate Extant 131 Hills et al., 1983 Gorilla Vertebrate Extant 25 Hills et al., 1983 Pongo Vertebrate Extant 24 Corruccini and McHenry, 1978 Homo sapiens Vertebrate Extant 57 Kieser and Groeneveld, 1988 Homo sapiens Vertebrate Extant 100 Ravosa and Daniel, 2010 Propithecus verreauxi Vertebrate Extant 74 Ravosa and Daniel, 2010 Propithecus diadema Vertebrate Extant 36 Ravosa and Daniel, 2010 Avahi laniger Vertebrate Extant 26 Ravosa and Daniel, 2010 Indri indri Vertebrate Extant 26 Ravosa and Daniel, 2010 Eulemur fulvus Vertebrate Extant 70 Ravosa and Daniel, 2010 Eulemur macaco Vertebrate Extant 24 Ravosa and Daniel, 2010 Eulemur mongoz Vertebrate Extant 29 Ravosa and Daniel, 2010 Eulemur rubriventer Vertebrate Extant 26 Ravosa and Daniel, 2010 Lemur catta Vertebrate Extant 44 Ravosa and Daniel, 2010 Varecia variegata Vertebrate Extant 43 Ravosa and Daniel, 2010 Hapallemur griseus Vertebrate Extant 61 Weston, 2005 Hippopotamus amphibius Vertebrate Extant 16 Wolpoff, 1981 Gorilla gorilla Vertebrate Extant 210 Wolpoff, 1981 Gorilla gorilla Vertebrate Extant 108 Wolpoff, 1981 Pan Vertebrate Extant 81 Wolpoff, 1981 Pan Vertebrate Extant 84 Wolpoff, 1981 Pongo Vertebrate Extant 51 Wolpoff, 1981 Pongo Vertebrate Extant 51 Wood, 1979 Gorilla gorilla Vertebrate Extant 20 Wood, 1979 Gorilla gorilla Vertebrate Extant 17 Wood, 1979 Pan Vertebrate Extant 13 Wood, 1979 Pan Vertebrate Extant 22

196

Referance Genus and Species Group Extinct? N KurtŽn, 1955 Ursus arctos Vertebrate Extant 151 Weston and Lister, 2009 Hippopotomus amphibius Vertebrate Extant 33 Weston and Lister, 2009 Hippopotomus lemerlei Vertebrate Extant 12 Weston and Lister, 2009 Hippopotomus madagascariensis Vertebrate Extant 12 Shea, 1985 Pan paniscus Vertebrate Extant 120 Shea, 1985 Pan troglodytes Vertebrate Extant 124 Shea, 1985 Gorilla gorilla Vertebrate Extant 135 Loponte et al., 2010 Pterodoras granulosua Vertebrate Extant 28 Holliday and Franciscus, 2012 Homo sapiens Vertebrate Extant 894 Holliday and Franciscus, 2012 Pan troglodytes Vertebrate Extant 35 Holliday and Franciscus, 2012 Gorilla gorilla Vertebrate Extant 36 Kaplan and Salthe, 1979 Amblystoma tigrinum Vertebrate Extant 15 Kaplan and Salthe, 1979 Amblystoma maculatum Vertebrate Extant 48 Kaplan and Salthe, 1979 Amblystoma opacum Vertebrate Extant 22 Kaplan and Salthe, 1979 Amblystoma opacum Vertebrate Extant 14 Wood and Wilso, 1986 Homo sapiens Vertebrate Extant 70 Wood et al., 1991 Homo sapiens Vertebrate Extant 75 Wood et al., 1991 Gorilla gorilla Vertebrate Extant 64 Wood et al., 1991 Pan troglodytes Vertebrate Extant 51 Wood et al., 1991 Pan pygmaeus Vertebrate Extant 43 Lnouye, 1991 Pan troglodytes Vertebrate Extant 107 Lnouye, 1991 Gorilla gorilla Vertebrate Extant 103 Lnouye, 1991 Pan paniscus Vertebrate Extant 29 Lnouye, 1991 Gorilla gorilla beringei Vertebrate Extant 23 Lnouye, 1991 Pan pygmaeus Vertebrate Extant 38 Gomez, 1991 Loris tardigradus Vertebrate Extant 24 Gomez, 1991 Nycticebus pygmaeus Vertebrate Extant 10 Gomez, 1991 Arctocebus calabarensis Vertebrate Extant 30 Gomez, 1991 Nyctucebus coucang Vertebrate Extant 24 Gomez, 1991 Perodicticus potto Vertebrate Extant 33 Blob, 2006 Larus californicus Vertebrate Extant 23 Blob, 2006 Larus californicus Vertebrate Extant 23 Blob, 2006 Larus californicus Vertebrate Extant 22 Blob, 2006 Lepus californicus Vertebrate Extant 49 Blob, 2006 Lepus californicus Vertebrate Extant 51 Blob, 2006 Lepus californicus Vertebrate Extant 20 Miller et al., 2009 Ursus americanus Vertebrate Extant 429 Miller et al., 2009 Ursus americanus Vertebrate Extant 502 Koumoundours et al., 1999 Dentex dentex Vertebrate Extant 368 Koumoundours et al., 1999 Dentex dentex Vertebrate Extant 398 Hernandez and Motta, 1997 Archosargus probatocephalus Vertebrate Extant 190 Cardini and O'Higgins, 2005 Marmota caligata Vertebrate Extant 64 Cardini and O'Higgins, 2005 Marmota caudata Vertebrate Extant 47 Cardini and O'Higgins, 2005 Marmota flaviventris Vertebrate Extant 80 Cardini and O'Higgins, 2005 Marmota himalayana Vertebrate Extant 55 Cardini and O'Higgins, 2005 Marmota marmota Vertebrate Extant 67 Cardini and O'Higgins, 2005 Marmota monax Vertebrate Extant 73 Leutenegger and Masterson, 1989 Pongo pygmaeus Vertebrate Extant 212

197

Referance Genus and Species Group Extinct? N Gisbert, 1999 Acipenser baeri Vertebrate Extant 143 Flores et al., 2003 Lutreolina crassicaudata Vertebrate Extant 43 Flores et al., 2006 Dasyurus albopunctatus Vertebrate Extant 31 Bergmann abd Motta, 2005 Cichlasoma urophthalmus Vertebrate Extant 204 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 57 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 57 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 41 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 42 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 56 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 29 Bergmann abd Motta, 2005 Mus musculus Vertebrate Extant 282 Graed et al., 2010 Sander vitreus Vertebrate Extant 20 Graed et al., 2010 Perca flavescens Vertebrate Extant 56 Hellig et al., 2010 Lepidiolamprologus elongatus Vertebrate Extant 40 Dodson, 1976 Protoceratops Vertebrate Extinct 24 Dodson, 1975 Corythosaurus casuarius Vertebrate Extinct 19 Dodson, 1975 Lambeosaurus lambei Vertebrate Extinct 13 Dodson, 1975 Lambeosaurus magnacristatus Vertebrate Extinct 2 Evans, 2010 Corythosaurus Vertebrate Extinct 18 Evans, 2010 Lamebeosaurus Vertebrate Extinct 16 Evans, 2010 Hypacrosaurus Vertebrate Extinct 10 Evans, 2010 Hypacrosaurus altispinus Vertebrate Extinct 7 Campman et al., 1981 Stegoceras validum Vertebrate Extinct 29 Lehman, 1992 Chasmosaurus mariscalensis Vertebrate Extinct 7 Lehman, 1992 Chasmosaurus mariscalensis Vertebrate Extinct 13 Lehman, 1992 Chasmosaurus mariscalensis Vertebrate Extinct 10 Currie and Carroll, 1984 Thadeosaurus colcanapi Vertebrate Extinct 14 Schott et al., 2011 Stegoceras validum Vertebrate Extinct 40 Houck et al., 1990 Archaeopteryx lithographica Vertebrate Extinct 6 Campione and Evans., 2011 Edmontosaurus regalis Vertebrate Extinct 9 Campione and Evans., 2011 Edmontosaurus annectans Vertebrate Extinct 14 Gould, 1973 Megaloceros giganteus Vertebrate Extinct 79 Devillers et al., 1984 Mesohippus Vertebrate Extinct 19 Devillers et al., 1984 Miohippus Vertebrate Extinct 6 Devillers et al., 1984 Parahippus Vertebrate Extinct 7 Devillers et al., 1984 Merychippus A Vertebrate Extinct 6 Devillers et al., 1984 Merychippus B Vertebrate Extinct 54 Devillers et al., 1984 Griphippus Vertebrate Extinct 18 Devillers et al., 1984 Cormohipparion Vertebrate Extinct 15 Devillers et al., 1984 Pliohippus Vertebrate Extinct 14 Devillers et al., 1984 Equus sp. (Alaska) Vertebrate Extinct 13 Benton and Kirkpatrick, 1989 Scaphonyx fisheri Vertebrate Extinct 13 Weston, 2003 Hexaprotodon liberiensis Vertebrate Extinct 19 Weston, 2003 Hexaprotodon harvardi Vertebrate Extinct 6 KurtŽn, 1955 Ursus spelaeus Vertebrate Extinct 110 Black et al., 2001 Nimbadon lavarackorum Vertebrate Extinct 26 Cavalcanti, 2002 Mesosaurus tenuidens Vertebrate Extinct 37 Cavalcanti, 2002 Stereosternum tumidum Vertebrate Extinct 13

198

Referance Genus and Species Group Extinct? N Cavalcanti, 2002 Brazilosaurus sanpauloensis Vertebrate Extinct 4 Bennet, 1995 Rhamphorhynchus muensteri Vertebrate Extinct 40 Bennet, 1995 Rhamphorhynchus muensteri Vertebrate Extinct 23 Witzmann, 2007 Archegosaurus decheni Vertebrate Extinct 96 Tomknins et al., 2010 Pteranodon longiceps Vertebrate Extinct 9 Werdelin and Long, 1986 Bothriolepis canadesis Vertebrate Extinct 104 Emerson, 1993 Oryctolagus cuniculus Vertebrate Extinct 17 Emerson, 1993 Lepus californicus Vertebrate Extinct 18 Purnell, 1993 Idiognathodus Vertebrate Extinct 59 Purnell, 1993 Gnathodus bilineatus Vertebrate Extinct 13 Kilbourne and Makovicky, 2010 Riojasaurus incertus Vertebrate Extinct 7 Kilbourne and Makovicky, 2010 Apatosaurus sp. Vertebrate Extinct 18 Kilbourne and Makovicky, 2010 Camarasaurus sp. Vertebrate Extinct 32 Kilbourne and Makovicky, 2010 Diplodocus sp. Vertebrate Extinct 17 Kilbourne and Makovicky, 2010 Miasaura peeblesorum Vertebrate Extinct 41 Kilbourne and Makovicky, 2010 Tenontosaurus tilletti Vertebrate Extinct 6 Kilbourne and Makovicky, 2010 Kentrosaurus authiopicus Vertebrate Extinct 14 Kilbourne and Makovicky, 2010 Centrosaurus apertus Vertebrate Extinct 6 Kilbourne and Makovicky, 2010 Allosaurus fragilis Vertebrate Extinct 49 Kilbourne and Makovicky, 2010 Tyrannosaurus rex Vertebrate Extinct 21 Kilbourne and Makovicky, 2010 Tarbosaurus bataar Vertebrate Extinct 5 Kilbourne and Makovicky, 2010 Gorgosaurus libratus Vertebrate Extinct 12 Kilbourne and Makovicky, 2010 Daspletosaurus torosus Vertebrate Extinct 6 Kilbourne and Makovicky, 2010 Gallimimus bullatus Vertebrate Extinct 5 Kilbourne and Makovicky, 2010 Herrerasaurus ischigualastensis Vertebrate Extinct 5 Kilbourne and Makovicky, 2010 Troodon formosus Vertebrate Extinct 7 Kilbourne and Makovicky, 2010 Massospondylus carinatus Vertebrate Extinct 16 Kilbourne and Makovicky, 2010 Riojasaurus incertus Vertebrate Extinct 8 Kilbourne and Makovicky, 2010 Apatosaurus sp. Vertebrate Extinct 12 Kilbourne and Makovicky, 2010 Camarasaurus sp. Vertebrate Extinct 27 Kilbourne and Makovicky, 2010 Diplodocus sp. Vertebrate Extinct 23 Kilbourne and Makovicky, 2010 Tenontosaurus tilletti Vertebrate Extinct 6 Kilbourne and Makovicky, 2010 Dyosaurus lettowvorbecki Vertebrate Extinct 27 Kilbourne and Makovicky, 2010 Hypacrosaurus stebingeri Vertebrate Extinct 12 Kilbourne and Makovicky, 2010 Miasaura peeblesorum Vertebrate Extinct 23 Kilbourne and Makovicky, 2010 Stegosaurus stenops Vertebrate Extinct 6 Kilbourne and Makovicky, 2010 Kentrosaurus aethiopicus Vertebrate Extinct 37 Kilbourne and Makovicky, 2010 Pachyrhinosaurus sp. Vertebrate Extinct 8 Kilbourne and Makovicky, 2010 Centrosaurus apertus Vertebrate Extinct 4 Kilbourne and Makovicky, 2010 Chasmosaurus mariscalensis Vertebrate Extinct 21 Kilbourne and Makovicky, 2010 Psittacosaurus lujiatunensis Vertebrate Extinct 5 Witzmann and Pfretzschner, 2003 Micromelerpeton credneri Vertebrate Extinct 32 Tsujino et al., 2003 Baculites tanakae Invertebrate Extinct 35 Tsujino et al., 2004 Baculites tanakae Invertebrate Extinct 23 Wei., 1994 Globoconella inflata Invertebrate Extinct 36 Wei., 1994 Globoconella subconomiozea Invertebrate Extinct 33 Wei., 1994 Globoconella puncticulata Invertebrate Extinct 30 Gould, 1966 Poecilozonites nelsoni Invertebrate Extinct 22

199

Referance Genus and Species Group Extinct? N Gould, 1966 Poecilozonites bermudensis Invertebrate Extinct 33 Gould, 1966 Poecilozonites cupula Invertebrate Extinct 27 Gould, 1966 Poecilozonites dalli Invertebrate Extinct 15 Brower, 1973 Glycymeris parilis Invertebrate Extinct 22 Scott, 1973 Globorotalia menardii Invertebrate Extinct 18 Scott, 1973 Globorotalia menardii Invertebrate Extinct 25 Scott, 1973 Globorotalia menardii Invertebrate Extinct 26 O'Brien and Caron, 2012 Syphusauctum gergarium Invertebrate Extinct 210 Brower, 2006 Euptychocrinus skopaios Invertebrate Extinct 11 Brower, 2006 Cincinnaticrinus varibrachialus Invertebrate Extinct 34 Brower, 2006 Ectenocrinus simplex Invertebrate Extinct 40 Brower, 2006 Cupulocrinus levorsoni Invertebrate Extinct 11 Brower, 2006 Praecupulocrinus conjugans Invertebrate Extinct 10 Brower, 2006 Eoparisocrinus crossmani Invertebrate Extinct 12 Brower, 2006 Cupulocrinus crossmani Invertebrate Extinct 19 Brower, 2006 Paracremacrinus laticardinalis Invertebrate Extinct 8 Brower, 2006 Cremacrinus ramifer Invertebrate Extinct 12 Brower, 2006 Euptychocrinus skopaios Invertebrate Extinct 11 Brower, 2006 Cincinnaticrinus varibrachialus Invertebrate Extinct 34 Brower, 2006 Ectenocrinus simplex Invertebrate Extinct 27 Brower, 2006 Ectenocrinus simplex Invertebrate Extinct 13 Brower, 2006 Cupulocrinus levorsoni Invertebrate Extinct 11 Brower, 2006 Praecupulocrinus conjugans Invertebrate Extinct 10 Brower, 2006 Eoparisocrinus crossmani Invertebrate Extinct 12 Brower, 2006 Cupulocrinus crossmani Invertebrate Extinct 19 Brower, 2006 Paracremacrinus laticardinalis Invertebrate Extinct 8 Brower, 2006 Cremacrinus ramifer Invertebrate Extinct 12 Brower, 2006 Cincinnaticrinus varibrachialus Invertebrate Extinct 34 Brower, 2006 Paracremacrinus laticardinalis Invertebrate Extinct 8 Brower, 2006 Cremacrinus ramifer Invertebrate Extinct 12 McKinney, 1984 Oligopygus phelani Invertebrate Extinct 226 McKinney, 1984 Oligopygus haldemani Invertebrate Extinct 733 McKinney, 1984 Oligopygus wetherbyi Invertebrate Extinct 131 Jones and Gould, 1999 Gryphaea dilatata Invertebrate Extinct 17 Jones and Gould, 1999 Gryphaea dilatata Invertebrate Extinct 24 Jones and Gould, 1999 Gryphaea dilatata Invertebrate Extinct 25 Jones and Gould, 1999 Gryphaea dilatata Invertebrate Extinct 13 Jones and Gould, 1999 Gryphaea dilatata Invertebrate Extinct 21 Jones and Gould, 1999 Gryphaea lituola Invertebrate Extinct 28 Jones and Gould, 1999 Gryphaea lituola Invertebrate Extinct 26 Jones and Gould, 1999 Gryphaea lituola Invertebrate Extinct 11 Jones and Gould, 1999 Gryphaea lituola Invertebrate Extinct 20 Jones and Gould, 1999 Gryphaea dilobotes Invertebrate Extinct 26 Jones and Gould, 1999 Gryphaea dilobotes Invertebrate Extinct 19 Jones and Gould, 1999 Gryphaea gigantea Invertebrate Extinct 16 Jones and Gould, 1999 Gryphaea gigantea Invertebrate Extinct 3 Jones and Gould, 1999 Gryphaea gigantea Invertebrate Extinct 10 Jones and Gould, 1999 Gryphaea mccullochi Invertebrate Extinct 19

200

Referance Genus and Species Group Extinct? N Jones and Gould, 1999 Gryphaea mccullochi Invertebrate Extinct 13 Jones and Gould, 1999 Gryphaea mccullochi Invertebrate Extinct 4 Jones and Gould, 1999 Gryphaea mccullochi Invertebrate Extinct 15 Jones and Gould, 1999 Gryphaea mccullochi Invertebrate Extinct 27 Jones and Gould, 1999 Gryphaea mccullochi Invertebrate Extinct 20 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 28 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 21 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 25 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 26 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 26 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 29 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 30 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 26 Jones and Gould, 1999 Gryphaea arcuata Invertebrate Extinct 28 Spight, 1973 Thais lamellosa Invertebrate Extinct 19 Spight, 1973 Thais lamellosa Invertebrate Extinct 17 Spight, 1973 Thais lamellosa Invertebrate Extinct 8 Spight, 1973 Thais lamellosa Invertebrate Extinct 9 Spight, 1973 Thais lamellosa Invertebrate Extinct 41 Spight, 1973 Thais lamellosa Invertebrate Extinct 27 Spight, 1973 Thais lamellosa Invertebrate Extinct 10 Spight, 1973 Thais lamellosa Invertebrate Extinct 22 Spight, 1973 Thais lamellosa Invertebrate Extinct 10 Spight, 1973 Thais lamellosa Invertebrate Extinct 13 Spight, 1973 Thais lamellosa Invertebrate Extinct 9 Spight, 1973 Thais lamellosa Invertebrate Extinct 20 Spight, 1973 Thais lamellosa Invertebrate Extinct 11 Spight, 1973 Thais lamellosa Invertebrate Extinct 7 Spight, 1973 Thais lamellosa Invertebrate Extinct 13 Spight, 1973 Thais lamellosa Invertebrate Extinct 10 Spight, 1973 Thais lamellosa Invertebrate Extinct 10 Spight, 1973 Thais lamellosa Invertebrate Extinct 60 Spight, 1973 Thais lamellosa Invertebrate Extinct 11 Spight, 1973 Thais lamellosa Invertebrate Extinct 10 Spight, 1973 Thais lamellosa Invertebrate Extinct 13 Spight, 1973 Thais lamellosa Invertebrate Extinct 17 Spight, 1973 Thais lamellosa Invertebrate Extinct 18 Spight, 1973 Thais canaliculata Invertebrate Extinct 8 Spight, 1973 Thais canaliculata Invertebrate Extinct 15 Spight, 1973 Thais canaliculata Invertebrate Extinct 10 Spight, 1973 Thais emaginata Invertebrate Extinct 10 Spight, 1973 Thais emaginata Invertebrate Extinct 15 Spight, 1973 Thais emaginata Invertebrate Extinct 11 Spight, 1973 Thais emaginata Invertebrate Extinct 12 Spight, 1973 Thais lamellosa Invertebrate Extinct 264 Brower, 1990 Eohalysiorinus typus Invertebrate Extinct 32 Chatterton et al., 1994 Cryptolithus tesselatus Invertebrate Extinct 130 Chatterton et al., 1994 Cryptolithus tesselatus Invertebrate Extinct 54

201

Referance Genus and Species Group Extinct? N Brower, 1987 Comannthus bennetti Invertebrate Extinct 24 Brower, 1987 Halysiocrinus tunicatus Invertebrate Extinct 16 Brower, 1987 Halysiocrinus dactylus Invertebrate Extinct 17 Brower, 1987 Halysiocrinus laticardinalis Invertebrate Extinct 12 Brower, 1987 Cremacrinus ramifer Invertebrate Extinct 12 Brower and Veinus, 1975 Hybocrinus punctatus Invertebrate Extinct 33 Webster, 2007 Nephrolenellus multinodus Invertebrate Extinct 254 Webster, 2007 Nephrolenellus geniculatus Invertebrate Extinct 367 Brower, 1988 Cremacrinus ramifer Invertebrate Extinct 12 Brower, 1988 Cremacrinus ulrichi Invertebrate Extinct 6 Brower, 1988 Paracremacrinus laticardinalis Invertebrate Extinct 6 Brower, 1988 Cremacrinus articulosis Invertebrate Extinct 5 Brower, 1988 Cremacrinus punctatus Invertebrate Extinct 4 Waters et al., 1985 Pentremites pyriformis Invertebrate Extinct 40 Waters et al., 1985 Pentremites conoideus Invertebrate Extinct 94 Waters et al., 1985 Pentremites robustus Invertebrate Extinct 87 Hopskins and Webster, 2009 Zacanthopis palmeri Invertebrate Extinct 184

202

Appendix B – Dataset of 108 specimens of Alligator mississippiensis, as well as their cranial measurements (23) used in this study.

203

Spec. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 UCMP no data 8 10 12 17 14 29 11 10 10 8 2 3 4 13 3 8 13 4 5 2 3 6 4 ROM R 7966 9 13 15 18 16 35 14 10 11 9 3 4 4 19 6 9 15 5 8 3 4 8 4 ROM R 7964 9 12 13 18 15 35 13 10 10 8 2 3 3 17 5 8 13 5 8 3 4 7 4 UCMP 142041 10 13 16 20 15 37 15 10 10 9 2 3 4 21 5 9 14 5 7 3 4 7 3 UCMP 142042 10 15 17 21 15 38 16 11 11 10 2 3 4 20 5 10 14 6 8 3 3 9 4 ROM R 7965 10 15 17 20 17 40 17 11 12 9 4 4 4 22 6 10 16 6 8 4 4 8 5 ROM R 6253 11 16 20 22 17 44 19 12 13 10 3 4 4 23 5 12 17 7 7 3 4 9 4 ROM R 6251 11 16 20 23 16 44 21 13 13 11 3 4 5 23 6 11 16 7 9 3 4 10 4 ROM R 6252 11 16 19 23 17 45 20 12 13 11 3 4 4 23 6 11 16 7 9 3 4 9 4 YPM 908 12 17 20 25 17 45 20 13 12 10 3 4 4 25 6 10 17 7 8 3 5 11 8 AMNH 46843 12 15 19 24 16 46 20 14 13 12 3 4 5 35 9 12 14 7 10 4 6 20 8 AMNH 67134 18 24 29 31 21 61 29 16 16 14 4 5 6 33 8 13 22 10 12 5 7 14 5 UCMVZ 222424 17 22 25 30 22 63 30 17 16 13 4 6 6 33 6 14 20 10 13 5 7 15 6 ROM R 0001 18 25 28 33 23 67 34 18 18 14 4 6 6 37 6 17 22 10 13 5 7 15 5 CMNAR 35755 19 25 29 34 25 70 34 20 18 14 5 7 6 37 8 18 24 11 15 5 8 17 6 dissected spec. 20 26 30 36 22 70 34 18 18 14 4 5 6 48 11 18 26 11 15 6 10 25 9 UCMVZ 95933 19 27 34 38 28 84 44 23 19 15 5 7 7 43 7 22 29 13 18 7 9 18 5 CMNAR 35753 22 30 35 45 33 85 42 23 22 16 6 7 7 46 8 22 27 14 19 7 9 18 7 ROM R 0008 25 35 41 47 32 97 51 26 25 16 7 8 9 53 11 23 32 16 18 7 10 20 8 ROM R 936 26 37 44 51 36 102 55 29 24 17 7 9 9 56 10 26 33 17 23 8 14 22 7 ROM R 8352 28 38 45 51 35 107 59 30 27 19 7 10 10 60 12 29 38 18 25 9 12 23 7 ROM R 8349 29 39 45 53 37 111 58 30 27 21 8 9 11 63 11 29 36 18 23 10 13 22 9 ROM R 8350 28 41 48 52 35 113 59 31 27 20 8 10 10 65 12 30 39 18 27 10 13 25 7 ROM R 7852 27 41 45 54 40 115 63 31 25 20 7 10 10 64 11 27 38 18 25 8 14 24 7 UCMVZ 65763 30 43 49 57 39 116 65 33 26 19 10 10 11 66 11 30 40 19 24 9 14 24 8 ROM no data 32 47 54 65 43 120 66 32 27 23 9 12 13 67 12 30 41 21 26 12 16 28 8 CMNAR 35968 32 45 57 77 54 128 64 42 28 23 10 18 16 67 16 36 50 24 20 10 19 36 10 ROM R 8039 39 53 63 69 48 140 74 47 28 23 7 12 12 74 13 32 45 25 32 12 16 34 9 ROM R 7853 36 52 61 74 50 146 80 40 32 26 9 14 14 77 12 38 50 23 31 12 17 32 9 ROM R 8355 40 55 64 73 49 150 82 42 33 23 10 14 14 86 16 41 52 24 34 14 20 32 9 ROM R 8354 37 50 58 64 43 151 76 38 34 24 9 11 11 81 14 32 44 22 28 11 17 29 10 785-1133 31 50 60 81 60 157 83 47 27 26 10 18 18 81 16 34 54 28 25 12 21 36 15 YPM 573 45 58 69 74 48 158 84 41 31 26 12 12 14 88 16 32 53 25 35 15 19 32 11 ROM R 8345 45 64 75 92 63 166 94 49 38 29 13 12 17 97 18 48 62 30 36 15 20 37 11 UCMVZ 129985 41 61 72 87 63 167 93 48 35 25 11 16 13 93 14 45 59 28 36 15 20 35 11 UCMP131080 43 65 76 92 64 172 93 52 35 27 13 17 15 94 15 47 60 31 39 17 24 35 11 AMNH 106269 40 58 71 88 57 172 96 47 36 29 10 18 16 91 17 36 53 28 41 15 23 43 11 ROM Herp 46540 43 62 71 82 57 182 101 50 39 26 14 17 16 97 13 41 53 28 42 14 22 34 10 UCMVZ 200608 47 71 85 95 67 183 102 55 35 30 12 16 15 101 16 50 65 34 43 18 26 40 11 ROM R 4405 54 68 75 91 62 188 106 55 37 26 12 19 16 101 15 44 55 28 39 15 24 39 12 UCMP 131688 50 68 80 97 68 189 100 58 40 28 14 17 16 102 19 49 68 34 43 19 27 43 14 ROM 4414 48 65 76 92 66 199 106 56 39 32 14 24 18 106 18 51 60 31 44 16 23 40 12 ROM R 8335 54 74 86 102 70 204 121 70 37 28 17 18 19 118 17 60 72 35 49 20 26 44 13 ROM R 1698 52 70 81 93 62 205 115 52 44 27 15 16 16 115 17 48 66 33 43 16 23 43 12 AMNH 40581 63 86 106 130 81 218 129 64 40 33 17 18 22 120 22 51 82 39 45 19 31 55 13 USNM 167541 lrg. 65 80 92 107 64 219 123 60 43 30 14 16 19 117 20 48 73 39 48 19 28 47 12 ROM R 5855 59 82 96 110 72 223 122 60 44 30 18 17 19 122 19 59 74 36 43 17 27 46 12 ROM R 8332 60 85 95 116 82 223 137 67 42 30 19 22 20 132 19 63 77 37 55 18 31 51 15 ROM R 4424 56 83 110 136 86 224 121 62 45 29 19 20 20 108 20 66 84 44 46 20 31 51 15 ROM R 4418 56 79 89 107 74 229 128 64 45 28 17 22 18 120 18 58 74 37 49 19 28 46 12 ROM R 4406 67 83 97 118 80 241 137 77 45 32 17 22 18 130 19 65 78 37 52 20 29 51 14 ROM R 8334 68 92 106 128 85 248 153 72 44 33 20 23 22 140 22 72 86 44 58 22 32 54 13 ROM R 8322 69 92 107 132 88 249 148 73 49 33 21 20 25 142 24 72 90 46 61 28 31 57 18 ROM R 4419 63 88 102 118 80 251 145 69 47 33 16 21 21 131 19 67 84 41 58 19 31 54 13 ROM R 4404 62 88 108 120 80 252 142 67 49 32 17 21 24 133 24 62 94 42 52 22 29 56 16 UCMVZ 191313 69 95 105 130 86 256 157 73 48 35 20 22 21 143 22 70 84 42 62 22 34 56 13 ROM R 4410 64 92 111 122 74 257 147 73 48 34 15 26 21 137 22 68 87 43 45 19 30 54 15 UCMVZ 191314 67 93 100 130 86 257 153 72 44 31 18 23 20 138 19 70 83 41 58 20 30 55 14 204

Spec. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ROM R 8347 74 97 115 127 86 257 155 73 50 35 20 18 21 151 24 71 95 46 55 24 34 53 16 ROM R 8323 68 90 109 130 86 261 153 77 56 36 20 23 24 151 27 74 90 44 65 23 35 54 15 ROM R 8346 75 102 115 142 90 267 160 79 53 37 20 23 24 155 22 79 94 47 65 23 36 61 14 ROM R 4420 69 94 106 124 81 267 144 74 51 35 18 23 23 141 22 69 88 45 56 24 30 57 14 UCMZ R 1975.229 71 97 111 142 95 272 161 77 49 36 21 26 23 144 21 77 90 47 62 25 29 56 14 ROM R 8341 74 102 114 138 93 279 178 81 51 32 23 27 26 158 23 83 92 47 64 24 38 56 15 ROM R 8331 78 112 125 149 98 281 168 82 47 40 20 25 27 164 25 86 108 52 66 27 39 60 16 ROM R 8344 74 103 121 149 98 282 173 89 53 35 21 25 21 164 28 85 99 49 67 26 40 62 16 AMNH 66643 89 112 131 172 105 285 177 73 45 32 24 21 26 151 26 76 107 51 61 22 42 69 17 ROM R 4421 77 107 121 148 102 287 156 79 51 32 24 24 23 156 23 82 98 50 58 27 37 59 15 ROM R 4407 81 115 134 154 104 288 170 81 45 38 23 26 26 152 25 76 103 53 52 22 39 60 17 ROM R 4409 80 115 129 158 102 298 180 82 54 40 21 26 27 160 21 81 104 54 61 25 37 69 16 TMP 1990.7.194 85 115 130 168 108 301 174 83 54 37 20 22 25 158 24 89 99 55 64 25 43 59 16 ROM R 4413 95 117 132 155 99 306 178 87 55 42 22 29 27 164 28 84 104 51 62 25 42 62 16 ROM R 8330 82 115 127 150 108 309 192 90 52 37 26 27 27 170 24 87 110 53 77 29 41 67 16 ROM R 388 96 133 160 206 137 315 176 92 65 47 23 31 32 161 32 99 134 72 69 39 41 79 17 AMNH 43316 86 111 133 175 110 315 175 92 51 50 27 24 27 162 29 73 112 53 67 29 44 68 17 ROM herp 46539 88 111 123 157 105 324 188 92 64 35 21 26 23 176 28 88 108 51 73 31 38 67 17 ROM R 4408 115 131 151 183 115 324 190 100 55 42 24 29 29 169 29 95 127 59 64 24 49 69 18 ROM 4412 79 112 131 154 95 325 172 86 56 38 22 28 24 169 24 83 102 52 59 25 39 70 17 AMNH uncat. 88 113 131 154 97 325 178 88 52 40 18 27 30 165 26 71 115 65 73 27 43 55 17 ROM R 690 85 111 126 159 108 331 186 85 64 36 22 27 23 174 29 88 109 52 73 30 39 67 16 ROM R 600 107 142 173 208 143 334 181 89 61 44 21 26 28 151 20 93 135 77 66 34 44 75 19 ROM R 8336 88 120 130 162 120 337 204 100 60 36 29 30 29 183 24 98 108 56 77 32 45 69 16 YPM 580 90 119 137 154 102 338 198 96 53 39 24 30 30 177 25 70 109 54 78 29 41 77 15 ROM R 8342 101 136 152 185 126 344 213 102 68 44 24 30 31 199 30 105 134 68 81 32 52 70 19 ROM R 4416 99 142 159 199 123 349 218 98 58 42 22 28 25 197 32 111 131 66 72 28 48 68 17 ROM R 4417 99 136 150 186 119 352 205 111 59 43 24 33 30 189 27 107 130 69 79 32 44 66 16 759 104 131 154 188 113 365 213 100 58 42 29 30 30 200 30 81 123 64 80 34 48 83 17 ROM R 8343 108 152 172 204 141 367 229 115 66 47 29 26 30 221 34 112 133 71 76 33 52 81 19 ROM R 8327 115 151 174 210 141 368 233 111 62 51 32 38 42 202 32 117 149 73 80 34 52 79 20 ROM R 8326 110 153 179 218 140 371 231 109 65 50 30 36 41 198 27 125 142 74 80 34 52 79 20 ROM R 4422 100 138 156 185 114 373 216 102 59 37 27 34 29 203 30 109 130 68 82 32 46 76 19 AMNH 71621 111 141 161 217 125 385 217 117 57 39 29 33 32 208 32 93 143 72 84 34 66 88 17 AMNH 46842 127 150 175 209 135 388 226 120 59 42 25 34 29 205 28 104 143 81 93 43 59 92 18 uncat. skull 119 133 153 190 117 390 230 111 63 40 30 31 33 220 36 90 131 65 90 35 51 84 20 AMNH 43315 113 154 181 240 160 390 227 118 47 47 29 34 39 216 34 103 153 83 92 40 59 99 20 ROM R 8329 115 162 171 226 146 391 254 115 64 51 31 39 37 218 32 132 148 73 86 36 52 75 19 ROM R 4402 131 149 172 218 133 398 231 115 70 52 26 29 32 214 32 128 152 75 92 34 56 91 19 ROM R 4401 112 162 180 210 139 399 229 115 62 46 30 32 31 206 32 121 145 78 86 35 52 93 18 579 122 148 163 196 115 410 248 114 57 41 28 36 31 230 29 103 134 66 94 31 52 87 15 ROM R 4415 124 163 179 232 155 432 265 122 67 50 26 35 36 228 31 125 155 80 86 35 57 79 19 ROM 4411 135 190 210 259 166 479 275 139 74 49 33 39 33 241 32 160 180 86 101 35 57 89 21 ROM R 494 140 196 212 282 182 497 302 148 71 45 41 39 39 262 34 153 191 95 99 47 66 99 23 AMNH 31563 149 186 212 259 168 550 287 141 72 51 40 38 39 255 35 110 181 93 97 46 68 101 22 ROM R 8324 178 237 249 320 186 563 372 186 86 54 45 49 47 317 48 191 229 126 150 69 86 128 23 YPM uncat. skull 181 225 270 350 225 610 315 180 117 88 40 63 66 295 68 157 255 125 115 47 94 143 37 USNM 06611 195 222 247 326 175 628 365 179 91 51 42 51 41 332 48 160 215 108 135 67 81 133 26 USNM 029573 193 240 257 338 199 632 388 185 85 64 43 56 56 331 37 173 210 113 137 58 88 111 25 ROM 51011 203 274 289 371 243 689 441 195 84 57 51 53 47 371 62 181 236 132 141 64 60 139 32 205

Appendix C – Allometric power plots for the 22 variables, illustrating how the conclusions of the best-fitting allometric trend change with increased sampling (results for random subsampling only). In each case, the x-axis is sample size and the y-axis is the percentage of replicates that best fit that allometric trend. White = isometry, green = positive allometry, red = negative allometry, grey = disagreement in allometric trend between RMA and OLS. Vertical dotted lines indicate the sample size at which 95% of the replicates show the same trend.

206

slope = 1.07 Variable 7 slope = 1.13 Variable 20 100 100 80 80 60 pos at 11 60 pos at 53 pos at 42 40 40 20 20 OLS

OLS RMA RMA 0 0

20 40 60 80 100 20 40 60 80 100

Variable 18 slope = 1.09 Variable 16 slope = 1.05 100 100 80 80 60 60 pos at 15 pos at 48

pos at 64

pos at 18 40 40 20 20 RMA

OLS OLS RMA 0 0

20 40 60 80 100 20 40 60 80 100

Variable 1 slope = 1.05 Variable 2 slope = 1.02 100 100 80 80

pos at 31 pos at 67 60 60

OLS pos at 77 OLS 40 40

RMA

neg at 76 20 20 RMA 0 0

20 40 60 80 100 20 40 60 80 100 207

slope = 1.04 Variable 11 Variable 8 slope = 1.01 100 100

OLS 80 80 RMA 60 60

iso at NA 40 40 pos at 95 OLS

RMA 20 pos at 75 20 0 0

20 40 60 80 100 20 40 60 80 100

Variable 19 slope = 1.03 Variable 17 slope = 0.996 100 1000 OLS

RMA 80 800

pos at 95 60 600 pos at 80 iso at 40

1 iso at 30 40 400

iso at 30

OLS 20 200

RMA OLS

RMA 0 0

20 40 60 80 100 20 40 60 80 100

Variable 4 slope = 1.02 Variable 3 slope = 1.003 RMA 100 100 OLS

OLS 80 80 60 60

RMA 40 40 20 20 0 0

20 40 60 80 100 20 40 60 80 100 208

Variable 14 slope = 0.991 Variable 5 slope = 0.945 100 100

OLS 80 80

RMA 60 60

iso at 65 40 40 neg at 61 neg at 76

neg at 65 neg at 95 20 20

OLS

RMA 0 0

20 40 60 80 100 20 40 60 80 100

Variable 12 slope = 0.969 Variable 13 slope = 0.916 100 100 RMA

OLS

neg at 54 80 80 60 60 neg at 43

iso at 1 40 40 OLS 20 20

RMA 0 0

20 40 60 80 100 20 40 60 80 100

Variable 22 slope = 0.959 Variable 15 slope = 0.784 100 pos at 98 100 RMA OLS 80 80

neg at 18 OLS 60 60 40 40 RMA 20 20

neg at 15 0 0

20 40 60 80 100 20 40 60 80 100 209

Variable 23 slope = 0.651 100 RMA OLS 80

neg at 7 neg at 8 60 40 20 0

20 40 60 80 100

Variable 10 slope = 0.666 OLS 100 RMA 80

60 neg at 6 neg at 7 40 20 0

20 40 60 80 100

Variable 9 slope = 0.737

100 OLS RMA 80 60 40

neg at 6 neg at 7 20 0

20 40 60 80 100 210

Chapter Four

Quantifying Stratigraphic Accuracy and Precision for Dinosaur Quarries in the Upper Belly River Group: A Case Study for Centrosaurinae

Caleb Marshall Brown1, David A. Eberth2 and David C. Evans1,3

1Department of Ecology & Evolutionary Biology, University of Toronto, 25 Willcocks Street,

Toronto, Ontario M5S 3B2, Canada

2Royal Tyrrell Museum of Palaeontology, Drumheller, T0J 0Y0, Canada.

3Department of Natural History, Palaeobiology division, Royal Ontario Museum, 100 Queen's

Park, Toronto, Ontario M5S 2C6, Canada

211

Abstract

The stratigraphic position of fossil specimens or sites within geological formations is of critical importance to understanding the temporal relationships of taxa. Independent issues of stratigraphic accuracy (correlating stratigraphic position across geography, or estimating from projections of a stratigraphic datum), and stratigraphic precision (confidence in measured position due to local lithology) can both affect the relative placement of specimens within a formation. In terrestrial systems that show mixed lithology of channel sands and floodplain deposits, the downcutting effect of large palaeochannel sandstones is of particular concern.

Here the known stratigraphic positions of centrosaurine quarries and bonebeds in the upper Belly River Group (Campanian) are used to quantify and compare the scale of these two sources of stratigraphic error. Utilizing a structural contour map, estimates of stratigraphic position relative to the Oldman/Dinosaur Park formational contact have a mean error of 2.5 m

(mean absolute error = 4.0 m). Adjusting specimens hosted in palaeochannel rhythms to a stratigraphic position equal to the highest preserved height of the rhythm results in mean offsets of 3.7 m, and a maximum of 14.5 m. Given the relatively restricted stratigraphic relief (~ 60 m) of this geologic interval, both error sources will likely significantly affect patterns of stratigraphic position of specimens, and therefore result in changing patterns of evolutionary analyses and faunal change.

212

Introduction

Palaeobiologists are often interested in the stratigraphic position of specimens within geological formations or members. These data are important for answering both ecological questions, such as faunal responses to climatic change, and evolutionary questions, such as evolutionary rates and modes within lineages.

Some of the best examples of palaeobiological research incorporating specimen-level stratigraphic data are those investigating fine-scale changes in morphology within lineages to elucidate modes and rates of evolution (Hallam, 1968; Gingerich, 1974, 1976; Raup, 1977; Raup and Crick, 1981; Williamson, 1981; Sorhannus et al., 1988; Sorhannus et al., 1991; Gingerich,

1993; Gingerich and Gunnell, 1995; Gingerich, 2003; Hunt et al., 2008). From a vertebrate perspective, preeminent amongst these is the pioneering work of Gingerich on the stratophenetics of mammal teeth within terrestrial deposits (eg., Willwood Fm.) in the Early

Eocene. This research has spurred much debate on both evolutionary theory, and methodologies and limitations of measuring evolutionary rates and testing evolutionary modes (Gould, 1984;

Gingerich, 1985; Gingerich, 1993). Building upon this work, researchers have further refined the testing of evolutionary modes utilizing model fitting approaches, and testing against a null model or treating different models as equal (Bookstein, 1987, 1988; Roopnarine et al., 1999;

Roopnarine, 2001; Roopnarine, 2005; Hunt, 2006; Hunt, 2007, 2008).

Although the methodologies for model fitting have been greatly refined, one of the greatest limitations to evolutionary work incorporating morphology and stratigraphic position lies in the stratigraphic error or uncertainty in the rocks themselves.

213

Sources of Stratigraphic Error in Alluvial-Paralic Systems

The two largest sources of stratigraphic error are firstly the error of estimating and/or correlating stratigraphic position across large geographic areas, and secondly the uncertainty arising from local host unit thicknesses. These issues are relevant for any terrestrial system, with the former becoming magnified when specimens from different sites are correlated across greater geographic distances, or as bedding deviates from horizontal, and the latter becoming magnified in fluvial-deltaic sandstone-dominated systems, and systems with alternating sand-dominated and mud-dominated lithologies. The presence of fluvial channels indicates that intraformational erosion resulting in the downcutting of existing underlying strata will limit the resolution of stratigraphic correlation within the formation (Eberth and Getty, 2005). Incised (down-cut) basal surfaces of palaeochannel rhythms indicate that the fossils contained in the rhythm were deposited in stratigraphically lower positions compared to adjacent, isochronous floodplain surfaces. Because rates of sedimentation are, by definition, greater in palaeochannels than on the adjacent floodplain, variations in stratigraphic position exhibited by fossils in palaeochannels are less significant biostratigraphically than those in the corresponding floodplain sections. Indeed, most of the stratigraphic range of a given rhythm equates stratigraphically to considerably less than one meter of section in adjacent floodplain deposits. Accordingly, specimens can be correlated consistently with the floodplain horizons adjacent to the top of a rhythm. In many cases, stratigraphic uncertainty and error are major, but often unquantified, limitations of these analyses. Additionally, time averaging can result in mixing of non-contemporaneous specimens, populations, or taxa within single sites (Behrensmeyer, 1982; Kidwell and Flessa, 1996).

214

In the case of Gingerich (1974), due to the large geographic extent of the field area, nearly horizontal strata, and historic section, stratigraphic positions for the majority of sites

(66.7%) were correlated with the stratigraphic section using altitude. These correlations were determined to the nearest 20 feet (6.1 m). Fortunately, given the large stratigraphic range over which the specimens are spread, the scale (6.1 m) of the inaccuracy of the correlations is likely not significant. This same degree of uncertainty manifested over more restricted stratigraphic intervals, however, may have large and unrealized effects on our understanding of the relative stratigraphic postion of the specimens.

Most evolutionary analyses rarely attempt to incorporate estimates of stratigraphic accuracy or precision, regardless if the scale will significantly affect the analysis (but see

MacLeod, 1991; Hunt, 2008 for exceptions). As a result, we have little understanding of how stratigraphic error affects these analyses. Hunt (2008) tested the effects that stratigraphic error and poor age control have on analyses of evolutionary mode in marine trilobite, radiolarians and foraminifera, and found that even extreme amounts of error have relatively little result on relative model support. It is important to note however, that the simulations of Hunt (2008) did not allow for the order of specimens to be changed due to stratigraphic error, and only the relative spacing of subsequent samples was varied. As some evolutionary models (e.g., stasis) are only concerned with the order of specimens (and not the span between them), stratigraphic error that affects the relative order within the sample may have much greater effect than those that merely change the spacing between samples (Hunt, 2008).

215

The Belly River Group

The upper Belly River Group (Dinosaur Park and Oldman formations) of the Campanian of

Alberta has a long history of collection and research and, as a result, represents one of the best- studied and most well sampled Mesozoic terrestrial assemblages (Dodson, 1971; Beland and

Russell, 1978; Currie, 2005; Currie and Koppelhus, 2005; Brown et al., 2013). In the past century more than 400 articulated skulls or skeletons of dinosaurs have been collected from the vicinity of Dinosaur Provincial Park (DPP) alone (Currie and Russell, 2005). Additionally, due to a foresighted and systematic quarry staking program instituted by C. M. Sternberg (Geological

Survey of Canada) and W. A. Parks (Royal Ontario Museum) in 1935, building upon initial work by L. Sternberg, (Sternberg, 1936; Sternberg, 1950; Tanke, 1994, 2005), and continued to this day (Tanke, 1994; Currie and Russell, 2005), the location (both geographically and stratigraphically) of dinosaur quarries have been recorded. Further, recent work has been conducted to systematically search for or identify both “lost quarries” (specimens without data on the quarry location) and “mystery quarries” (known sites with a lack of specimen data)

(Tanke, 2005). As a result, as of 2005, the majority of the 468 articulated or associated dinosaur specimens are now known from precise geographic and stratigraphic occurrences within DPP

(Currie and Russell, 2005). In addition to the quarry-staking program, a multi-year initiative was started in 1999 to precisely map each quarry utilizing high resolution differential GPS (Pryor et al., 2001; MacDonald et al., 2005). More than 650 quarries, bonebeds, and uncollected skeletons were marked using differential GPS between 1999 and 2003 (MacDonald et al., 2005).

The contact between the Oldman Formation and the overlying Dinosaur Park Formation is a regionally extensive and diachronous disconformity on the scale of southern Alberta (Eberth,

2005). On the scale of the Dinosaur Provincial Park region, however, this contact represents an

216 isochronous datum (Eberth and Getty, 2005). As such, this contact has been used extensively as an isochronous datum for the stratigraphic positions of specimens and quarries in Dinosaur

Provincial Park (Eberth and Getty, 2005, Ryan and Evans, 2005; Ryan et al., 2007; Mallon et al.,

2012). This contact, however, shows both regional structural dip (30 m on the scale of DPP) and topography (Eberth and Getty, 2005). Building on previous work (Dodson, 1971; Eberth 1990) examining regional structure trends, and incorporating 274 marked localities of the Dinosaur

Park/Oldman contact using the differential GPS, allowing for the creation of a structural contour map of the Dinosaur Provincial Park area, with the formation contact as the datum (Eberth, 2005;

Eberth and Getty, 2005).

When combined with altitudinal data for the recorded quarries, positions of each quarry relative to the Oldman/Dinosaur Park formational contact can be estimated, allowing for stratigraphic positions of a large number of quarries to be determined without measuring section for each site (Currie and Russell, 2005). Individual stratigraphic sections for each site represent a more accurate approach, but this requires a great deal of time investment. Measured stratigraphy, overlain with quarry positions, have been drafted based on subsets of the dinosaur sample

(Eberth and Getty, 2005; Ryan et al., 2007), but this has yet to be performed on the complete sample.

As a result of the combined work of generations of researchers, the upper Belly River

Group of Alberta hosts a huge sample of dinosaur skeletons with unparalleled data on their individual localities. This represents the best system in which to test ideas of dinosaur palaeobiology that incorporate aspects of stratigraphy. To date, these studies of dinosaur biostratigraphy in the Belly River Group have largely described the patterns of faunal change through time or as a result of environmental change. Initial reports of stratigraphic separation of

217 taxa within the Dinosaur Park Formation (DPF) (Sternberg, 1950) have been supported with the collection of further specimens (and discovery of lost quarries), refined stratigraphic work, and the use of more refined quantitative techniques (Eberth and Hamblin, 1993; Currie and Russell,

2005; Eberth, 2005; Eberth and Getty, 2005; Ryan and Evans, 2005; Tanke, 2005; Evans, 2007;

Ryan et al., 2012; Mallon et al., 2013; Evans et al., in press). These analyses have indicated distinct stratigraphic segregation of multiple dinosaur taxa within the upper Belly River Group, and relatively high rates of faunal turnover within the formation. These results are mainly found for the large ornithischian dinosaurs, due to their well-resolved taxonomy and large sample size of quarries with known stratigraphic position.

Due to the large datasets involved, rather than measuring the position of each site, some work incorporating the stratigraphic position of specimens has utilized the estimation of stratigraphic position relative to the Oldman/Dinosaur Park formational contact based on the regional structural contour maps (Currie and Russell, 2005; Mallon et al., 2013). As these estimates are derived from the difference between the quarry elevation and the projected elevation of the Oldman/Dinosaur Park formational contact, they have two potential sources or error: the quarry and the structural contour. The differential GPS altitude readings have been shown to be very internally accurate (95% of readings within 10cm, and 98% of readings within

50cm), and with average differences of 2.7m and 3.6m from previous altitude estimates, derived from an aneroid barometer and a combination of air photos and topographic maps respectively

(MacDonald et al., 2005). Despite this accuracy in altitude, the accuracy of the stratigraphic position relative to the formational contact remains unclear.

This paper aims to quantify two distinct, but interrelated, aspects of stratigraphic knowledge for these dinosaur quarries. First, we test the accuracy of currently known estimates

218 of stratigraphic position by contrasting those based on the structural contact against those derived from measurements of individual stratigraphic sections. This will indicate how representative the estimated positions are to the true positions, and will reveal the average deviations and any systematic errors. This quantification will serve as a proxy for the error associated with stratigraphic estimation when the section is not measured. Obviously, this source of error can be

(and often is) eliminated by measuring the relevant stratigraphic section. It is quantified here to illustrate the scale of error if measured sections cannot be performed, and to compare this scale to the overall vertical relief, as well as stratigraphic offset resulting from channel downcutting.

Secondly, and more importantly, we aim to use the individual stratigraphic sections to test the precision of stratigraphic placement given host unit thickness and lithology. The vast majority of the centrosaurine specimens collected, as with those of most dinosaurs within DPP, are hosted by sandstone-dominated deposits that record deposition in ancient river channel thalwegs and point bars (Dodson, 1971; Wood et al., 1988; Eberth, 2005; Eberth and Currie,

2005; Eberth and Getty, 2005). Here we use the term “rhythm” in reference to upward-fining sandstone-dominated units. Many of the palaeochannel deposits in DPP comprise stacked rhythms that are bounded by channel-wide erosive surfaces and lags comprised of intraclasts, ironstone deposits, and organic debris (including vertebrate, invertebrate, and plant fossils)

(Wood, 1989). Rhythms exhibit crude-to-well-developed upward-fining grain sizes suggestive of deposition during the waning stages of a flood event. Stacked rhythms form multistoried deposits and indicate a palaeochannel that experienced multiple flood events and that, over time, may have migrated laterally many tens of meters within a single braid- or meander belt. Individual rhythms comprise a variety of facies including trough cross-bedded sandstone, inclined heterolithic strata, and ripple-laminated sandstone and siltstone. More detailed descriptions and interpretations are available in Wood (1989) and Eberth (2005).

219

Materials and Methods

As part of a study examining within-lineage morphological change through stratigraphy (i.e., stratophenetics) in Centrosaurinae, stratigraphic data were collected for nearly all known centrosaurine quarries and bonebeds in the region of Dinosaur Provincial Park (Chapter 5). There are 59 sites (32 individual specimens and 27 monodominant bonebeds) of centrosaurines known from the area of DPP. Of these sites, 45 (27 individual specimens and 18 monodominant bonebeds) were visited and measured relative to the Oldman/Dinosaur Park formational boundary in the summers of 2011 and 2012 (Appendix A and B). Additionally, the stratigraphy of several of these sites (eg., BB 42, 91, and 138) had previously been documented, and did not require additional measurement. Remaining sites were not visited due to time constraints or extensive distance from exposures of the contact datum.

For each site, the height above the Oldman/Dinosaur Park formational contact was determined by measuring up section from the nearest outcropping contact using a Jacob’s staff equipped with a clinometer. In most cases this outcrop was found within 100 meters of the site, but in a few cases the height was correlated across large distances (eg., 1.5 km for Q 248). In addition to the height above section, thickness and lithologies of the host units and overlaying/overlying units were recorded.

Comparison of the measured stratigraphic positions to those estimated from the GPS data and structural contour map allows for an evaluation of the estimation accuracy of the position based on the GPS data. This represents a proxy for accuracy of GPS estimates of stratigraphic positions, when measured section either have not or cannot be collected. This difference

220 between measured stratigraphic position and that estimated from differential GPS is here termed

‘stratigraphic estimation accuracy’.

In order to establish a reliable and accurate biostratigraphic framework for specimens from the strata of DPP, specimens in palaeochannel rhythms are adjusted to the stratigraphically highest preserved level of each rhythm. Whereas such an approach can reduce stratigraphic position error for specimens preserved in palaeochannels, it cannot eliminate it completely. Rhythms are often incompletely preserved as a result of downcutting by overlying rhythms. Thus, reassigning specimens (hosted by palaeochannels) to the preserved top of their host rhythms indicates only the lowest real stratigraphic occurrence of the specimen for purposes of consistent biostratigraphic comparison utilizing all facies. Where rhythms are completely preserved, reassignment of the specimen to the top of the rhythm may position the specimen slightly higher than it occurred in coeval floodplain deposits, but that error cannot be greater than the difference between the equivalent downcutting and uppermost surfaces in the floodplain section, which are typically considerably less than one metre in the sections we have examined. This approach is deemed particularly beneficial where channel rhythms are thick; the thicker the sandstone rhythm, the larger the potential stratigraphic offset that is possible and the less accurate the biostratigraphic data.

Quantification of thickness of host channel-sandstone rhythms, as well as overlying sandstone rhythms, and reassignment of channel-hosted specimens to the preserved top of their host rhythms will allow for a conservative stratigraphic adjustment of channel-hosted specimens.

The distribution of thicknesses of complete rhythms and those that are downcut by an overlying rhythm are also compared to test if this conservative adjustment is underestimating the stratigraphic height of specimens hosted in multi-rhythm stacked channel sequences. This offset

221 based on channel thickness is here termed ‘stratigraphic precision’. This attempt to adjust for channel thickness does not take into account other stratigraphic issues, such as compaction or mud-filled incised valleys, but it attempts to quantify and adjust for the largest factor affecting stratigraphic position of quarries and specimens.

The relative scale of these two sources of error is compared to determine which of these is a larger factor in affecting the stratigraphic position of specimens, and if one of these sources is negligible compared to the other.

Results

Accuracy of Stratigraphic Position from Differential GPS

The stratigraphic positions estimated from differential GPS data and the structural contour map are highly correlated (r2 = 0.914, p-value << 0.001), with the stratigraphic positions based on the measured sections (Fig, 1). The slope of this relationship is similar to 1.00 (0.9031), though the

95% confidences intervals do not include 1.00. Analysis of the residuals (Fig. 1B) shows them to be normally distributed (Shapiro-Wilk p-value = 0.5549) and not correlated with stratigraphic position. Based on the correlation coefficient, slightly more then 90% of the variance in measured position is explained in the GPS estimates.

The distribution of the estimation error (Fig. 2A) is also normally distributed (Shapiro-

Wilk p-value = 0.5588), with a mean of 2.5 m and a median of 2.9 m (Table 1). Fitting this to a normal distribution, 95% of further observations should be expected to fall between -5 m and 9.6

222 m. When the absolute difference (absolute value) of the estimation error is considered, the mean is 4.0 m and the median is 3.9 m (Table 1, Fig. 2B).

A plot of estimation accuracy (estimated – measured stratigraphic height), as a function of measured stratigraphic height illustrates no significant correlation (positive or negative) of estimation accuracy with stratigraphy (Fig. 3A). As a result, average estimation accuracy is expected to neither increase nor decrease up-section, and estimates high in section should have a similar average error as those low in section. In contrast low (r2 = 0.0867, and 0.1982) but significant (p-value = 0.0400, and 0.0013) correlations are seen between estimation accuracy and geographic position, for both longitude (Fig. 3B) and latitude (Fig. 3C). Although correlations of geography and estimation accuracy are quite low, they can been seen on a regional map where estimates further southwest are bluer (indicating stratigraphic over-estimates), whereas those further northwest are more yellow-red (indicating stratigraphic underestimates).

Host Rhythm Thickness and Stratigraphic Resolution

The average thickness of sandstone rhythms hosting centrosaurine specimens is 3.7 m, with a maximum of 14.5 m (Table 2). Given that the stratigraphic range over which these specimens are positioned is around 60 m, the average channel cut represents ~6.2% of the entire stratigraphic range. The largest host channel thickness (14.5 m) represents nearly one quarter (24.2%) of the entire stratigraphic sequence.

When the thicknesses of completely preserved rhythms versus those that are incomplete due to downcutting by overlying rhythms are compared, they show a similar distribution (Fig. 5B and C), and are not statistically different (Kolmogorov-Smirnov p-value = 0.8668). Although not

223 significantly different, the complete rhythms show a greater range and variance than the incomplete rhythms (Table 2), and with larger sample sizes significant differences may be present.

Unlike the estimation accuracy, the thickness of host rhythms (precision) does not correlate with either latitude (p-value = 0.614) or longitude (p-value = 0.960). Also, unlike estimation accuracy, the thickness increases (decreased precision) up-section (p-value = 0.021) when complete rhythms are examined (Fig. 6). This same trend is not observed when all host rhythms (regardless of completeness) are included.

Comparison of Stratigraphic Precision to Estimation Accuracy

Comparison of the relative magnitudes of estimation accuracy and stratigraphic precision shows a great deal of overlap. When the raw (negative and positive) estimation errors are used, the distributions are statistically different due to the higher variance and negative values

(Kolmogorov-Smirnov p-value = 0.034), despite the higher degree of overlap (Fig. 7A). When the absolute values are used, the two distributions are statistically indistinguishable from each other (Kolmogorov-Smirnov p-value = 0.303) (Fig. 7B).

Of the 59 sites examined, more than half, 54% (n = 32), show that the situation where the estimates from the GPS fall outside the ranges of stratigraphic uncertainty indicated by the rhythm thickness (Fig. 8A). Another 15% (n = 9) show that the situation where the estimates from the GPS fall within the ranges of stratigraphic uncertainty indicated by the rhythm thickness (Fig. 8B). Finally, 31% (n = 18) of sites lack data on one or both of the two factors and the relative influence cannot be determined.

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Discussion

This study represented a novel program attempting to quantify, on a site-specific scale, the potential stratigraphic errors seen in the Belly River Group. These methods and results likely have application beyond this geologic system, and are likely applicable to any analyses incorporating stratigraphy within an alluvial-paralic system.

Although the patterns of estimated stratigraphic positions are highly correlated (>>0.001) with, and predictive (r2 = 0.914) of, measured positions, average errors (2.5 m or 4 m) are relatively large given the limited stratigraphic range of the sample. These errors represent approximately 4.2% and 6.7% of the entire stratigraphic range of the entire series, and 8–20% of the total stratigraphic ranges of Centrosaurus or Styracosaurus. As a result, the placement of individual species within the series is likely to be affected by this source of error. Fortunately, this source of error is relatively easily eliminated, and only requires that individual specimens be measured from the contact, the isochrnous datum.

The correlation of stratigraphic estimation accuracy and geography (both latitude and longitude) are likely due to errors or oversimplifications in the structural contour map, as these systemic errors are not seen when plotted against stratigraphy. This pattern, and the previously illustrated internal consistency of the GPS altitude readings, suggests that the major source of estimation error is associated with the structural contour map, and not the altitude data for each quarry. This result is unsurprising, as it is expected that values at the extremes of the represented data, or extrapolations beyond the represented data, will show higher error than well-sampled intervals in the middle of the represented range of the data.

225

Given the scale of the stratigraphic range of the entire sample of specimens (60 m), the scale of the average thickness of host sandstone rhythms (3.7 m) is significant (6.2%). This degree of error is actually greater than the default alpha value (0.05) for the statistical comparisons. The largest stratigraphic offset due to channel thickness (14.5 m) represents nearly one quarter (24.2%) of the entire stratigraphic range of the fossil sample. For comparison, this is greater than, or similar to identified palynological and megaherbivorous faunal zones within the park. For example, an error range of 14.5 m represents nearly the entire known stratigraphic range of Styracosaurus albertensis. Stratigraphic error of this magnitude can therefore have huge effects on the placement of individual specimens within the formation and will likely affect large-scale patterns of biostratigraphy.

There is a sea-level transgression recorded in the Dinosaur Park Formation, with the lower sandy zone dominated by alluvial palaeochannels, and the upper muddy zone dominated by overbank deposits grading into the Lethbridge Coal Zone (Eberth, 2006). In this context, the average channel thickness increased up-section (Fig. 6), with larger rhythms on average in the muddy zone than the sandy zone. This could be due to an artifact of increased number of stacked

(multistory) sandstone rhythms, and therefore more incomplete rhythms, in the lower portion of the section. This explanation does not hold, however, as the trend is seen when only the complete rhythms are included, and not observed when all rhythms are included. This trend could potentially be explained by the small sample size (n=27), but Type I error should not be associated with low samples size like Type II error.

When the two different sources of stratigraphic error, estimation accuracy and host rhythm thickness, are considered, and the relative importance of the two sources are compared, three general possibilities exist: If host rhythms are thin, their dimensions consistently smaller

226 than the estimation error – as in Figure 8A, then the uncertainly due to host rhythm thickness can generally be ignored as it will be overwhelmed by the estimation error (Fig. 8D). Conversely, if the host rhythms are large, much larger on average than the estimation accuracy – as is the case in Figure 8B, then the estimation accuracy can generally be ignored, as it will be overwhelmed by the estimation error (Fig. 8E). Finally, the condition may exist where the scale of the estimation error and rhythm thickness are similar – as in Figure 8C, in which case both factors contribute to the overall error, and neither can by systematically ignored. The data from the quarries surveyed here indicate that the third case, similar contribution to error, is likely the situation in the Belly River Group. This is supported by similar relative scale of error due to GPS estimation and channel thickness (Fig. 7). When compared on a site-by-site basis, however, 54% of GPS estimates occur outside the range of the rhythm thickness, while only 15% of GPS estimates occur within the rhythm thickness. This may suggest that while these phenomena act at a similar general scale, error introduced by GPS estimation is slightly greater than that introduced by channel thickness.

Fortunately, the more severe of the two sources of error is also the easier of the two to eliminate. All sources of error in the GPS estimates can be eliminated by measuring sections of each quarry noting their heights above the contact. Although measurement of sandstone rhythms and reassignment of specimens to the top of host rhythms makes initial and conservative attempts to compensate for this source of error, it does not completely eliminate it.

Given that the Oldman and Dinosaur Park formations have limited stratigraphic relief in the area of Dinosaur Provincial Park (~60 m), the proportion of this relief that is equal to the average channel thickness is relatively large (~6.2%) compared to that of other study areas (Fig.

9, Table 3). In classic studies of evolutionary modes in marine invertebrates (Kellogg, 1975;

227

Raup and Crick, 1981; Malmgren et al., 1983), the stratigraphic resolution of individual intervals is often less than 1% of the total stratigraphic range (Fig. 9). Work on lacustrian deposits persevering an adaptive radiation of sticklebacks also shows error on a scale of less than 1%

(Bell, 1988; Bell et al., 2004; Hunt et al., 2008), and in Gingerich’s work on Eocene mammals the error is around 1% (Gingerich, 1974; Gingerich and Gunnell, 1995). By comparison, the relative effect of stratigraphic error in the Belly River Group is more than five times greater than that seen in any of the exemplar samples. This illustrates the importance of the documentation and adjustment of this stratigraphic error, and highlights the potential limitations that this may have for evolutionary analyses in this system.

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Table 1 – Summary statistics of the differences between the GPS stratigraphic estimates and the measured position for both the raw values (A) and the absolute differences (B).

Estimation Error Absolute Est. Error

N 47 47

Mean 2.55 m 3.97 m

Median 2.93 m 3.86 m

Max 11.00 m 11.00 m

Min -6.89 m 0.01 m

St. Dev. 4.12 2.74

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Table 2 – Summary statistics for the thickness of sandstone palaeochannels rhythms for A) all rhythms, B) Complete (non-eroded) rhythms, and B) eroded (incomplete) rhythms.

All Rhythms Complete Rhythms Eroded Rhythms N 44 27 13 Mean 3.69 m 3.69 m 3.69 m Median 3.38 m 3.5 m 2.5 m Min 0.3 m 0.3 m 1.0 m Max 14.5 m 14.5 m 9.0 m St. Dev. 2.99 3.67 2.58

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Table 3 – Summary statistics and stratigraphic data for exemplar studies of evolutionary modes.

Genus Globorotali Eucyrtidium Kosmoceras Gasterosteus Hyopsodus Hyopsodus Centrosaurus Pleistocene to Late Age Miocene to Recent Middle Jurassic Miocene Eocene Eocene Recent Cretaceous Taxon Foraminifera Radiolarian Ammonite Teleost Mammal Mammal Dinosaur Temp Range 15,000 1,000,000 1,000,000 21,500 5,000,000 2,000,000 1,000,000 (years) Strat range 140 140 14 7 515 750 60 (m) Accuricy (m) .1-.3 0.2 0.1 0.02 6.1 10 4 Precision (m) 3.7 Sites (n) 105 235 400 44 45 Speciems (n) 3,150-7875 3,035 5000 4,000 1094 Error (%) 0.1428 0.1428 0.7143 0.2857 1.1845 1.3333 6.6667 Malmgren et al., Raup and Crick, Gingerich, Gingerich and Reference Kellog, 1995 Hunt, 2008 Thesis 1983 1981 1974 Gunnell, 1995

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Figure 1 – A) Correlation of estimated stratigraphic position based on differential GPS and contour map with stratigraphic position measured with Jacobs Staff. B) Residuals of A (but with vertical and horizontal axes switched) plotted against measured stratigraphy showing no correlation and a normal distribution.

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Figure 2 – Histograms showing the distribution of estimation error (GPS estimate minus measured height) for all quarries with both positive and negative error (A), and the absolute value error (B).

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Figure 3 – Correlation of stratigraphic estimation accuracy with stratigraphic position (A),

Longitude Easting (B), and Latitude - Northing (C). Estimation accuracy is not correlated with stratigraphic position, but is significantly, but poorly, correlated with geography.

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Figure 4 – Graphical representation of stratigraphic estimation accuracy with geography. Green sites have high accuracy, whereas blue to black are overestimates and yellow to red are underestimates of stratigraphic position. Circles represent monodominant bonebeds and triangles represent individual specimens.

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Figure 5 – Histograms illustrating distribution of host sandstone rhythm thicknesses for all rhythms (A), complete rhythms – those without an erosive upper contact (B), and incomplete rhythms – those with an erosive upper contact (C).

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Figure 6 – Correlation between stratigraphic position of a quarry and the thickness of the host rhythm for all rhythms and only complete (non-erosive) rhythms. No correlation is present when all rhythms are included, but thickness appears to increase up-section when only complete rhythms are included.

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Figure 7 – Comparison of the magnitudes (in metres) of the stratigraphic precision (uncertainty due to rhythm thickness) and stratigraphic accuracy (estimation error) of the sample of sites. The raw accuracy (A) shows slight, but significant (Kolmogorov-Smirnov p-value = 0.034), differences in the distribution – but with high overlap, while the absolute value for the accuracy

(B) show no statistical differences (Kolmogorov-Smirnov p-value = 0.303). In both cases, the solid lines indicated the density distribution, while the dashed lines represent the density distribution reflected in the x-axis for comparison.

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Figure 8 – Hypothetical stratigraphic sections illustrating the relative effect of accuracy of stratigraphic estimation and stratigraphic precision. A) Estimation accuracy is low and precision is high. B) Estimation accuracy is high and precision is low. C) Estimation accuracy and stratigraphic precision are similar. Consistent occurrence of A will result in condition D.

Consistent occurrence of B will result in condition E. Consistent occurrence of C will result in condition F.

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Figure 9 – Exemplars of evolutionary studies and the scale of the realized or assumed stratigraphic resolution relative to the entire sample.

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Appendix A – Dataset of stratigraphic and geographic data for Centrosaurinae quarries and bonebeds in the area of Dinosaur Provincial Park. For each site, the species, elevation, calculated stratigraphic height (from structural contour map), measured stratigraphic height, upper and lower bounds of the host rhythm, whether the upper contact is downcut or eroded, formation, and

UTM coordinates are listed.

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Quarry Inst. Spec. Genus Species Elev. Calc. Ms. BB138 TMP Centrosaurus brinkmani 648.73 -11.271 -14.2 Q022 ROM 43214 indet. indet. 651.34 -8.66 Q127 TMP 1980.23.3 indet. indet. 645.71 -8.292 Q083 CMN 347 Centrosaurus apertus 659.99 -0.01 0 BB91D indet. indet. 679.67 0 BB062 TMP Centrosaurus apertus 659.62 -0.385 0.5 BB162 TMP Centrosaurus apertus 654.24 -1.76 1 BB178 Centrosaurus apertus 658.18 2.182 1 BB168 TMP Centrosaurus apertus 668.63 6.63 1 Q148 TMP 1980.54.1 Centrosaurus apertus 667.43 7.43 1 Q091 CMN 8795 Centrosaurus apertus 662.68 2.684 1.5 Q217 TMP 1994.164.1 Centrosaurus apertus 669.58 7.584 1.75 Q081 CMN 12229 Centrosaurus apertus 664.53 4.53 2 Q135 UALVP 11735 Centrosaurus apertus 665.27 5.272 2 BB180 TMP indet. indet. 669.88 9.88 2 BB041A Centrosaurus apertus 660.84 0.84 2.6 BB041 TMP Centrosaurus apertus 660.91 0.91 3 BB091 TMP Centrosaurus apertus 672.90 8.9 4 BB166 TMP 79.11 Centrosaurus apertus 654.39 4.388 3 Q001 ROM 767 Centrosaurus apertus 671.45 9.446 4.5 Q044 ROM 1426 Centrosaurus apertus 672.32 10.319 5 BB061A TMP Centrosaurus apertus 664.41 2.405 5.2 Q181 TMP 1982.16.11 Centrosaurus apertus 665.83 5.832 5.5 Q057 AMNH 5239 Centrosaurus apertus 652.38 4.375 6 BB030 TMP Centrosaurus apertus 671.25 9.246 6 Q224 TMP 1997.85.1 Centrosaurus apertus 654.65 6.654 6.5 BB061 TMP Centrosaurus apertus 673.87 11.874 7 BB152 TMP Centrosaurus apertus 675.05 13.054 8 BB091A TMP Centrosaurus apertus 679.67 13.67 8 Q231 ROM 1427 Centrosaurus apertus 674.18 14.178 8 BB128 TMP Centrosaurus apertus 667.99 9.991 10 Q204 TMP 1992.82.1 Centrosaurus apertus 659.81 11.814 10 BB091C TMP Centrosaurus apertus 681.03 19.03 10 BB091B TMP Centrosaurus apertus 681.31 15.31 10 BB206 Centrosaurus apertus 21 10 BB043 TMP Centrosaurus apertus 663.76 3.76 10.5 Q102 CMN 8798 Centrosaurus apertus 676.66 14.659 10.5 Q249 YPM 2015 Centrosaurus apertus 11 BB165 TMP Centrosaurus apertus 679.07 19.07 13 BB188 TMP Centrosaurus apertus 668.12 12.119 13.5 Q042 USNM 8897 Centrosaurus apertus 690.11 26.11 22 Q109 CMNH 11374 Centrosaurus apertus 684.62 24.624 22.75 BB156 TMP indet. Spike 669.76 21.76 23 259

Quarry Inst. Spec. Genus Species Elev. Calc. Ms. Q136 UALVP 16248 Centrosaurus apertus 668.11 18.107 25 Q124 BM R8648 Centrosaurus apertus 669.76 21.758 23 Q078 BM R4859 Centrosaurus apertus 669.14 21.142 25 Q033 La Plata 79/XI/23/2 Centrosaurus apertus 675.42 27.418 Q105 AMNH 5351 Centrosaurus apertus 688.75 28.747 26.7 Q070 NMC 8896 693.92 29.918 BB042 TMP Styracosaurus albertensis 698.90 38.902 29 Q183 TMP 1987.52.1 Styracosaurus albertensis 38 32.5 Q248 TMP 2005.12.58 Styracosaurus albertensis 700.16 38.164 37 TMP 2003.12.168 Styracosaurus albertensis 41 Q179 TMP 1986.126.1 Styracosaurus albertensis 700.33 42.333 Q016 CMN 344 Styracosaurus albertensis 708.85 48.85 43.5 BB130 TMP indet. indet. 704.90 42.896 44.25 Q240 TMP 2002.76.1 Pachyrhinosaurus sp. 714.56 56.56 48 TMP 2009.80.01 Styracosaurus albertensis 48 BB167 Styracosaurus albertensis 707.58 49.577 260

Quarry Lr. Upr Err. Diff. EU? Th. Fm. UTM E. UTM N. BB138 -14.2 -12.7 1.5 2.929 no 1.5 OMF 463030 5621890 Q022 DPF 469731 5623254 Q127 OMF Q083 0 7 7 -0.01 yes 7 DPF 470807 5622370 BB91D 0 2 2 yes 2 DPF 462960 5620170 BB062 0.5 3.25 2.75 -0.885 no 3.25 DPF 466880 5622850 BB162 2.76 DPF 462780 5624620 BB178 -1.182 DPF 463350 5624290 BB168 -5.63 DPF 463550 5621230 Q148 1 2 1 6.43 no 1 DPF 471179 5622351 Q091 1.5 3.5 2 1.184 no 2 DPF 463597 5622148 Q217 1.75 5.5 3.75 5.834 no 3.75 DPF 462721 5621121 Q081 2 4 2 2.53 ? 1.2 DPF 471952 5623021 Q135 2 11.25 9.25 3.272 yes 7 DPF 465430 5622230 BB180 2 2.25 0.25 7.88 no 0.3 DPF 471130 5622270 BB041A 2.6 2.9 0.3 -1.76 no 0.3 DPF 469290 5623610 BB041 -2.09 DPF 470000 5623290 BB091 4 4 0 4.9 no 0.3 DPF 462790 5620620 BB166 1.388 DPF 458010 5630720 Q001 4.5 10.5 6 4.946 no 6 DPF 466997 5622503 Q044 5 9.5 4.5 5.319 ? 4.5 DPF 467170 5622376 BB061A 5.2 14.2 9 -2.795 ? 9 DPF 466750 5622580 Q181 5.5 6.75 1.25 0.332 no 1.25 DPF 464439 5622416 Q057 6 8 2 -1.625 no 4.5 DPF 457638 5630455 BB030 6 6.3 0.3 3.246 no 0.3 DPF 464260 5621660 Q224 6.5 6.5 0 0.154 no 0.3 DPF 458014 5630713 BB061 7 9 2 4.874 ? 2 DPF 466580 5622530 BB152 8 8 0 5.054 no 1.5 DPF 463400 5621380 BB091A 8 11.5 3.5 5.67 no 3.5 DPF 462960 5620170 Q231 8 8 0 6.178 NA 0 DPF 463348 5621391 BB128 -0.009 DPF 465480 5622750 Q204 10 17.5 7.5 1.814 no 7.5 DPF 458065 5630880 BB091C 10 11 1 9.03 no 5 DPF 463280 5620140 BB091B 10 11 1 5.31 no 2.5 DPF 463020 5620000 BB206 11 DPF 464260 5621660 BB043 10.5 15 4.5 -6.74 no 4.5 DPF 465860 5622570 Q102 10.5 17.5 7 4.159 no 14.5 DPF 463828 5621808 Q249 11 11 0 no 2 DPF 463867 5621507 BB165 6.07 DPF 471490 5622280 BB188 13.5 17.5 4 -1.381 no 4 DPF 465420 5623390 Q042 22 25 3 4.11 yes 2 DPF 464148 5620621 Q109 22.75 23.75 1 1.874 yes 1 DPF 460028 5621847 BB156 23 26.5 3.5 -1.24 yes 3.5 DPF 459450 5627150 261

Quarry Lr. Upr Err. Diff. EU? Th. Fm. UTM E. UTM N. Q136 25 27 2 -6.893 yes 2 DPF 459171 5629534 Q124 23 26.5 3.5 -1.242 yes 3.5 DPF 459453 5627146 Q078 25 30.25 5.25 -3.858 no 5.75 DPF 457811 5630255 Q033 DPF 456775 5627009 Q105 26.7 28.7 2 2.047 yes 2 DPF 460125 5621687 Q070 DPF 463915 5620666 BB042 29 30 1 9.902 yes 1 DPF 465170 5620970 Q183 32.5 40.5 8 5.5 yes 8 DPF 459304 5622636 Q248 37 41.75 4.75 1.164 no 3.5 DPF 460563 5619572 41 43.5 2.5 yes 2.5 DPF 475367 5625614 Q179 DPF 477273 5629801 Q016 43.5 47.5 4 5.35 yes 4.5 DPF 472168 5622233 BB130 44.25 50.25 6 -1.354 no 6 DPF 461460 5620680 Q240 48 58 10 8.56 no 10 DPF 473291 5623474 48 52.4 4.5 no 4.5 DPF 461295 5617724 BB167 DPF 474700 5626640 262

Appendix B – Stratigraphic sections for 18 of the surveyed quarries and bonebeds. Thicknesses of host sandstone rhythms, and potential stratigraphic offset of the specimen are indicated with brackets and vertical bars.

263

BB 91 A 91 BB

BB 91 BB

BB 91 B 91 BB

BB 91 D 91 BB BB 91 C 91 BB 5 6 7 8 9 1 2 3 4 0 15 11 19 10 16 12 13 14 17 18 C Abandoned Abandoned Fill Channel M F Si Cl 5 6 7 8 9 1 2 3 4 0 BB 91B 11 10 12 13 C Abandoned Abandoned Fill Channel M F Si Cl 5 6 7 8 9 1 2 3 4 0 11 10 12 13 BB 91 C 0 m 8 m C 11.5 m 11.5 M 91 D 91 A BB 91 Series F Si Cl BB 91 A & D A BB 91 5 6 7 8 9 1 2 3 4 0 11 10 12 13 C M F Si Cl BB 91 5 6 1 2 3 4 0 Mudstone Sandstone Massive Sandstone Graded Sandstone Crossbedded Inclined Heterlithic (IHS) Sandstone

Root Traces Ripple Lamellae Meters Above DPF/OMF Contact DPF/OMF Above Meters Q 102 - CMN 8798 19 19

18 18 17.5 m 17 17

16 16

15 15

14 7 m Q 249 Q 231 14 Q 102

13 13 13 13

12 12 12 12

11 11 11 m 11 11

10.5 m Q 231 10 10 10 10 Meters Above DPF/OMF Contact Meters 9 9 Fe 9 9

8 8 8 8 m 8 Q 231 7 7 7 7 Q 91 - CMN 8765 6 6 6 6

5 5 5 5 5

4 4 4 4 4 3.5 m 3 3 3 3 3 2 m

2 2 2 2 2 Q 91 1.5 m 1 1 1 1 1

0 0 0 0 0

Cl Si F M C Cl Si F M C Cl Si F M C Cl Si F M C 264 Q 231 BB 152 Q 217 Q 181

13 13 13 13

12 12 12 12

11 11 11 11

10 10 10 10

9 9 9 9

8 8 8 8 8 m 7 7 7 7

6 6 6 6 1.25 m 5.5 m 5.5 m 5 5 5 5

4 4 4 4 3.75 m 3 3 3 3

2 2 2 2 1.75 m 1 1 1 1 Meters Above DPF/OMF Contact Meters 0 0 0 0 Cl Si F M C Cl Si F M C Cl Si F M C Cl Si F M C 265 BB 188 Q 204 19 19

18 Fe 18 17.5 m 17.5 m 17 17 Planar 16 16 4 m Planar 15 15 IBS BB 135 14 14 7.5 m 13.5 m 13 13 13 Meters Above DPF/OMF Contact Meters 12 12 12

11 11 11

10 10 10 10 m

9 9 9 Fe 8 8 8

7 7 7

6 6 6

5 5 5

4 4 4

3 3 3

2 2.0 m 2 2

1 1 1

0 0 0 Cl Si F M C Cl Si F M C Cl Si F M C 266 Q 42 38 BB 156 Q 248 Q 124 37 30 50 28.5 m 36 29 below 49 lignite lignite 35 28 48 34 27 47 12 m below 33 26 46 lignite 32 25 45 31 24 44 30 23 43 Q 43 29 22 42 22 m 28 21 41 27 40.5 m 20 40 26 19 39 3.5 m 25 18 38 24 17 37 37 m 23 16 36 22 15 35 21 14 34 20 13

Cl Si F M C Cl Si F M C Cl Si F M C 267 268

Chapter Five

Morphological Variation and Evolutionary Trends in Centrosaurine

Ceratopsids from the Belly River Group (Campanian) of Alberta

Caleb M. Brown1

1Department of Ecology & Evolutionary Biology, University of Toronto, 25 Willcocks Street,

Toronto, Ontario M5S 3B2, Canada

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Abstract

The roles of anagenesis and cladogenesis in the diversification of dinosaurs, particularly the horned ceratopsid dinosaurs, are hotly debated. Due to their large sample sizes, well-documented stratigraphic positions, highly diagnostic ornamentation, and unique taphonomic mode of monodominant bonebeds (representing populations), centrosaurine dinosaurs from the Belly

River Group (Campanian) of Alberta make an ideal model system for testing the predictions of these two divergent evolutionary modes. A dataset of complete skulls, as well as isolated nasal, postorbital, and parietal elements from bonebeds are quantified for their completeness, variation, asymmetry and morphospace occupation. Principal axes describing taxonomically informative morphology were then correlated with stratigraphy to test for evolutionary trends and fit to models of morphologic evolution. Stratigraphic uncertainty resulting from downcutting of channel sands was corrected for, and simulations of this uncertainty were developed to test its effect on stratigraphic overlap of taxa and evolutionary model fitting.

The ornamentation (nasal horncore, postorbital horncore, and parietal) has significantly higher levels of both variation and asymmetry than the rest of the skull, suggesting they are both diagnostic to species, and functioned in sexual display. Height of the nasal horncore is positively allometric, and the shape changes from equilateral to recurved to procurved through ontogeny and fusion. The height of the postorbital horncore is isometric (or possibly negatively allometric) due to development of resorption pits following fusion of the peripheral elements. The diagnostic adult morphology of Centosaurus apertus and Styracosaurus albertensis is achieved through allometry of the parietal spikes. PCA of the nasal, postorbital, and parietal morphology illustrates low taxonomically informative morphology in the nasal and postorbital, and high taxonomically informative morphology in the parietal. Stratigraphic overlap of centrosaurine taxa depends on a

270 single specimen/site, even when the current method of adjusting for channel thickness is taken into account. Regression of PC scores for all elements suggests low correlation of morphology and stratigraphy within species ranges. Evolutionary time series analyses suggest random walk and stasis as the best-supported model for evolution of all variables, with no support for generalized random walk (trend). Hypotheses of anagenesis within centrosaurine lineages are not supported, and results are consistent with either cladogenesis or ecological replacement as the mechanism of taxonomic turnover in the formation.

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Introduction

Evolutionary Modes

Speciation and the resulting increase in biodiversity are some of the most basic and enduring aspects of evolutionary biology. The study of speciation events, and their mechanism, mode, and timing, occupies much of the research of evolutionary biologists (White, 1968; Bush, 1975;

Grant, 1981; Coyne and Orr, 2004). The majority of the research investigating speciation events concentrates on extant taxa, as the mechanisms thought to be responsible for causing speciation have underpinnings in behaviour, ecology, or, perhaps most importantly, genetics, data that are generally unavailable for extinct taxa.

Despite this, some of the strongest evidence for biological evolution and speciation is found in the fossil record (Simpson, 1944; Gingerich, 1987; Benton and Pearson, 2001; Gould,

2002). Although limited to observation as opposed to manipulative experiments, the study of fossils reveals a record of the history of life on a geological time scale in which speciation events can be readily viewed, a unique quality unavailable with living taxa. In this context much of the work of palaeobiologists has been the documentation of empirical evolutionary patterns revealed in the fossil record (Simpson, 1944; Hallam, 1968; Eldredge and Gould, 1972; Raup and Crick,

1981; Williamson, 1981; Gingerich, 1985; Sorhannus et al., 1988; Sepkoski and Ruse, 2008).

This research has revealed multiple examples of speciation, and a great diversity of patterns associated with these events. These data served as the basis for much theory and debate regarding the potential mechanisms and patterns of speciation, and had wide ranging implications to future studies, not only in palaeobiology, but evolutionary biology in general

(Eldredge and Gould, 1972; Gingerich, 1976; Gingerich, 1985; Hallam, 1998; Gould, 2002;

272

Sepkoski and Ruse, 2009). Perhaps most prominent amongst these debates is the relative ubiquity of the evolutionary modes of phyletic gradualism vs. punctuated equilibrium, and the related question of anagenesis vs. cladogenesis (Eldredge and Gould, 1972; Hallam, 1998). The paramount distinction between these viewpoints is whether evolutionary changes are concentrated at speciation events, or are widespread across non-splitting histories of lineages.

Fossil lineages with good stratigraphic sampling therefore offer exciting possibilities of testing these evolutionary ideas.

Despite these theoretical advances, and due partly to a lack of defined models, the same empirical records may be interpreted differently by different scientists, and may reflect the mindset of the scientist more than the empirical data (see Hunt, 2007). More recently several authors have worked on developing formalized models and methodologies for objectively quantifying intraspecific morphological change in biological lineages through constrained periods of geologic time (Raup, 1977; Raup and Crick, 1981; Bookstein, 1987, 1988; Foote and

Raup, 1995; Roopnarine et al., 1999; Roopnarine, 2001; Roopnarine, 2005; Hunt, 2006;

Valentine et al., 2006; Hunt, 2007, 2008; Hunt et al., 2008). These methods often only seek to explain evolutionary trends when the observed pattern cannot be reasonably explained by a null model of an unbiased random walk (Raup, 1977; Raup and Crick, 1981; Bookstein, 1987, 1988), or more recently, treat multiple explicit evolutionary models as equal, and compare their likelihood using model fitting (Gingerich, 1993; Roopnarine, 2001; Roopnarine, 2005; Hunt,

2006; Hunt, 2007, 2008). These most recent methods provide a robust and objective manner of evaluating patterns of morphological change within a lineage through time.

To date, and due to the available fossil record, these analyses have largely been concentrated on marine invertebrate taxa (Roopnarine, 2001; Hunt, 2007, 2008), with few

273 investigations on terrestrial vertebrates. This distinction between the utility of the invertebrate and vertebrate fossil record for evolutionary analysis is likely due to multiple factors, including smaller sample sizes for vertebrate fossil datasets (see Chapter 3) and decreased stratigraphic resolution in many terrestrial environments. One notable series of studies on terrestrial vertebrates is the work of Gingerich (Gingerich, 1974, 1976; Gingerich, 1985; Gingerich, 1993;

Gingerich and Gunnell, 1995), who quantified evolutionary rates and patterns of mammals in the

Eocene. Though relying mainly on tooth measurements and ratios, and criticized methodologically (Gould, 1984), these analyses have revealed much about the rates and patterns of mammalian evolution, and stand as one of the few examples for terrestrial vertebrates.

Evolutionary Modes in Dinosaurs

The relative roles of anagenic and cladogenic processes in dinosaur diversity are hotly debated

(Horner et al., 1992; Sampson, 1995; Scannella and Fowler, 2009; Evans, 2010; Sampson and

Loewen, 2010; Scannella, 2010; Padian and Horner, 2011b, a), with most of these debates concentrating on the ornithischian (specifically ceratopsid) fossil record. Horner and colleagues

(1992) suggested that the dinosaur record from the Two Medicine Formation (Campanian) of

Montana provided evidence for anagenesis as the dominant evolutionary mode at this time. This was based on putative transitional/intermediate forms of ceratopsids, lambeosaurids, pachycephalosaurids and tyrannosaurids, with the most robust support provided by the ceratopsids. Within ceratopsids, an anagenic model of evolution from Styracosaurus to

Pachyrhinosaurus via 3 intermediate forms (A-C) was put forward (Fig. 1A). The intermediate or transitional status of these forms was supported by both transitional morphology (an apparent lack of autapomorphies, and stepwise acquisition of derived states) and intermediate stratigraphic

274 position (specimens in specific stratigraphic order without overlap). A lack of evidence for cladogenesis was used to suggest there was nothing to rule out anagenesis as the evolutionary mechanism (Horner et al., 1992). Sampson (1995, 1997), however, indicated that due to small sample size, the centrosaurine fossil record in the Two Medicine Formation was equivocal with regard to anagenesis and preferred a cladogenic pattern, with distinct lineages represented (Fig.

1B).

These different evolutionary modes make testable predictions regarding the dataset used in support of each claim, as well as other datasets. In spite of the level of debate and testable predictions, this analysis marks the first quantitative analyses of intraspecific morphological change through stratigraphy in dinosaur lineage. Furthermore, the possibility exists to contrast evolutionary patterns seen in structures hypothesized to function in sexual display/species recognition (and therefore may be under sexual selection) with those that are hypothesized to be under natural selection alone.

Belly River Group Centrosaurines as an Evolutionary Model System

The Belly River Group (BRG), specifically the Dinosaur Park Formation (DPF), represents one of the best-understood and most well-sampled Mesozoic terrestrial assemblages (Dodson, 1971;

Beland and Russell, 1978; Currie, 2005; Currie and Koppelhus, 2005; Brown et al., 2013). Not only are several taxa represented by large samples of specimens, but also a foresighted quarry- staking program has recorded the exact geographic and stratigraphic position of the majority of these specimens (Sternberg, 1936; Sternberg, 1950; Tanke, 1994; Currie and Russell, 2005;

275

Tanke, 2005). Within the Dinosaur Park sample, the centrosaurine fossil record is an excellent model clade for testing hypotheses of dinosaur evolution (Fig. 2).

Centrosaurines (particularly Centrosaurus apertus) are represented by large sample sizes, surpassed only by the duck-billed hadrosaurs (Currie and Russell, 2005) in number of articulated specimens. In more then a century of collection, more than 40 articulated or associated skeletons or skulls have been collected (Currie and Russell, 2005). In addition to isolated skeletons, and more importantly, centrosaurines are preserved in large, often heavily sampled, monodominant bonebeds, representing many contemporaneous or penecontemporaneous individuals (Currie and

Dodson, 1984; Ryan, 1992; Ryan et al., 2001; Ryan, 2003; Eberth and Getty, 2005). Within the

Park, this appears to be a unique taphonomic mode of centrosaurines, as most other dinosaur taxa are mainly known from isolated specimens (Dodson, 1971; Currie and Russell, 2005). These bonebeds are most often regarded as mass death assemblages of populations (Currie and Dodson,

1984; Ryan et al., 2001), and as such can preserve population level variation in morphological traits. Most importantly, these bonebeds allow for the collection of large, element-specific datasets at the population level. They may also allow for independent testing of evolutionary models between different taphonomic modes.

In addition to their manner of preservation, centrosaurine dinosaurs also possess a unique morphology that increases their usefulness in evolutionary studies. They possess large and dramatic ornamentation structures, some of the largest cranial ornamentations ever to evolve

(Lull, 1933; Sampson, 1997; Dodson et al., 2004), that likely functioned in sexual selection or species recognition (Sampson, 1995, 1997; Sampson and Loewen, 2010; Padian and Horner,

2011b). Not only may this allow for contrasting evolutionary patterns and rates between features under sexual selection versus those under natural selection, but our current knowledge of the

276 taxonomy of Dinosaur Park Formation centrosaurines suggest that the only morphological features diagnosing species are located in the parietal, and possibly the nasal and postorbital

(Sampson, 1995). Additionally, these elements are also taphonomically robust, and often abundant in the monodominant bonebeds, allowing for easy targeting of taxonomically informative elements.

Finally, as with the other large ornithischians, centrosaurines experience a well- documented faunal turnover within their highly sampled stratigraphic horizon, associated with a transgression (Ryan and Evans, 2005; Ryan et al., 2007; Ryan et al., 2012b; Mallon et al., 2013).

In the Dinosaur Provincial Park region, Coronosaurus brinkmani occurs in the Oldman

Formation, Centrosaurus apertus occurs in the lower ~27 m of the Dinosaur Park Formation

(DPF), Styracosaurus albertensis occurs between 28 m and 46 m of the DPF, and a single specimen of an indeterminate Pachyrhinosaurus-like taxon occurs in a large channel fill 21 m below the Dinosaur Park-Bearpaw contact, and pertains to the Lethbridge Coal Zone or higher.

Spinops sternbergorum also occurs at an uncertain stratigraphic position within the upper

Oldman or lower Dinosaur Park formations (Farke et al., 2011).

Hypotheses

Centrosaurine dinosaurs from the Dinosaur Park Formation not only provide an unparalleled dataset for testing hypotheses regarding evolutionary modes in dinosaurs, but also hypotheses regarding stratigraphic occurrence, as well as morphological variation and allometry. The hypotheses tested here are subdivided into three general categories: morphological (those using

277 morphometric, and not stratigraphic data), stratigraphic (those using only stratigraphic, not morphometric data), and evolutionary (those utilizing both morphometric and stratigraphic data).

Morphology

A large intraspecific and interspecific morphological dataset for both individual specimens and bonebeds allows for testing hypotheses regarding morphological variation, asymmetry, allometry, and morphospace occupation in Centrosaurinae. Higher amounts of both variation and asymmetry in the cranial ornamentation, interpreted as secondary sexual display characters relative to the rest of the skull is predicted (Geist, 1966; Dodson, 1975; Dodson, 1976;

Packer, 1983; Dodson, 1990). Higher interspecific variation, but not necessarily intraspecific variation or asymmetry, is a necessary prediction for regions with species-diagnostic morphologies (i.e., parietal). The height or length of ornamentation structures is also predicted to be positively allometric with respect to skull length or basal width/length of structure.

Variation

H0-Variation. Regions of the skull with cranial ornamentation (nasal, postorbital, parietal) will show similar amounts of morphological variation as those regions without cranial ornamentation.

HA-Variation. Regions of the skull with cranial ornamentation (nasal, postorbital, parietal) will show higher amounts of morphological variation than those regions without cranial ornamentation.

278

Asymmetry

H0-Asymmetry. Regions of the skull with cranial ornamentation (nasal, postorbital, parietal) will show similar amounts of asymmetry as those regions without cranial ornamentation.

HA-Asymmetry. Regions of the skull with cranial ornamentation (nasal, postorbital, parietal) will show higher amounts of asymmetry than those regions without cranial ornamentation.

Allometry

H0-Allometry. Heights of nasal and postorbital horncores, and heights parietal ornamentation, will be isometric with respect to their relative lengths/widths.

HA-Allometry. Heights of nasal and postorbital horncores, and heights parietal ornamentation, will be positively allometric with respect to their relative lengths/widths.

Multivariate Morphometrics

H0-Multivariate. Principal component analysis of morphometric variables will not result in the segregation of different taxa in morphospace.

HA-Multivariate. Principal component analysis of morphometric variables will result in the segregation of different taxa in morphospace.

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Stratigraphy

The non-random distribution of centrosaurine taxa (as well as other ornithischian taxa) has long been noted (Sternberg, 1950), and recently quantified (Currie and Russell, 2005; Eberth and Getty, 2005; Evans, 2007; Ryan et al., 2012b; Mallon et al., 2013). What has not been tested, however, is the effect that stratigraphic uncertainty has on our understanding of the position of centrosaurine specimens (and species) within the Belly River Group. A direct ancestor- descendant anagenetic relationship from Centrosaurus apertus to Styracosaurus albertensis predicts not only a non-random distribution of specimens, but also a lack of overlap in the stratigraphic or temporal range of each species. Faunal turnover due to cladogenesis or ecological shifts do not, however, make predictions regarding potential stratigraphic overlap between taxa. Previous work has suggested the stratigraphic overlap between the Belly River centrosaurine species is either non-existent, or is based on a single specimen (Ryan et al., 2007).

Here we test both rank correlations between taxonomic identity and stratigraphic position, as well as taxonomic overlap at the specimen level given stratigraphic uncertainty.

Stratigraphic Correlation

H0-Stratigraphic correlation. The taxonomic identity of specimens will not be correlated with the stratigraphic position.

HA-Stratigraphic correlation. The taxonomic identity of specimens will be correlated with the stratigraphic position.

280

Stratigraphic Overlap

H0-Stratigraphic overlap. The stratigraphic range of species will overlap.

HA-Stratigraphic overlap. The stratigraphic range of species will not overlap.

Evolution

The taxonomic turnover from Coronosaurus brinkmani to Centrosaurus apertus to

Styracosaurus albertensis can be explained by either an environmental or evolutionary hypothesis. An environmental explanation does not require speciation or extinction events to be preserved in the sample, but rather immigration and emigration of the respective taxa.

Evolutionary explanations of the taxonomic turnover make the prediction that the evolutionary events linking the taxa are preserved in the sampled interval. These events could be explained alternately by gradual within-lineage change (anagenesis), or a punctuated and branching change

(cladogenesis). The opposing evolutionary modes of anagenesis and cladogenesis make different predictions regarding the patterns of morphological change through stratigraphy. Anagenesis predicts morphological change within a species, correlated with stratigraphy, such that later occurring specimens of the ancestor should be more similar morphologically to the descendant, and earlier occurring specimens of the descendant should be more similar to the ancestor (Fig.

3B & D). Cladogenesis predicts lack of direct morphological change (stasis) within species through stratigraphy, with those latest occurring being as similar to the whole sample as those occurring earliest, and distinct change in morphology between the last specimen of the earlier

281 species and the fist specimen of the later species (Fig. 3A & C). Morphological change within

(and between) species is tested here firstly by correlation of morphology and stratigraphy, and secondly by a time-series analysis of morphology and stratigraphic position, both incorporating uncertainty.

Morphological Correlation

H0-Morphological Correlation. Morphological traits (or PC scores based on traits) within a species will not be correlated with stratigraphy.

HA-Morphological Correlation. Morphological traits (or PC scores based on traits) within a species will be correlated with stratigraphy.

Time-Series Analysis

This test is not a paired null and alternate hypothesis, rather support for three models

(Trend “Generalized Random Walk”, Random Walk “Unbiased Random Walk”, and Stasis) are compared using Akaike Information Criterion, log likelihoods, and Akaike weight.

Institutional Abbreviations

AMNH, American Museum of Natural History, New York, New York, USA; CMN, Canadian

Museum of Nature, Ottawa, Canada; FMNH, Field Museum of Natural History, Chicago,

282

Illinois, USA; NHM, Natural History Museum, London, UK; ROM, Royal Ontario Museum,

Toronto, Ontario, Canada; TMP, Royal Tyrrell Museum of Palaeontology, Drumheller, Alberta,

Canada; UALVP, University of Alberta Laboratory of Vertebrate Paleontology, Edmonton,

Alberta, Canada; USMN; National Museum of Natural History (Smithsonian), Washington;

YPM, Yale Peabody Museum, New Haven, Connecticut, USA.

Materials

Nearly all significant centrosaurine specimens from the Oldman and Dinosaur Park formations were examined and measured. The morphometric analyses were derived from two different centrosaurine datasets, one based on partial or complete skulls from isolated specimens and the other based on elements derived from monodominant centrosaurine bonebeds. For some analyses

(e.g. completeness, variance, allometry) these two datasets were analyzed separately. For other analyses (e.g. time-series analysis) these datasets were pooled.

Measurements greater then 150 mm were taken using a fiberglass measuring tape, and measurements less than 150 mm were taken using digital calipers. All measurements were taken to the nearest millimeters. For bilaterally symmetrically elements without distortion, measurements were taken from both sides to either evaluated asymmetry or to average the values. When one side was taphonomically distorted, measurements were only taken from the non-distorted side. Estimates, and incomplete measurements, are those potentially distorted, indicated in the dataset with a “*”, “#”, and “$” respectively.

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Individual Skull Dataset

The dataset based on individual skulls consisted of measurements and coding of all known centrosaurine skulls derived from the Belly River Group, with the exception of the partial skull now housed at the La Plata Museum in Argentina. This consisted of 45 specimens, which break down into 35 Centrosaurus apertus (either confidently or tentatively assigned to C. apertus), six

Styracosaurus albertensis, three Spinops sternbergorum, as well as one Monoclonius lowei

(Appendix 1). These skulls nearly exclusively consisted of large, adult or nearly adult animals, and do not contain small-bodied specimens. For each skull 104 measurements and seven counts/codings were taken when possible (Fig. 4). Of these, 76 measurements and six of counts/codings pertained to paired elements, so the total number of measurements and counts/codings was 181 and 13 (for a total of 194). The majority of these measurements were derived from Ryan (1992) (modified from Dodson, 1990), with additional measurements added to underrepresented areas (Table 1). Not all of these measurements were used, but this represented the range of measurements available for analyses.

The vast majority of the skulls were examined and measured in person, however a small number of skulls, (mainly those housed at the AMNH) are not longer available for study. In these cases the measurements of Ryan (1992) were used, or measurements were taken from casts. The final dataset consisted of 3,188 measurements from 45 skulls.

Given the incompleteness of the majority of the skulls, the amount of missing data in the entire dataset is 63% (i.e., 37% completeness), and 55% (i.e., 45% completeness) if paired elements are considered (i.e., one side was preserved, but not the other side). This level of incompleteness is due to the lack of entire regions of many of the skulls. Although this level of incompleteness is representative for the whole dataset, due to the paring down of variables and

284 specimens for each analysis, the amount of missing data in the analyses was much lower. The greatest cause of missing data was due to incompleteness of specimens, but this can also be caused by obscured or limited access, lack of preparation, or method of display.

Bonebed Datasets

In addition to the dataset based on individual articulated skulls, morphometric datasets were also created for specific individual elements derived from monodominant bonebeds, or collected individually, within the Belly River Group. These datasets attempted to survey all possible specimens of these elements, and across all collected bonebeds. The elements surveyed include the nasal, postorbital, and parietal, and were chosen due to their numerical abundance in the bonebeds, and/or potential taxonomic signal. These datasets included specimens of Centrosaurus apertus, Styracosaurus albertensis, and Coronosaurus brinkmani, and sampled a large range of element sizes representing ontogenetic series.

Nasal Dataset.The nasal dataset consists of morphometric data from 10 measurement variables and 84 specimens (Appendix 2). In addition to the measurements, the status of the nasal-nasal midline suture was documented as unfused, partially fused, or fully fused. Cases where unfused paired elements were both collected, but the association has been lost, will result in double sampling of an individual, but this is likely a rare occurrence. Including the data from the individual articulated skulls, the total number of nasals used here is 109. The completeness of the entire dataset is 69.2% (30.8% incompleteness), and is 86.3 % (13.7% incompleteness) if only the nasal horncore is included. Taxonomically, the dataset consists of 43 Centrosaurus apertus,

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12 Styracosaurus albertensis, 12 Coronosaurus brinkmani, one Monoclonius lowei, one Spinops sternbergorum, and 38 nasals of uncertain affinity.

Postorbital Dataset. The postorbital dataset consists of morphometric data from 10 measurements of 166 specimens (Appendix 3). In addition to the measurements, the status of the element’s fusion to the peripheral elements (palpebral, frontal) was recorded as unfused, partially fused, or fully fused. The presence or absence of a resorption pit was also recorded. Cases where both postorbitals from the same individual are present, but without association, will result in double sampling of that individual, but this is likely a rare occurrence. Including the data from the individual articulated skulls (with paired postorbitals averaged), the total number of elements sampled is 196. The completeness of the entire dataset is 76.1% (23.9% incompleteness), and is

94.6 % (3.4% incompleteness) if only the nasal horncore is included. Taxonomically, the dataset consists of 105 Centrosaurus apertus, 9 Styracosaurus albertensis, 14 Coronosaurus brinkmani,

10 from Milk River Ridge, one Monoclonius lowei, one Spinops sternbergorum, and 54 postorbitals of uncertain affinity.

Parietal Dataset. The parietal dataset consists of morphometric data of 46 measurements from

36 specimens, or 78 specimens when the individual specimens are added (Appendix 4). The completeness of the entire dataset is 49.7% (51.2% incompleteness), and is 65.1% (33.9% incompleteness) if only parietal lengths and P1-3 variables are included. Taxonomically, the dataset consist of 46 Centrosaurus apertus, 21 Styracosaurus albertensis, 4 Coronosaurus brinkmani, two Spinops sternbergorum, and one each of Monoclonius lowei, Monoclonius crassus, and Rubeosaurus ovatus.

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Methods

All values were log transformed (base 10) prior to quantitative analysis. For all analyses not investigating asymmetry, and where paired elements were equally well preserved, the mean value of the paired elements was used. For comparison of morphological properties between cranial ornamentation and non-ornamental skull elements, the parietal and parietal ornamentation, postorbital horncore, and nasal horncore were categorized as ‘ornamentation’, while all other regions were categorized as ‘skull’.

Completeness

To understand the patterns of completeness both between skulls and between elements (within skulls), the completeness of each element/region (or skull) was quantified. This was approximated by documenting the representation of each measurement (i.e. how often it could be recorded), and subsequently averaging this completeness across an element (or skull). Values for paired elements were averaged across the skull. Completeness will inform on which elements have the greatest sampling (and therefore greatest statistical power), as well as taphonomic factors resulting in incompleteness within skulls. The completeness analysis was performed on all isolated skulls that contained measurements for more than one skull region.

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Variation

In order to determine which regions of the skull were most and least variable, and to quantify their variability, the standard deviation for each variable across the dataset was recorded. Values for paired elements were averaged across the skull first, so this metric of variation is not affected by asymmetry. These values were then averaged across elements or regions to reveal the large- scale patterns of variation within the skull. This analysis was performed on both the entire dataset of articulated skulls (i.e., all taxa), as well as only skulls of Centrosaurus apertus, to give an idea of where both the intraspecific and interspecific variation is located.

Asymmetry

To understand the patterns of asymmetry within the skulls all paired measurements were quantified using Equation 1.1, and averaged across skull elements or regions. By recording the asymmetry as a percentage, the absolute magnitude of the measurement was standardized. This analysis was performed on the dataset of complete skulls (i.e., all taxa), as well as only skulls of

Centrosaurus apertus (Appendix 1).

Eq. 1.1

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Allometry

As the dataset of articulated skulls is nearly exclusively represented by large, adult or nearly adult specimens, and due to issues of missing data and small sample size (high Type II error), informative allometric analyses could not be performed on this dataset. Allometric analyses could, however, be performed on the element-based datasets derived from monodominant bonebeds (Appendices 2-4), for the nasal, postorbital, and parietal. As is the case of allometric analyses based on bonebed samples (Lehman, 1990), variables could not be compared to an ideal skull or skeletal reference variable, but rather had to be compared based on within-element variables – in this case usually height against basal length or thickness. These analyses were performed both on the entire multitaxic dataset and a subset containing only specimens of C. apertus.

Slope and 95% confidence intervals of the slope were determined for each variable pair using the lm function for Ordinary Least Squares (OLS) regression and the sma function in the smatr package (Warton et al., 2011) for Reduced/Standardized Major Axis (RMA) Regression in the R programming language (R Development Core Team, 2009). Measurements were determined to be allometric if the 95% confidence intervals do not include 1, and isometric is the

95% confidence intervals include 1.

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Multivariate Analysis

In addition to bivariate analysis of growth and allometry, principal component analysis (PCA) was used to visualize and quantify patterns of shape and size in the complete multivariate dataset for the nasal, postorbital and parietal.

As PCA can only function using complete datasets, the missing data within the individual datasets were estimated using the Bayesian PCA estimator (function bpca) in the package pcaMethods (Stacklies et al., 2007). This method has been found to impose the least amount of error and was most consistent of the methods tested (see Chapter 1). Additional research (Arbour and Brown, in review) supports the estimation of missing data-points, as it has less of an effect on the results than the listwise deletion of incomplete specimens. The amount of missing data estimated was 14% (86% complete) for the nasal horncore, 31% (69% complete) for the complete nasals, 3% (97% complete) for the postorbital horncore, 24% (76% complete) for the complete postorbital, 51% (49% complete) for the entire parietal, and 34% (65% complete) for

P1-P3 of the parietal. In each analysis both co-variation and correlation matrices were utilized though generally only the results of the co-variation matrix are presented. Although measures of the loading and scores differ between these analyses, the general patterns of relative position and separation of the specimens/species are consistent.

Stratigraphic Position

The stratigraphic position of the majority of the articulated specimens and bonebed samples (but not isolated specimens) were able to be determined. Caleb Brown and/or Dave Eberth visted the

290 vast majority of these quarries in the summers of 2010-2013, and their stratigraphic position was measured relative to the Oldman/Dinosaur Park Formation contact using a Jacob’s staff. In addition to the height, the lithology and the lower and upper stratigraphic bounds of the host unit were recorded (See Chapter 4). For a small subset that were either unable to be visited, or were too far from the outcrop of the contact, the stratigraphic position based on the structural contour map and differential GPS altitude to the quarry were used (see Chapter 4). Specimens without known stratigraphy, or located outside of the DPP region, were unable to be utilized in the stratigraphic or evolutionary analyses.

Simulation of Stratigraphic Uncertainty

The relatively large thicknesses of many of the host channel sandstone units within the Dinosaur

Park Formation suggests there are often significant amounts of uncertainty in the true stratigraphic height of sandstone hosted specimens, which represent the vast majority of the specimens (see Chapter 4 for full discussion). This high level of uncertainty, combined with the restricted vertical exposure in the study region, means that the chance of the unit thicknesses changing the relative stratigraphic position of the quarries, and therefore having effects on potential evolutionary patterns, is significant. To test the effects of this stratigraphic uncertainty, two different simulation methods were used to either randomize the position of the specimens within known upper and lower bounds, or add a specified amount of error to measured positions.

In each case, simulations randomizing the position of the specimens within the given parameters were performed with 1000 replicates and the results compared to analyses with no stratigraphic error term. When binning of samples was performed for time series analysis, the simulation of

291 stratigraphic uncertainty was conducted first, followed by binning, and then the time series analyses. This allowed for stratigraphic uncertainty to have an effect on bin membership.

Four different methods of stratigraphic position were used and are discussed below:

Lower. The stratigraphic position of each specimen was regarded as the measured (or estimated) position, with no attempts to adjust or compensate for local host unit lithology or thickness (Fig.

5A). This is by far the simplest method and does not require any information on the lithology of the quarry. This is also the least accurate method as specimens in channel sands are likely located too low due to downcutting of underlying sediments (Chapter 4).

Upper. For sandstone hosted specimens, the positions of the specimens were changed to be equal to the stratigraphically highest preserved level of each host rhythm, regardless of downcutting of the host rhythm (Fig. 5B) (See Chapter 4 for discussion). This marks a simple method of attempting to adjust the stratigraphic height of channel hosted specimens by taking into account the minimum vertical relief seen in their host channel sand.

Method 1 - Normal Distribution Around a Mean. To simulate a consistent amount of stratigraphic error across all quarries, a function was developed to produce a normal probability distribution using the known position (“Lower”) as the mean and a set amount of variance (Fig.

5C). The error input by this method is not site specific and will input the same amount of uncertainty for every site. The degree of variance can be derived from quantification of the amount of error seen across the measured sites, and can also manipulated to test the effect of increasing amount of uncertainty. This same method could be performed using the “Upper” position as the mean, but this was not explored here.

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Method 2 – Uniform Distribution Within Upper and Lower Bounds. This method takes into account the measured host unit thicknesses of each site and creates a random uniform distribution of possible positions between these upper and lower bounds (Fig. 5D). In this simulation, the lower bound was the position of the specimen, and the upper bound was the height of the top of the host unit (regardless of whether the host unit was downcut by an overlying unit or not). This allows for the position of the specimens to be randomized within the stratigraphic extent of the host unit, which will be different for each site.

Two different methods were used to test for correlations and patterns of morphological change through stratigraphy.

Correlation of Morphology and Stratigraphy

Pearson’s product moment correlation was utilized to test for simple correlations between morphological features and stratigraphic occurrence (e.g., Fig. 3). The morphological variable used could be either individual morphological variables (e.g., nasal height, P1 length), or eigenscores from PCA. In this analysis the majority of tests utilized PCA eigenscores, as they summarized multiple variables and often provided the best taxonomic metric. Each specimen for which both the morphological feature could be determined, and the stratigraphic position was known, was plotted and correlation coefficients determined. This was done both across the entire taxonomic and stratigraphic range (including: Coronosaurus, Centrosaurus, and Styracosaurus; or just Centrosaurus and Styracosaurus), as well as only for specimens of Centrosaurus apertus.

The multitaxic approach will indicate those features that change consistently through the stratigraphic range and faunal turnover. The intraspecific approach will test if these same

293 characters change within the stratigraphic range of a particular species (Centrosaurus apertus).

Analyses were performed both without a stratigraphic error term, and incorporating both methods of stratigraphic uncertainty.

Time Series Analysis of Evolutionary Modes

Correlations of morphology and stratigraphy do not necessarily indicate driven evolutionary trends, as these can also be produced by random walks (Hunt, 2007, 2008). Beyond simple correlations, a time series analysis approach using the functions as.paleoTS and fit3models in the package paleo-TS was also undertaken to compare the relative fit of evolutionary models (e.g., random walk, stasis, trend) to the morphologic and stratigraphic data (Hunt, 2011). This function compares the means and variances of time successive populations to those produced under explicit models of stasis, generalized random walk (GRW - i.e., directional trend), and unbiased random walk (URW - i.e., true random walk). For a complete discussion of the methodology of the paleo-TS package see Hunt (2008; 2008). Analyses were performed both without a stratigraphic error term, and incorporating both methods of stratigraphic uncertainty. In each case the relative support for each evolutionary mode are compared using Akaike Weights. This metric derives from the Akaike Information Criterion (AIC), which allows for comparison of goodness- of-fit of a model (log likelihood) while taking into account model complexity.

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Results

Morphological Variation and Asymmetry

General patterns of completeness, variation, and asymmetry in centrosaurine (and Centrosaurus apertus) skulls are illustrated in Figure 6 and shown in Table 2.

Completeness of Articulated Skulls. Measurements of the articulated skulls allow for an analysis of the pattern of completeness in the skull of centrosaurines. The skulls range in completeness from 4% to 91%, with a mean of 36% and median of 28%. This range is 6% to

95% with a mean of 45% and a median of 37%, when paired elements are considered. When partitioned between skeletal elements/regions there are clear patterns in completeness (Figs. 6A

& 7, Table 2). More massive or robust elements and those closer to the midline (e.g., nasal, postorbital, parietal midline bar) are more likely to be preserved than those elements that are thinner, more peripheral, or not sutured to the skull (e.g., maxilla, jugal, dentary). There is also a significant difference in the distribution of the completenesses between ornamental structures

(nasal horncore, postorbital horncore, and parietal) and the rest of the skull, with ornamental structures being represented more consistently than other elements (Kolmogorov-Smirnov test p- value << 0.001, Fig. 7).

Variation within Centrosaurus Crania. Comparing measurements between skulls allows for documentation of the level of variance between cranial elements or regions. Averaging the variance between lefts and rights of paired elements allows examination of which elements or regions are most highly variable between skulls, and is not affected by any patterns of asymmetry in the skull. When specimens from all taxa are analyzed, the elements/regions with the highest variation between skulls are the nasal horncore, frontal fontanelle, postorbital

295 horncore and most prominently, the parietal horns, with low variation in the remaining skull and lower jaw (Figs. 6B, 8, Table 2). Within the parietal, the ornamentation closest to the midline

(P1-P4) shows the highest variation. The amount of variance for the ornamental elements (nasal horncore, postorbital horncore, and parietal horns) is significantly higher than that seen in the rest of the skull (Kolmogorov-Smirnov test p-value << 0.001, Fig. 6B, 8). Across multiple species, the null hypothesis (Ho-Variation) of equal variation between ornamental and non- ornamental is rejected, as ornamental structures consistently show significantly more variation.

When only specimens of Centrosaurus apertus were considered the pattern is similar, but most of the variation in the parietal is largely restricted to P1 and to a lesser extent P2 (Figs. 6C,

9, Table 2). Again, the variances between the ornamental structures and the rest of the skull and lower jaw show distinctly different distributions, with ornamentation being more variable

(Kolmogorov-Smirnov test p-value << 0.001, Fig. 6C, 9, 10A). For Centrosaurus apertus, as with the result across multiple species, the null hypothesis (Ho-Variation) of equal variation between ornamental and non-ornamental is rejected, due to ornamental structures showing consistently more variation.

The pattern of variation within the skulls of Centrosaurus apertus can be further broken down by comparing rates of variation between different skull elements. Figure 10A shows the distribution of variances in the nasal horncore, postorbital, parietal, and remaining skull as well as the lower jaw in C. apertus, when the paired elements are not averaged. The skull and lower jaw show low variation (means of 0.076 and 0.082 standard deviations). The nasal, postorbital, and parietal show increasing variation (means of 0.119, 0.134, and 0.178 standard deviations, respectively). Interesting, within the parietal, the epimarginal P1 has the highest variance (mean of 0.243), followed by P2 (0.180), then P3-7 (ranging from 0.163-0.137), then the midline

296 parietal. All parietal epimarginal elements are more variable than either the postorbital and nasal, but non-ornament measurements (mainly the midline bar and fenestrae) show less variation than the nasal or postorbital and a similar amount of variation as the rest of the face (Fig. 10A).

Figure 10B illustrates the distribution of variation in the ornamentation when broken down between variables. Variables that account for some aspect of height or amplitude have both the highest ranges of variation, and the higher mean variation. Measurements that quantify length/width/thickness or circumference show similar and low variation, while aspects of position are usually the least variable, with a few highly variable exceptions (Fig. 10B).

Asymmetry Within Skulls. When skulls preserve left and right halves that are not clearly distorted, the average asymmetry between various paired elements can be compared (Table 2).

For the entire sample of Belly River Group centrosaurine specimens, the average asymmetry within paired elements can been seen in Figure 11. Asymmetry is low across the majority of the skull, is higher in the postorbital and jugal, and is highest in the parietal, particularly in the epiparietal elements (Fig. 6D, 11). Within the parietal ornamentation, P1 shows less asymmetry than those of the more lateral ossifications. As with the variance, the distribution of asymmetry in the ornamentation is significantly higher than that of the rest of the skull. When only skulls of

Centrosaurus apertus are considered, the pattern of asymmetry is nearly indistinguishable from that for the entire sample (Figs. 6E, 12). Based on these data, the null hypothesis (H0-

Asymmetry) of equal asymmetry between ornamental and non-ornamental regions is rejected both between species, and within C. apertus.

Correlation of Variation with Asymmetry. Due to the treatment of the paired elements, values for the variation (between skulls) and symmetry (within skulls) are mathematically independent, and as a result, correlations between these two factors can be tested. A plot of asymmetry as a

297 function of variation for the measurements from all BRG specimens reveals a strong (r2=0.532) and significant (p-value << 0.001) correlation (Fig. 13A). A similar (r2=0.447) correlation is seen when only specimens of C. apertus are included (Fig. 13B). In both cases, measures of ornamentation are both more variable, and more asymmetric than those of the rest of the skull.

When the measurements are averaged across elements, and the correlation of variation and asymmetry is examined on the scale of cranial elements, the correlation is also strong (r2 =

0.598 and 0.485, p-value << 0.001, Fig. 14). In both the entire sample (Fig. 14A) and the C. apertus sample (Fig. 14B) the ornamentations are the most variable and asymmetric elements.

The nasal shows high variation and low asymmetry, the postorbital is both variable and asymmetric, and the most variable and asymmetric elements are those of the parietal. Within the parietal, the midline and lateral bars (exclusive of the epiossifications) show low asymmetry and variation, relative the epiparietal ossifications. Of the epiparietal ossifications, P1 shows less asymmetry than would be expected given the amount of variation, P2 and P7 show similar amounts of variation and asymmetry, and P3-6 are all more asymmetrical then expected given the amount of variation. Interestingly, of the epiparietals, P1 shows the highest amount of variability between skulls, but the lowest about of asymmetry within skulls (Fig. 14).

Regression and Allometry

Nasal Allometry. Plots of nasal horncore height as a function of anteroposterior basal length of the nasal horncore in Centrosaurus apertus show a slope and 95% confidence intervals greater than 1, indicating positive allometry of the nasal height relative to nasal length (Fig. 15A). The null hypothesis of isometry for the height of the nasal horn (H0-Allometry) is rejected.

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Unsurprisingly, the smallest nasals are unfused, and they become partially fused then fully fused as the nasal becomes larger. Not only does the height increase faster than basal length, but a curved fit line on a plot of anterior curve height as a function of posterior curve height shows a distinct change in shape through size (Fig. 15B). The smallest unfused nasals are initially equilateral, with the anterior curvature increasing faster than the posterior curve, resulting in an increased height and posterior displacement of the tip, and a recurved horn (Fig.

15B). At the approximate size for which fusion of the paired nasals starts, growth of the posterior height outpaces the anterior height, and the horn straightens and then becomes procurved. The degree of curvature in the adult nasal is highly variable, but the general pattern appears to be of increasing procurvature with increasing size. The amount of variation of curvature is greater for fused nasals than unfused nasals, and indicates increased variation with growth/age, but the reason for this variation is unclear.

Postorbital Allometry. A plot of postorbital horncore height as a function of anteroposterior basal length for all specimens illustrates an interesting pattern of postorbital growth (Fig. 16A,

B). Although the sample of very small specimens is limited, the initial phase of growth in height appears to be relatively fast, up to the point that the peripheral elements fuse onto the postorbital

(Fig. 16A). Following fusion of the peripheral elements, the growth in height slows relative to length, and resorption pits begin to appear on the horncores (Fig. 16B). When only specimens of

C. apertus are plotted, horncore height is isometric with respect to horncore length (Fig. 17A), with a best-fit slope less than 1 (0.959). The null hypothesis of isometry for the height of the postorbital horn (H0-Allometry) is not rejected.

When all specimens are included, the regression shows negative allometry of horncore height relative to length, with a slope of 0.759 (Fig. 17B). Specimens of C. brinkmani and from

299 the Milk River Ridge Reservoir show relatively higher postorbitals compared to C. apertus, whereas S. albertensis show relatively lower postorbitals compared to C. apertus.

In both of these cases however, due to heavily biased sampling towards large specimens

(and poor sampling of small specimens), regression of horncore height against horncore basal length may not accurately represent the growth curve. To adjust for poor sampling of small specimens, the spectrum of horncore basal lengths was binned in 0.1 log10 mm intervals, and the means and standard deviations of height plotted (Fig. 18). The results show a rapid increase in height relative to basal length in the first half of the bins, “Phase I”. The second half of the bins show a plateau of height relative to basal length that also corresponded to an increase in the proportion of specimens showing both fusion and resorption pit “Phase II”.

Parietal Allometry. Regression of height/length of each epiparietal ossification against its basal length can been seen in Fig. 19. Best-fit lines indicating the slope and 95% confidence intervals for a linear model (OLS) for C. apertus and S. albertensis are recorded in Table 3. Due to small sample sizes (high Type II error) the null hypothesis of isometry is difficult to reject, with only two of the fourteen regressions (P1 and P2 for C. apertus) being significantly allometric (i.e., slopes and 95% confidence intervals greater than 1.00). The slopes of the best-fit lines for

Centrosaurus and Styracosaurus are also illustrated on each plot (Fig. 19A-G), and slopes summarized in Figure 19H. For P1 and P2, Centrosaurus is more strongly allometric than

Styracosaurus, with the reverse being true for P1-P7 (Fig. 19H). Within Centrosaurus, P1 and P2 are strongly positively allometric with respect to their base, whereas the other parietal ossifications (P3-P7) do not deviate greatly from isometry. Within Styracosaurus, P1 and P2 show the lowest deviation from isometry, P4 and P5 are the most positively allometric (though not statistical due to sample size), and P3, P6, and P7 are reasonably positively allometric.

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Within Centrosaurus the sample size decreases laterally, and in Styracosaurus P3 has the highest sample, with the sample decreasing slightly medially and greatly laterally (Table 3).

Multivariate and Morphospace Analysis

Nasal. Results of the Principal Component Analysis (PCA) of the nasal and nasal horncore are illustrated in Figure 20. The first three axes explain 93.1% of the variation (Table 4), and only these axes are illustrated here (Fig. 20A, B). As is the case with most PCAs based on linear measurements, the first axis is largely associated with general size (Fig. 21), and explains a large proportion for the variance (81.0%). The loadings of axis 2 (8.2%) indicate it is largely an axis defined by height relative to width, with positive scores being relatively long (anteroposteriorly) at the base and dorosventrally short, and negative scores being anteroposteriorly short and dorsoventrally high (Fig. 21). Axis 3 (3.9%) is largely defined by negative loading of variables associated with transverse width relative to the other measurements. In plots of both PC1 vs PC2

(Fig. 20A), and PC2 vs PC3 (Fig. 20B), there is a great deal of overlap in the morphospace occupation of each of the three well-sampled species (Coronosaurus brinkmani, Centrosaurus apertus, and Styracosaurus albertensis). Visual depictions of the distributions of these species in each axis are above and to the right of each PC plot.

Postorbital. Results of the Principal Component Analysis (PCA) of the postorbital and postorbital horncore are illustrated in Fig. 22. The first three axes explain 93.0% of the variation

(Table 6), and only these axes are illustrated here (Fig. 22A&B). As is the case with the nasal, the first axis is largely associated with general size (Fig. 23), and explains a large proportion of the variance (65.4%). PC2 (16.2%) is characterized by negative loadings for variables of horncore height (dorsoventral) as well as posterior length, and positive loading for horncore basal length and width (Fig. 23). PC3 (11.4%) is characterized by negative loading for variables

301 of horncore height (dorsoventral), and positive loading for all other variables (Fig. 23). In plots of both PC1 vs PC2 (Fig. 22A), and PC2 vs PC3 (Fig. 22B), there is slightly less overlap in the morphospace occupation of each of the three well-sampled species than seen in the nasal. The greatest taxonomic separation appears to be in PC2, where C. brinkmani, and the Milk River

Ridge taxon are more negative than the large sample of C. apertus and smaller sample of S. albertensis. The distributions of these species in each axis are above and to the right of each PC plot.

Parietal. Results of the Principal Component Analysis (PCA) of the parietal are illustrated in

Figure 24. The first three axes explain 82.0% of the variation (Fig. 25A, Table 8), and only these axes are illustrated here (Fig. 24A&B). Unlike the results for the nasal and postorbital, PC1 explains less than half of the variation (46.7%), is not simply a general size axis, and is almost exclusively loaded by the epiparietals P1 and P2. The highest loading is on length variables of P1 and P2, although all variables, with the exception of ‘distance from midline’, show the same direction of loading (negative) as the length variables. PC2 explains 29.2% of the total variation and shows strong negative loading for height of P3-P6, and thickness of P1-P6, as well as positive loading for P2 distance from midline. PC3 explains only 6.1% of the total variation and is characterized mainly by positive loading of variables of the parietal midline and fenestrae, as well as distance to the midline for P1-P6. As a result, PC1 is largely represented by size of ornamentation at loci P1 and P2 and shows strong negative loading for Centrosaurus apertus,

PC2 is largely represented by length of ornamentation at loci P3 to P6 and thickness of all loci and shows negative loading for Styracosaurus albertensis, and PC3 is represented by general parietal shape not highly influenced by ornamentation, and does not show good taxonomic separation. Separation of Centrosaurus and Styracosaurus in PC1 and PC2, but not PC3, is highly significant (both Kolmogorov-Smirnov p-values << 0.001).

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Stratigraphy

Stratigraphic Correlation. Tests for correlation between taxonomy and stratigraphic rank show strongly significant results (all p-values < 0.001) regardless of which stratigraphic method or error simulation was utilized (Table 10). In these tests, changing stratigraphic position from the lowest occurrence, to the upper limit of each rhythm, or introducing both set and variable amounts of error had no effect on the significance of the results. The null hypothesis of random position of taxa within the formations (H0:Stratigraphic correlation) is constantly rejected regardless of attempts to simulate stratigraphic error.

Stratigraphic Overlap. Tests of stratigraphic overlap of Belly River Group centrosaurine taxa are shown in Table 11 and Figure 26. The stratigraphic ranges of Coronosaurus brinkmani and

Centrosaurus apertus do not overlap when the estimated position or measured lower and upper positions are used and only show minimal change of overlap (0-2.4%) when stratigraphic error is simulated (Table 11, Fig. 26). Contrasting this pattern, the stratigraphic ranges of Centrosaurus apertus and Styracosaurus albertensis consistently overlap regardless of the stratigraphic datum or simulation used (Table 11, Fig. 26). Closer inspection reveals that nearly all this overlap is due to a single specimen (TMP 1998.68.33), from a poorly sampled bonebed (BB 156 or Quarry

124) (Fig. 27). When this specimen is removed, no overlap is seen of estimated, lower, and upper positions, and overlap is minor in two of the three simulations of stratigraphic overlap (Table 11,

Fig. 26).

Evolution

Visual examples of both the correlation test and time series analysis using PC1 for the parietal, and incorporating methods 1 and 2 for stratigraphic simulations is shown in Figure 28.

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All other results reported here are the summary statistics of evolutionary runs, and do not show direct plots of morphology and stratigraphy.

Correlation of Morphology and Stratigraphy

Nasal. When all nasal specimens are examined, there are slight correlations of PC1 scores and stratigraphic position across all stratigraphic methods (Table 12, Fig. 29). This correlation does not hold when only specimens of Centrosaurus apertus + Styracosaurus albertensis, C. apertus, or S. albertensis are analyzed. PC1 is largely a size axis, and this slight correlation may be reflective of unequal size/ontogenetic sampling across the taxa (i.e., many juveniles of C. brinkmani and C. apertus and fewer of S. albertensis). PC2 and PC3 show no significant correlations with stratigraphy regardless of taxa included or stratigraphic methodology (Table 12,

Fig. 29). The lack of strong correlation of the major PC axes with stratigraphy when all taxa are included (taxa that do correlate with stratigraphy) suggests that the nasal is of limited taxonomic utility and is either constant, or changes randomly, with respect with stratigraphy.

Postorbital. Similarly to the nasal, PC1 for the postorbital is marginally significantly correlated with stratigraphy when all taxa are considered, but not when individual taxa are examined (Table

13, Fig. 30). This can likely also be explained by unequal size/ontogenetic sampling between species. PC2 shows not correlation with stratigraphy. Interestingly PC3 shows a strong correlation with stratigraphy across all stratigraphic methods and for all taxa, C. apertus + S. albertensis, and in C. apertus alone. The loadings of PC3 suggest it is described by robusticity of the posterior postorbital relative to the size of the horncore (Table 6). This indicates direct change in this trait through the stratigraphic range of C. apertus, although this may represent another aspect of size not represented in PC1.

304

Parietal. Contrasting the pattern seen in the nasal and postorbital, for the parietal both PC1 and

PC2 are highly correlated (all p-values < 0.002) with stratigraphy when all taxa are concerned and across all stratigraphic methods (Table 14, Fig. 31). This suggests, as would be expected of a diagnostic element, that the parietal changes distinctly through the stratigraphic range of the

Belly River Group. Neither PC1 nor PC2 are correlated when Centrosaurus and Styracosaurus are considered independently, suggesting the parietal does not change distinctly within each species. PC3 shows no correlation regardless of taxonomy or stratigraphic method used.

Time Series Analysis of Evolutionary Modes

Nasal. Model fitting of time series methods to the morphometric and stratigraphic data for the nasals results in the best fitting models of stasis for PC1 and PC2, and an unbiased random walk for PC3, when all taxa are included and when the lowest stratigraphic level is used (Fig. 33A,

Table 15). Using method 2 of simulated stratigraphic error resulted in the same model selection, under the majority of the replicates (Fig. 32E). Under method 1, however, PC1 and PC3 are best explained by an unbiased random walk, and PC2 by stasis (Fig. 32C, Table 15).

When only specimens of C. apertus and S. albertensis are examined, lowest stratigraphic occurrence results in an unbiased random walk being the preferred model for all three PC axes

(Fig. 32B, Table 16). Under both methods of stratigraphic error simulation, PC1 and PC3 are still best explained by a random walk, but PC2 is best explained by stasis (Figs. 32D, F, Table 16).

Under none of the situations, and in none of the 4000 replicates, is a generalized random walk

(directional trend) selected as the most likely model.

305

Postorbital. Evolutionary model fitting for the postorbital, for all taxa, and lower stratigraphic position, results in support for stasis in PC1, strong support for stasis in PC2, and support for unbiased random walk in PC3 (Fig. 33A, Table 17). Implementation of both methods of stratigraphic error does not change these overall results (Fig. 33C, E).

When only specimens of C. apertus and S. albertensis are used, PC1 and PC2 are both explained best by stasis, and PC3 is best fit by an unbiased random walk when lower stratigraphic position is used (Fig. 33B, Table 18). Both methods of stratigraphic error have no significant effect on model choice for PC2 and PC3, but results in PC1 being best supported by an unbiased random walk, instead of stasis (Figs. 33D, F, Table 18). Under none of the situations, and in none of the 4000 replicates, is a generalized random walk (i.e., trend) selected as the most likely model.

Parietal. Evolutionary model fitting for parietal morphology results in very strong support for an unbiased random walk for all PC axes, when all taxa are included and the stratigraphically lowest point is used (Fig. 34A, Table 19). Simulation of stratigraphic error using method 1 does not significantly change this result (Fig. 34E). Simulation of stratigraphic error using method 2, however, results in PC3 being best supported by stasis (Fig. 34C).

When only specimens of C. apertus and S. albertensis are considered the result is the same, with support for an unbiased random walk for all PC axes when the lowest stratigraphic position is used (Fig. 34B, Table 20). Stratigraphic error simulation using both method 1 and 2 result in PC1 and PC2 being both explained by an unbiased random walk, and PC3 being best explained by stasis (Fig. 34D,F). As with the nasal and postorbital, no significant support for a generalized random walk (i.e., a trend) is found in any combination of taxa selected and stratigraphic method, nor in any of the 4000 replicates.

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Discussion

Completeness and Element Representation

The pattern of completeness within the articulated skull is a result of multiple compounding factors. The largest of these is likely taphonomy, with robust and central elements being less prone to destruction and loss than thin and peripheral elements. This pattern of higher sampling of robust and central elements is indeed seen, as the best-sampled elements are the nasal, postorbital and medial parietal (Fig. 6A, 7). These values (Table 2, Figure 6A and 7) may therefore been seen as rough approximations of how likely different elements within the skull are to be preserved. This pattern is not fully due to taphonomy, however, as it is also affected by multiple human-mediated factors. The first of these is measurement choice; elements with multiple difficult measurements may be underrepresented due to measurement difficulty, and not completeness. Secondly, choices in preparation and display may limit regions available for study, independent of their actual preservation. Finally, as most interest in centrosaurine dinosaurs is concentrated in their ornamentation, there are likely to be collection or preparation biases, as well as research biases towards skulls with these elements preserved. The statistically higher representation of ornamentation elements relative to skull elements many be explained by this, but may also be explained by their robusticity. Deciphering the relative contributions of these two factors is not possible given the current data. Regardless of the cause of this pattern, it is a positive result, as those features of most interest for research (i.e., ornamentation), are best represented in the sample allowing for the highest statistical power.

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Variation and Taxonomic Utility

The statistically higher amounts of variation in the ornamentation (specifically in the nasal, postorbital and parietal) compared to the rest of the skull in centrosaurines should not be surprising. In datasets including multiple species, those morphological traits useful for differentiating taxa will also show higher amounts of variation, as is the case here with the nasal, postorbital, and parietal (Fig. 8, Table 2). Evidence for higher variation in the multitaxic sample is therefore evidence that these structures do differentiate the taxa. Additionally, the higher the variation in a structure, the more likely that the structure is diagnostic to the species. This would suggest that the parietal is much more diagnostic to species than either the postorbital or nasal, which is consistent with most current ideas regarding diagnostic morphology within centrosaurines (Sampson, 1995; Sampson et al., 1997a; Ryan and Russell, 2005; Ryan et al.,

2012a). Not only is this pattern noted between the ornamentation and non-ornamental structures, but distinct differences are also seen with individual elements. Within the parietal, those parietal epiossifications that are most diagnostic to C. apertus (P1 and P2) are also the most variable relative to the less informative epiossifications (P3-P7). The sample size for other species (e.g.,

S. albertensis) is not large enough to test, but if this pattern were consistent across taxa, it would indicate that variation in an element might be a reasonable predictor of the taxonomic informativeness of that element.

Analyses of centrosaurine evolution should be directed towards quantifying the change in morphology of the parietal, and to a lesser extent the nasal and postorbital, through stratigraphy.

Regrettably, measurements of the epiossifications of the squamosal were not included in the morphometric variables collected in this study. As such, the relative taxonomic informativeness

308 of the squamosal ossifications, relative to the parietal, cannot be tested given this dataset, and may represent an area of future research.

Variation and Sexual Display

It is predicted that within a species, elaborate secondary sexual display structures should show higher levels of variation than those of rest of the organism, and this phenomenon has been noted in many extant and extinct vertebrate and invertebrate taxa (Geist, 1966; Hartnoll, 1974; Dodson,

1975; Dodson, 1976; Packer, 1983; Alatalo et al., 1988; Dodson, 1990; Moller and

Pomiankowski, 1993; Pomiankowski and Moller, 1995; Cuervo and Moller, 1999). In this analysis, higher variation in the cranial ornamentation is seen within the C. apertus sample alone

(Figs. 9, 10A, Table 2). This may be used as evidence that these elements likely did have a role in sexual display. Most current debate regarding potential functions of these so-called ‘bizarre’ structures in ornithischians centers around sexual display versus species recognition (Farlow and

Dodson, 1974; Sampson, 1997; Knell and Sampson, 2011; Padian and Horner, 2011b).

Unfortunately the relative differences in patterns of variation (if present) in sexual display structures versus species recognition structures have not been adequately established to allow the current data to be informative.

Asymmetry Within Skulls

Asymmetry within the ornamentation has long been noted, with the holotype of C. apetrus

(CMN 971) showing a well developed parietal hook at the P1 locus of the right side with no

309 evidence for ornamentation on the right side (Lambe, 1910). Further work documented and discussed asymmetry, usually in the frill, of centrosaurine taxa (Sampson et al., 1997b;

McDonald and Horner, 2010; Tanke et al., 2010), but this marks the first real attempt to quantify it.

Although methodologically independent of the pattern of variation between skulls, the pattern of left-right variation between paired elements within skulls (asymmetry) shows a very similar pattern to that seen in variation between skulls (Figs. 6, 11, 12). The most asymmetric elements are those of the parietal, followed by the jugal and postorbital, with the parietal ornamentation showing the highest amount of asymmetry. Ornamental elements show a significantly higher amount of asymmetry than the rest of the skull.

Levels of asymmetry in secondary sexually characters have been shown to be much higher than those of the rest of the animal in many extant taxa (Moller, 1990, 1992; Moller and

Pomiankowski, 1993; Cuervo and Moller, 1999). This provides further evidence that the ornamental elements of centrosaurines likely functioned as secondary sexually characters. This increased asymmetry in secondary sexual characters is, among other ideas, thought to be related to a disruption of developmental homeostasis related to strong directional selection in sexual traits, compared to stabilizing selection in non-sexual traits (Moller and Pomiankowski, 1993;

Cuervo and Moller, 1999, and references therein).

Correlation of Variation and Asymmetry

Strong and significant correlations between variation and asymmetry, both at the variable (Fig.

13) and element (Fig. 14) scale, illustrate these two concepts are strongly linked. This may

310 superficially suggest a common, or at least linked, mechanism responsible for the creation of both variation and asymmetry in these skulls. This, however, is opposite the pattern seen in most studies of fluctuating asymmetry, where rates of plasticity (variation) and asymmetry do not correlate (Cuervo and Moller, 1999), and plasticity and asymmetry are thought to be under independent genetic or mechanistic regulation (Scheiner et al., 1991; Yampolsky and Scheiner,

1994; Tarasjev, 1995). Future work, more explicitly testing for differences between variation and asymmetry in these traits, may help to reveal subtle differences in the distribution of these two phenomena.

Allometry

The result of a positively allometric nasal horncore (for height relative to basal length) is to be expected given the shape change seen through ontogeny, from roughly equilateral shape to a much taller isosceles or scalene shape (Fig. 15A). Plots of nasal curvature, however, reveal a pattern of growth of the horncore that has not previously been quantified (Fig. 15B). Change in the degree of nasal curvature, previously used as a taxonomic indicator (Lull, 1933), is here interpreted as ontogenetic, consistent with Sampson et al. (1997b), with an increase in procurvature with size. Relative growth of anterior and posterior lengths is quite constrained for unfused and partially fused nasals, but is much less constrained (i.e., higher variation) in fully fused nasals. This indicates higher variation in nasal curvature in adults than subadults and juveniles, although the implications for this variation are not known.

Given the well-documented acquisition of resorption pits in the postorbital through ontogeny (Sampson et al., 1997b; Ryan et al., 2001; Tanke and Farke, 2007), the fact that its

311 height is isometric, or possibly even negatively allometric, with respect to basal length, is expected (Figs. 16, 17). There appear to be two phases of postorbital growth in C. apertus, a poorly sampled Phase I characterized by fast growth in height, and a heavily sampled Phase II characterized by slowing of height growth, fusion of the postorbital to the peripheral elements and development of a resorption pit (Fig. 18). The slow growth in horncore height (relative to basal length) in Phase II may be accomplished by either decrease in rate of height growth or increase in robusticity and length of the postorbital base, or both.

Regression of height against basal horn length for the parietal epiossifications in both

Centrosaurus apertus and Styracosaurus albertensis reveals some interesting patterns. Although sample sizes are small (especially for S. albertensis), and therefore statistical allometry is difficult to achieve, comparison of the slopes of the best fits lines reveals distinct differences between C. apertus and S. albertensis (Fig. 19H). In C. apertus, P1 and P2 are strongly and significantly positively allometric, whereas P3-P7 are isometric. Contrasting this, in S. albertensis, P4 and P5 show the highest degree of allometry, followed by P3, P6 and P7, with P1 and P2 having the lowest slopes. Based on this, it can easily be demonstrated how the species- diagnostic adult parietal morphology of C. apertus and S. albertensis can arise from a putatively similar juvenile starting point (Sampson et al., 1997b). In C. apertus, the diagnostic long (and anteriorly directed) P1 and (medially directed) P2 show positive allometry, while the other epiossifications grow isometrically. In S. albertensis, the characteristic morphology would be achieved by high rate of growth in the lengths of P4 and P5, slower growth in P3, P6 and P7, and potentially isometric growth in P1 and P2.

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Multivariate Morphometrics

The large degree of overlap among all taxa in the first three axes of the PCA of nasal shape suggests that the nasal does not preserve a large degree of species-diagnostic morphology (Fig.

20). The greatest separation between C. apertus and S. albertensis is seen in axis 2, where S. albertensis has negative loading. This axis corresponds to the relative size of the horncore to the rest of the nasal, with negative loading corresponding to larger horncores. Although two extreme specimens of S. albertensis fall outside the range of variation of C. apertus, the majority fall within. The discovery of future S. albertensis nasals will show if these may be differentiated from C. apertus. All nasal specimens of C. brinkmani fall within the variation of C. apertus in all axes, and as such these two taxa do not appear to differ greatly in nasal morphology.

Multivariate analysis of the postorbital reveals slightly more morphospace segregation than seen in the nasal (Fig. 22). Once again, PC1 is largely explained by size and all taxa fall within the size range of C. apertus. All specimens of S. albertensis fall within the range of C. apertus in each axis. The largest segregation between C. apertus and C. brinkmani occurs in

PC2, which is characterized by height of the horncore (especially medial height) relative to the other variables. Negative loading of C. brinkmani suggests higher and more laterally curved horncores than C. apertus. This is consistent with the original diagnosis of C. brinkmani as having inflated postorbital horncores that project laterally over the orbit.

Unsurprisingly, the multivariate analysis of parietal shape shows the greatest segregation of taxa in morphospace (Fig. 24). This suggests that among the ornamentation elements, the parietal has the most utility for taxonomic discernment. PC1, having strong (negative) loadings for all aspects of P1 and P2 magnitude, is mainly an axis of C. apertus (and S. sternbergorum) morphology. PC2, having strong (negative) loadings for height and thickness of P3-P6 is largely

313 an axis of S. albertensis morphology. The loadings of PC3 are largely aspects of general shape of the parietal, and this axis is not useful for taxonomic differentiation. PC1 and PC2 are therefore useful univariate metrics to describe general C. apertus vs. S. albertensis morphology.

Stratigraphy

Strong, and highly significant, correlations of taxonomy and lowest stratigraphic position (Table

10) are consistent with previous results indicating a non-random distribution of taxa within the formations, and a distinct faunal turnover (Currie and Russell, 2005; Eberth and Getty, 2005;

Evans, 2007; Ryan et al., 2012b; Mallon et al., 2013). Furthermore, results from both methods of simulating stratigraphic error indicated that this pattern is robust to stratigraphic error at the scale seen in the formation, a unique result not previously tested.

Although there is clear evidence for non-random distribution, specifically a taxonomic turnover from C. brinkmani, to C. apertus, to S. albertensis, the potential for overlap between these successive taxa is less clear. Given the current dataset, there is clear lack of overlap between C. brinkmani and C. apertus, with C. brinkmani only occurring in the Oldman

Formation and C. apertus only occurring in the Dinosaur Park Formation (in the region of

Dinosaur Provincial Park). Stratigraphic overlap (or lack thereof) between C. apertus and S. albertensis is less clear, and depends on interpretation of a single specimen. If the partial parietal

TMP 1998.68.33 (Fig. 26) from Bonebed 156 is assignable to S. albertensis then all methods of stratigraphic position (Lower, Upper, Method 1, and Method 2) show overlap between the two taxa. If this specimen is disregarded, however, most instances of overlap between the taxa disappear.

314

Two possible explanations exist for this specimen/site that allow for non-overlap between these taxa, one taxonomic and one stratigraphic.

Morphological Explanation of Overlap. Although interpreted here as assignable to S. albertensis, it is possible that TMP 1998.68.33 may be assignable to C. apertus. The specimen is characterized by a curved section of parietal bar (~230 mm long) with one complete (~230 mm long) and straight parietal spike positioned next to the base of a second equally sized but either broken or unfused parietal ossification (Fig. 27). Both spikes are ovoid, and originate on the same side of the parietal bar, with the complete spike showing no sign of curvature. The parietal bar is teardrop shaped in cross section with both spikes originating from the thicker margin.

TMP 1998.68.33 was originally interpreted as pertaining to S. albertensis, and subsequently was regarded by Ryan et al. (2007) as a C. apertus P2 spike (Ryan et al., 2007, p960). This taxonomic referral is unclear, however, as the authors also regard the same element as a S. albertensis P3 spike in the same paper (Ryan et al., 2007, p956). The authors also discuss how this single specimen is the only case of stratigraphic overlap between the two taxa, and conclude that if the specimen is assignable to C. apertus, then no overlap occurs between the taxa (Ryan et al., 2007).

Based on comparison with other parietals, the best interpretation for the location and orientation of the partial parietal bar (TMP 1998.68.33 ) is as a section of the posterior or posterolateral bar, with the thinner margin representing the margin of the parietal fenestrae (Fig.

27G). This indicates the complete spike projects posteriorly or posterolaterally, a morphology inconsistent with C. apertus but consistent with S. albertensis. In C. apertus, not only do the two large parietal spikes (P1 and P2) show strong curvature, they also do not originate from the same side of the posterior parietal bar, the condition seen in TMP 1998.68.33. If assignable to S.

315 albertensis, two spikes would then represent loci P3 and P4, or P4 and P5, or possibly P5 and P6.

This is consistent with the original interpretation, as well as figure 12 of Ryan et al., (Ryan et al.,

2007). Collection of further diagnostic material from this site will undoubtedly aide taxonomic identification, and excavation for BB 156 should therefore be regarded as a priority.

Stratigraphic Explanation. BB 156 is located at the base of a multistory, stacked sandstone complex, with a total height of 13 meters across three successive and downcutting channel rhythms (Fig. 35). The error simulation of Method 2 only takes into account the height of the host (lowest - I) of the rhythms. If this lower rhythm was significantly downcut by the overlying

(second - II) channel rhythm then the possibility does exist that the specimen derived from a location that is stratigraphically higher than the upper bound of the simulation. The stratigraphic error simulation of Method 2 was intended to be a conservative estimate for the possible stratigraphic uncertainty caused by channel downcutting, and it is entirely possible that this method will underestimate possible positions of specimens hosted in multistory sandstone complexes that are themselves downcut by overlying sands. This possibility is not directly testable, but can be inferred from systematic differences in the thickness of downcut

(incomplete) and complete sandstone rhythms. Current data (Chapter 4) suggests no systematic differences, but collection of a larger sample may allow for differences to be determined, and for the adjustment of stratigraphic error simulations for specimens hosted by downcut sandstone rhythms to be made.

Until both the morphology of specimens for BB 156 can be unambiguously determined and stratigraphic error simulation of downcut hosted specimens can be further refined, the potential stratigraphic overlap between C. apertus and S. albertensis is questionable. If future research finds more concrete evidence for statigraphic and temporal overlap of C. apertus and S.

316 albertensis the hypothesis of anagenesis between the two will be rejected, in favor of either an evolutionary hypothesis of budding cladogenesis (nor a non-sister taxon relationship) or and ecological hypothesis of regional replacement of contemporaneous but allopatic species.

Evolution

Horner and colleagues (1992) suggested, based on putative transitional/intermediate morphology and stratigraphy of species from four different dinosaur clades from the Two Medicine

Formation (Campanian) of Montana, that anagenesis characterized dinosaur evolution in the Late

Cretaceous, and was driven by range restrictions during sealevel transgressions. The strongest case for this was made by centrosaurine ceratopsids where an anagenic model of evolution from

Styracosaurus to Pachyrhinosaurus via three intermediate forms (A-C) was forwarded (Fig. 1A).

This suggestion makes testable predictions about the nature of the fossil record for both dinosaurs in the Two Medicine Formation, and those in contemporaneous formations. Under anagenesis, within the stratigraphic range of well-sampled species, morphological change is expected to correlate with stratigraphy. Contrasting this, suggestions that cladogenesis via allopatric speciation, and ecological replacement, better characterize the evolutionary mode of these dinosaurs would predict no correlations of morphology and stratigraphy (Sampson, 1995).

Using correlation of morphology and stratigraphy, almost no evidence is found for directional change within centrosaurine taxa, with the only exception being that PC3 of the postorbital analysis changes within the record of C. apertus. This pattern for PC3 of the postorbital is robust enough to be seen when all taxa, C. apertus + S. albertenesis, and just C. apertus are sampled, and is present regardless of stratigraphic sampling method (Fig. 30).

317

Though statistically significant, the biological significance of this correlation is unclear. PC3 for the postorbital describes the height of the postorbital horncore relative to all other measurements of the postorbital and, given the negative loading and positive correlation, suggests the postorbital horncore gets progressively shorter through the stratigraphic range of C. apertus.

Whether this is due to evolutionary change or potentially due to unequal sampling of ontogenetic stages through the stratigraphic range of C. apertus is unclear.

Strong correlations of parietal morphology (PC1 and PC2) and stratigraphy when both C. apertus and S. albertensis are sampled are expected due to diagnostic material and taxonomic turnover. These same correlations do not occur, however, when the individual species are sampled. These results indicate that the parietal PCA accurately captures taxonomically informative morphological changes, and that these changes do not occur within the preserved stratigraphic ranges of the individual species.

When a time-series analysis is undertaken, which compares relative likelihood support for three main evolutionary models, the best-supported models are invariably unbiased random walk and stasis. No strong support is found for a generalized random walk (i.e., trend), either across multiple species, or within single species. Further, PC1 and PC2 of parietal shape, which are the best morphologic indicators of taxonomy, show very strong support for an unbiased random walk regardless of stratigraphic method. These results are consistent with those of the correlation analysis, and suggest that there is no detectable and directed changed in morphology within the individual centrosaurine species.

Hypotheses of an ancestor-descendant relationship between C. apertus and S. albertensis would therefore require no distinct morphological change within the 28–30 m and 20–25 m ranges (respectively) of the individual taxa, and all distinct morphological changes to occur in

318 the 0–2 m gap between the subsequent species (Fig. 26). Although unsampled intervals of time almost certainly exist within the preserved stratigraphic range of the Dinosaur Park Formation, and the rate of sedimentation is not constant, there is no indication that there is any more missing time between the ranges of the two taxa than there is within the ranges of either taxon (Braman and Koppelhus, 2005; Eberth, 2005). A hypothesis of ancestor-descendant relationship would therefore necessitate a long period of stasis in the well sampled intervals of both C. apertus and

S. albertensis, divided by an unsampled (and presumably unsampleable) period that encompasses all of the quantifiable evolutionary change.

This lack of morphological change within either C. apertus leading up to the faunal turnover, or S. albertensis after the turnover, during a well-documented transgressive event

(Eberth, 2005), contrasts with the evolutionary scenario proposed by Horner and colleagues

(Horner et al., 1992). The results are inconsistent with predictions of a gradual anagenetic change, but are consistent with the taxonomic turnover resulting from either a punctuated evolutionary event or ecological replacement via migration of allopatric species (Sampson,

1995). Further research, and the discovery of new species of centrosaurine, have resulted in attempts to reconstruct the phylogenetic relationships of Centrosurinae failing to recover a sister taxon relationship between C. apertus and S. albertensis, or other relationship that may be consistent with an anagenetic mode of evolution from one to the other (Ryan et al., 2012a;

Sampson et al., 2013).

Data from the geographic occurrence of Centrosaurus apertus indicate it (and other ornithischians) had an extensive geographic range, extending over hundreds of kilometers and into neighboring depositional basins (Ryan et al., 2010; Tokaryk et al., 2012; Evans et al., in press). Given this large geographic range, it is quite likely that speciation events resulting in the

319 diversity we see as C. apertus and S. albertensis occurred outside of the geographic and temporal sampling of the Dinosaur Provincial Park area.

A similar, but more prolonged, pattern of faunal turnover in the Horseshoe Canyon

Formation shows well supported ecological correlates, and have been interpreted as being due to immigration/emigration of dinosaur taxa as a result of habitat tracking, specifically rainfall

(Eberth et al., 2013). Tests of correlation between the faunal turnover and ecological proxies in the Dinosaur Park Formation, however, have failed to find significant results (Mallon et al.,

2013). Rather the turnover is more correlated with the patterns of fossil palynomorphs (Mallon et al., 2013).

Ecological replacement as a mechanism for the taxonomic turnover requires allopatry of distinct but closely related taxa. Although we have reasonably strong evidence against widespread sympatry of C. apertus and S. albertensis in the Belly River Group, we also have little evidence either for or against allopatry of these two taxa on a regional scale. This is at least partially due to limited sampling of contemporaneous but geographically disparate formations and regions. Current work extending the geographic sampling of contemporaneous rocks will hopefully inform of the presence or lack of allopatry.

320

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Table 1 – List of 104 linear measurements, and seven counts/codeings, used for morphometric analyses of centrosaurine skulls. Measurements are ordered by element, and indicate the abbreviation, whether it is a paired measurement, and whether it is considered ornamentation.

The majority of measurements were derived from Ryan (1992) (modified from Dodson, 1990), with additional measurements added to underrepresented areas.

Bone Measurement/Coding Abbrev. Paired Ornam. Total Snout-condyle length SCL N N Total Total length of skull (including ornament) TL N N Total Preorbital length PRL Y N Total Skull dorsoventral height at postorbital SKH-PO Y N Total Transverse width between postorbital horncore tips POW N N Rostral Dorsoventral height of rostral on face RSTH Y N Rostral Anteroposterior length of rostral on face RSTL Y N Rostral Transverse width of rostral across face RSTW N N Premaxilla Transverse thickness of ventral premaxilla PM-WD Y N Premaxilla Distance between premaxillary ventral PM-TW N N Premaxilla Length of posterior premaxillary phlange PMPL Y N Premaxilla Dorsoventral height of nares NRSH Y N Maxilla Anteroposterior maxilla length MAXL Y N Maxilla Dorsoventral maxilla height MAXH Y N Maxilla Number of alveoli in maxilla #ALV MAX Y N Maxilla Anteroposterior length Length of maxillary tooth row MAX TRL Y N Nasal Height of nasal horn (perpendicular to base) H1 N Y Nasal Anterior length of nasal horn NH-AL N Y Nasal Posterior length of nasal horn NH-PL N Y Nasal Circumference of nasal horn at base NH-CIR N Y Nasal Anteroposterior length of nasal horn core at base LBASE Y Y Nasal Transverse thickness of nasal horn core at base TK1 N Y Nasal Distance from nasal horn to frontal fontenelle AFL N N Nasal Distance from nasal horn to posterior margin of nasal PL N N Nasal Length of anterior process of nasal ANL N N Nasal Height of ventral process of nasal VNL Y N Nasal Transverse nasal half width WTH-1/2-N Y N Fontenelle Midline length of frontal fontanelle FFL N N Fontenelle Max transverse width of frontal fontanelle FFMW N N Fontenelle Maximum depth of frontal fontanelle FD N N Orbit Orbital max height ORBMXH Y N Orbit Orbital min length ORBMXL Y N Postorbital Dorsalventral height of postorbital horn core HCH Y Y Postorbital Dorsalventral height of postorbital posterior to horn PPT Y N Postorbital Medial length of postorbital horn core OHC-ML Y Y Postorbital Lateral length of postorbital horn core OHC-LL Y Y Postorbital Anteroposterior length of supraorbital horn core base HCL Y Y Postorbital Transverse width of supraorbital horn core at base HCW Y Y Postorbital circumference of supraorbital horn core at base OCH-CIR Y Y Postorbital resorption pit on supraorbital horn core RPIT? Y Y Palpebral Anteroposterior length of palpebral PALL Y N Palpebral Transverse width of palpebral PALW Y N Jugal Maximum length of jugal at midline MXL Y N Jugal Minimum width of jugal MNW Y N Jugal Thickness of jugal at midpoint thickness MDPTX Y N Jugal Epijugal present EJ? Y N Jugal Infratemporal fenestra maximum width ITFMXW Y N Jugal Infratemporal fenestra miimumn width ITFMNW Y N Predentary Anteroposterior length of predentary PD-TL Y N Predentary Transverse width of predentary PD-WP N N

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Bone Measurement/Coding Abbrev. Paired Ornam. Dentary Anteroposterior length of length of dentary DNT-L Y N Dentary Anteroposterior length of dentary tooth row DNT-TRL Y N Dentary Dorsoventral height of dentary ad mid tooth row DNT-H Y N Dentary Number of dentary alveoli #ALV DNT Y N Dentary Dorsoventral height of coronoid process CPH Y N Dentary Anteropsterior ength of coronoid process CPL Y N Dentary Transverse thickness of coronoid process CTK Y N Braincase Transverse width of paroccipital wings EXOPW Y N Braincase Transverse width of basioccipital BASI N N Braincase Occipital condyle max diameter OCMXD N N Braincase Occipital condyle max height OCH N N Braincase Occipital condyle max width OCW N N Braincase Occipital condyle girth OCG N N Squamosal Squamosal maximum length SQMTL Y N Squamosal Squamosal lateral margin length SQMML Y N Squamosal Number or squamosal epiossifications SQ# Y N Parietal Midline parietal bar midpoint thickness PBMTK N N Parietal Total parietal length (including ornamentation) PTL N N Parietal Sagittal length of parietal bar PBTL N N Parietal Parietal transverse half width P1/2W Y N Parietal Number of sagittal bumps on parietal bar PB# N N Parietal Number or posterolateral epiossifications PL# Y N Parietal Parietal bar midpoint transverse width PBW N N Parietal Dorsoventral thickness of lateral parietal bar PLTK Y N Parietal Parietal fentesta transverse width PFW Y N Parietal Parietal fenestra anteroposterior length PFL Y N Parietal Inner curve height of P1 P1-OL Y Y Parietal Inner curve height of P1 P1-IL Y Y Parietal Height of P1, perpendicular to base P1-L Y Y Parietal Transverse width of P1 at base P1-LT Y Y Parietal Thickness of P1 (anteroposterior) at base P1-TK Y Y Parietal Circumference of P1 at base P1-CIR Y Y Parietal Distance of tip of P1 from midline P1-dist Y Y Parietal Inner curve height of P2 P2-OL Y Y Parietal Inner curve height of P2 P2-IL Y Y Parietal Height of P2, perpendicular to base P2-L Y Y Parietal Transverse width of P2 at base P2-LT Y Y Parietal Thickness of P2 (anteroposterior) at base P2-TK Y Y Parietal Circumference of P2 at base P2-CIR Y Y Parietal Distance of tip of P2 from midline P2-dist Y Y Parietal Height of P3, perpendicular to base P3-HT Y Y Parietal Transverse width of P3 at base P3-LT Y Y Parietal Thickness of P3 (dorsoventral) at base P3-TK Y Y Parietal Distance of tip of P3 from midline P3-dist Y Y Parietal Height of P4, perpendicular to base P4-HT Y Y Parietal Transverse width of P4 at base P4-LT Y Y Parietal Thickness of P4 (dorsoventral) at base P4-TK Y Y Parietal Distance of tip of P4 from midline P4-dist Y Y Parietal Height of P5, perpendicular to base P5-HT Y Y Parietal Transverse width of P5 at base P5-LT Y Y Parietal Thickness of P5 (dorsoventral) at base P5-TK Y Y Parietal Distance of tip of P5 from midline P5-dist Y Y Parietal Height of P6, perpendicular to base P6-HT Y Y Parietal Transverse width of P6 at base P6-LT Y Y Parietal Thickness of P6 (dorsoventral) at base P6-TK Y Y Parietal Distance of tip of P6 from midline P6-dist Y Y Parietal Height of P3, perpendicular to base P7-HT Y Y Parietal Transverse width of P3 at base P7-LT Y Y Parietal Thickness of P3 (dorsoventral) at base P7-TK Y Y Parietal Distance of tip of P3 from midline P7-dist Y Y

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Table 2 – Results for test of completeness (%), variation (St. Dev. - between skulls) and asymmetry (% - within skulls) for all linear measurements across the dataset of complete and partial skulls. Averaged values (grey and bold) for either elements or regions are shown. For graphical representation see Figures 6-9, 11, and 12.

All BRG taxa C. apertus All BRG taxa C. apertus Abbr. Complet. (%) Var. (SD) Var. (SD) Assym. (%) Assym. (%) SCL 29.09 0.0339 0.0358 NA NA TL 27.27 0.0528 0.0415 NA NA PRL 20.00 0.0423 0.0469 0.4516 0.4226 SKH-PO 20.91 0.0877 0.0881 0.9482 1.0703 POW 36.36 0.0845 0.0891 NA NA Rostral 20.00 0.0615 0.0615 1.1132 1.1132 RSTH 20.91 0.0407 0.0407 0.6902 0.6902 RSTL 20.91 0.0579 0.0579 1.5363 1.5363 RSTW 18.18 0.0860 0.0860 NA Premaxilla 24.32 0.0773 0.0775 1.3236 1.4962 PM-WD 23.64 0.0805 0.0814 2.5830 3.0344 PM-TW 25.45 0.1128 0.1135 NA NA PMPL 23.64 0.0464 0.0495 0.7582 0.8119 NRSH 24.55 0.0696 0.0656 0.6297 0.6424 Maxilla 16.36 0.0456 0.0474 1.1239 1.1489 MAXL 17.27 0.0372 0.0389 0.8360 0.7986 MAXH 19.09 0.0846 0.0900 1.7388 1.9486 #ALV MAX 8.18 0.0265 0.0277 NA NA MAX TRL 12.73 0.0342 0.0331 0.7969 0.6996 Nasal 26.86 0.1152 0.1113 1.1480 1.1360 Nasal Orn. 31.45 0.1272 0.1268 NA NA Nasal Non. 23.03 0.1008 0.0927 NA NA H1 36.36 0.1957 0.1998 NA NA NH-AL 32.73 0.1718 0.1697 NA NA NH-PL 32.73 0.1838 0.1776 NA NA NH-CIR 27.27 0.0484 0.0517 NA NA LBASE 28.18 0.0671 0.0701 0.5034 0.2692 TK1 32.73 0.0964 0.0919 NA NA AFL 29.09 0.1008 0.1128 NA NA PL 12.73 0.1758 0.1358 NA NA ANL 20.00 0.0492 0.0386 1.3682 1.4943 VNL 20.00 0.0630 0.0606 1.5723 1.6444 WTH-1/2-N 23.64 0.1151 0.1155 NA NA Fontanelle 28.48 0.1935 0.1906 NA NA FFL 34.55 0.1306 0.1198 NA NA FFMW 32.73 0.2717 0.2723 NA NA FD 18.18 0.1783 0.1799 NA NA Orbit 35.00 0.0780 0.0818 2.3144 2.3489 ORBMXH 34.55 0.0872 0.0920 1.8564 1.8216 ORBMXL 35.45 0.0688 0.0716 2.7724 2.8762 Postorbital 34.68 0.1252 0.1265 2.1616 2.2206 Post. Orn 38.03 0.1328 0.1344 2.3234 2.3373 Post. Non. 14.55 0.0797 0.0791 1.1905 1.5203 HCH 41.82 0.1873 0.2009 3.2587 3.6438 PPT 14.55 0.0797 0.0791 1.1905 1.5203 OHC-ML 34.55 0.1423 0.1422 3.0070 3.0699 OHC-LL 34.55 0.1668 0.1726 3.1718 3.4073 HCL 41.82 0.0922 0.0928 1.5710 1.3389 HCW 40.00 0.1048 0.0971 1.9345 1.6748 OCH-CIR 35.45 0.1036 0.1007 0.9975 0.8894 RPIT?

337

All BRG taxa C. apertus All BRG taxa C. apertus Abbr. Complet. (%) Var. (SD) Var. (SD) Assym. (%) Assym. (%) Palpebral 29.55 0.0589 0.0566 1.9621 1.8511 PALL 28.18 0.0494 0.0528 1.2969 1.3002 PALW 30.91 0.0685 0.0605 2.6272 2.4019 Jugal 24.36 0.0936 0.0952 2.5416 2.5822 MXL 26.36 0.0464 0.0476 0.7242 0.7822 MNW 27.27 0.1000 0.1035 0.9988 0.9163 MDPTX 17.27 0.1396 0.1399 3.9320 4.2449 EJ? ITFMXW 25.45 0.0647 0.0612 2.5888 2.2043 ITFMNW 25.45 0.1176 0.1238 4.4644 4.7636 Predentary 9.39 0.0696 0.0696 0.5052 0.5052 PD-TL 9.70 0.0580 0.0580 0.5052 0.5052 PD-WP 9.09 0.0811 0.0811 NA Dentary 10.76 0.0652 0.0715 0.6562 0.6212 DNT-L 13.64 0.0465 0.0499 0.5387 0.5214 DNT-TRL 11.82 0.0317 0.0337 0.5199 0.5754 DNT-H 12.73 0.0474 0.0490 0.7154 0.7154 #ALV DNT 9.09 0.0183 0.0184 NA NA CPH 8.18 0.0966 0.1080 NA NA CPL 7.27 0.1670 0.1928 NA NA CTK 10.91 0.0488 0.0485 0.8507 0.6727 Braincase 25.00 0.0584 0.0589 1.2627 1.3533 EXOPW 22.73 0.1065 0.1074 1.2627 1.3533 BASI 23.64 0.0773 0.0760 NA NA OCMXD 23.64 0.0379 0.0379 NA NA OCH 21.82 0.0355 0.0367 NA NA OCW 23.64 0.0329 0.0328 NA NA OCG 34.55 0.0602 0.0626 NA NA Squamosal 24.24 0.0518 0.0520 1.2082 1.2082 SQMTL 23.64 0.0561 0.0596 1.5032 1.5032 SQMML 25.45 0.0476 0.0444 0.9131 0.9131 SQ# 23.64 Parietal 30.91 0.0973 0.0948 3.1934 3.2243 PBMTK 23.64 0.1642 0.1642 NA NA PTL 53.64 0.0789 0.0660 NA NA PBTL 54.55 0.0796 0.0768 NA NA P1/2W 34.55 0.0507 0.0513 1.6144 1.6144 PB# 40.00 PL# 17.30 PBW 45.45 0.0757 0.0728 NA NA PLTK 26.36 0.1192 0.1101 4.9771 4.9771 PFW 30.00 0.1288 0.1346 4.9328 5.0565 PFL 30.00 0.0815 0.0825 1.2494 1.2494 P1 31.43 0.2622 0.2387 2.8850 2.7804 P1-OL 32.73 0.4028 0.3667 2.8932 2.7211 P1-IL 25.45 0.3855 0.3148 2.2029 1.6937 P1-L 31.82 0.4315 0.3849 3.8696 3.7336 P1-LT 38.18 0.1283 0.1328 1.9569 2.1595 P1-TK 30.91 0.1876 0.1767 3.9538 3.7050 P1-CIR 24.55 0.1396 0.1407 1.5795 1.6631 P1-dist 36.36 0.1599 0.1543 3.7393 3.7865 P2 32.34 0.2602 0.1934 4.7285 4.4674 P2-OL 31.82 0.3741 0.2697 3.8612 3.0480 P2-IL 30.91 0.3112 0.1779 5.9714 5.3440 P2-L 31.82 0.3575 0.2214 5.4372 3.5918 P2-LT 39.09 0.1494 0.0724 2.8064 2.6920 P2-TK 30.91 0.1499 0.1500 5.0615 5.3752 P2-CIR 27.27 0.1175 0.0871 2.1895 1.7410 P2-dist 34.55 0.3615 0.3752 7.7723 9.4794 P3 32.27 0.1906 0.1468 4.5702 4.9882 P3-HT 32.73 0.3469 0.2446 7.3280 8.0608 P3-LT 34.55 0.1121 0.1057 3.6247 3.8727 P3-TK 29.09 0.2051 0.1500 5.8387 6.5534 P3-dist 32.73 0.0982 0.0870 1.4895 1.4660 P4 32.27 0.2030 0.1546 6.0252 6.3587 P4-HT 33.64 0.3851 0.2618 14.4476 15.3877 P4-LT 34.55 0.1203 0.1180 2.9793 2.9598 P4-TK 29.09 0.2158 0.1566 4.7286 5.1127

338

All BRG taxa C. apertus All BRG taxa C. apertus Abbr. Complet. (%) Var. (SD) Var. (SD) Assym. (%) Assym. (%) P4-dist 31.82 0.0908 0.0819 1.9451 1.9746 P5 30.45 0.1694 0.1375 4.8777 4.8777 P5-HT 31.82 0.3248 0.2556 10.8881 10.8881 P5-LT 32.73 0.0899 0.0847 3.5940 3.5940 P5-TK 27.27 0.1950 0.1489 3.3132 3.3132 P5-dist 30.00 0.0681 0.0608 1.7155 1.7155 P6 29.32 0.1696 0.1633 5.6899 5.6899 P6-HT 30.00 0.3416 0.3337 10.3652 10.3652 P6-LT 31.82 0.1128 0.1145 5.4833 5.4833 P6-TK 26.36 0.1690 0.1498 5.0562 5.0562 P6-dist 29.09 0.0550 0.0551 1.8550 1.8550 P7 15.23 0.1497 0.1508 3.5929 3.5929 P7-HT 16.36 0.3046 0.3287 6.1878 6.1878 P7-LT 16.36 0.0966 0.0964 3.2320 3.2320 P7-TK 13.64 0.1531 0.1353 4.1827 4.1827 P7-dist 14.55 0.0444 0.0429 0.7690 0.7690 P height 29.92 0.3605 0.2872 6.9583 6.4565 P length 32.12 0.1157 0.1035 3.6556 3.4276 P thick. 26.36 0.1822 0.1525 4.5907 4.7569 P dist 29.87 0.1098 0.1072 2.4107 2.6308

339

Table 3 – Results of allometric analyses (OLS) of the height of the parietal ornamentation relative to their basal width for Centrosaurus apertus and Styracosaurus albertensis. For graphical representation see Figure 20.

Centrosaurus apertus Slope Lower Upper Trend n P1 2.153 1.612 2.694 pos. allo. 37 P2 1.972 1.512 2.433 pos. allo. 35 P3 1.454 0.955 1.953 iso 28 P4 0.524 -0.061 1.108 iso 29 P5 1.289 0.587 1.991 iso 28 P6 1.324 0.484 2.164 iso 28 P7 0.715 -0.980 2.410 iso 16 Styracosaurus albertensis Slope Lower Upper Trend n P1 1.622 0.387 2.858 iso 11 P2 1.439 -0.218 3.095 iso 12 P3 1.884 0.547 3.221 iso 14 P4 2.602 -2.108 7.311 iso 6 P5 2.869 -3.229 8.967 iso 8 P6 1.800 -0.356 3.956 iso 9 P7 1.859 0.102 3.615 iso 7

340

Table 4 – Results (PC loadings, and proportion of variance explained by each axis) of the

Principal Component Analysis of ten linear measurements of nasal shape in Centrosaurinae. For graphical representation, see Figure 22. For PC scores see Figure 21

Comp.1 Comp.2 Comp.3 HT -0.477 -0.346 0.225 AL -0.434 -0.238 0.181 PL -0.508 -0.283 -0.131 LT -0.239 0.169 0.146 TK -0.297 0.385 -0.476 CIR -0.268 0.32 - LTH.post - 0.51 0.605 LTH.vent - 0.203 0.23 LTH.ant -0.189 0.287 - WTH.1/2 -0.207 0.285 -0.479 Variance 0.810 0.082 0.039 Cum. Var. 0.810 0.893 0.931

341

Table 5 – Results of Kolmogorov-Smirnov tests for differences of sample distributions between taxa in the PC scores for the nasal. Significant differences are shown in bold.

Comp.1 C. brinkmani S. albertensis C. apertus << 0.001 0.9044 C. brinkmani - 0.0079 Comp.2 C. brinkmani S. albertensis C. apertus 0.0447 0.4006 C. brinkmani - 0.0314 Comp.3 C. brinkmani S. albertensis C. apertus 0.1452 0.2710 C. brinkmani - 0.0996

342

Table 6 – Results (PC loadings, and proportion of variance explained by each axis) of the

Principal Component Analysis of ten linear measurements of postorbital shape in

Centrosaurinae. For graphical representation, see Figure 24. For PC scores see Figure 23.

Comp.1 Comp.2 Comp.3 HT -0.392 -0.276 -0.374 LH -0.386 -0.173 -0.249 MH -0.411 -0.361 -0.305 LTH -0.320 0.216 0.202 WTH -0.404 0.297 0.199 CIR -0.342 0.220 0.177 post - -0.661 0.638 Orb.lth - - 0.243 ant.tk -0.114 0.333 - post.tk -0.359 0.172 0.350 Variance 0.654 0.162 0.114 Cum. Var. 0.654 0.816 0.930

343

Table 7 – Results of Kolmogorov-Smirnov tests for differences of sample distributions between taxa in the PC scores for the postorbital. Significant differences are shown in bold.

Comp.1 C. brinkmani S. albertensis MRR C. apertus 0.6423 0.0015 0.0014 C. brinkmani - 0.0030 0.0010 S. albertensis - - 0.1353 Comp.2 C. brinkmani S. albertensis MRR C. apertus << 0.001 0.1259 0.2411 C. brinkmani - 0.2461 0.1247 S. albertensis - - 0.7396 Comp.3 C. brinkmani S. albertensis MRR C. apertus << 0.001 0.0687 0.0012 C. brinkmani - 0.0008 0.7516 S. albertensis - - 0.0014

344

Table 8 – Results (PC loadings, and proportion of variance explained by each axis) of the

Principal Component Analysis of 45 linear measurements of parietal shape in Centrosaurinae.

For graphical representation, see Figure 26. For PC scores see Figure 25.

PC1 PC2 PC3 SCL - - - TL - - 0.123 PBMTK - - - PTL - -0.115 0.307 PBTL - - 0.177 P1.2W - - 0.123 PB. - - - PL. - - - PBW - - 0.162 PLTK - - - PFW -0.101 - 0.35 PFL - - 0.203 P1.OL -0.431 - -0.175 P1.IL -0.402 - -0.18 P1.L -0.438 - -0.225 P1.LT -0.135 - - P1.TK - -0.125 - P1.CIR -0.115 - - P1.dist - - - P2.OL -0.317 - - P2.IL -0.295 - 0.145 P2.L -0.292 - - P2.LT -0.122 - - P2.TK -0.12 -0.13 - P2.CIR -0.108 - - P2.dist - 0.174 0.413 P3.HT - -0.438 - P3.LT - - - P3.TK - -0.218 - P3.dist - - 0.243 P4.HT 0.121 -0.537 - P4.LT - - - P4.TK - -0.228 - P4.dist - - 0.282 P5.HT - -0.41 - P5.LT - - - P5.TK - -0.19 - P5.dist - - 0.185 P6.HT - -0.272 -0.121 P6.LT - - - P6.TK - -0.126 - P6.dist - - 0.165 P7.HT - - -0.23 P7.LT - - - P7.TK - - - P7.dist - - - Variance. 0.467 0.292 0.061 Cum. Prop. 0.467 0.759 0.820

345

Table 9 – Results of Kolmogorov-Smirnov tests for differences of sample distributions between taxa in the PC scores for the parietal. Significant differences are shown in bold.

Comp.1 C. brinkmani S. albertensis C. apertus 0.02935 << 0.001 C. brinkmani - 0.151 Comp.2 C. brinkmani S. albertensis C. apertus 0.0447 << 0.001 C. brinkmani - 0.2557 Comp.3 C. brinkmani S. albertensis C. apertus 0.91 0.4254 C. brinkmani - 0.8217

346

Table 10 – Kendall rank correlation of taxonomy and statigraphic order. All correlations are highly significant.

Estimated: Metric All Sites Excluding BB156 Lower p-value 9.67E-06 1.88E-05 Method 1 p-value 8.47454E-05 0.000279732 sd 5.32201E-06 1.29678E-05 Measured: Lower p-value 9.65E-06 1.88E-05 Upper p-value 9.39E-05 0.0002676 Method 1 p-value 9.68636E-05 0.000280676 sd 2.30858E-05 2.84349E-05 Method 2 p-value 8.25391E-05 0.000270096 sd 4.78157E-06 NA

347

Table 11 – Occurrence (yes/no), or proportion for replicates (%), for which stratigraphic overlap occurs between taxa. Lack of (or rare) overlap is indicated in bold.

C. brinkmani C. apertus C. apertus Estimated: C. apertus S. albertensis S. albertensis (ex. BB156) Lower No Yes No Method 1 2.4% 99% 3.6% Measured: Lower No Yes No Upper No Yes No Method 1 0.1% 97.2% 63.5% Method 2 0% 100% 15.9%

348

Table 12 – Results of significance tests of Pearson correlation between stratigraphic positions and the first three PC axes of nasal shape. Method 1 and 2 show the average of 1000 replicates.

Significant values are shown in bold.

Lower All Taxa Centro./Styrac. Centro. Styrac. PC1 p-value 0.017 0.193 0.484 0.815 PC2 p-value 0.895 0.775 0.851 0.987 PC3 p-value 0.429 0.233 0.467 0.597 Method 1 PC1 p-value 0.022 0.225 0.534 0.739 sd 0.012 0.104 0.241 0.193 PC2 p-value 0.850 0.762 0.715 0.798 sd 0.110 0.161 0.204 0.167 PC3 p-value 0.446 0.262 0.528 0.577 sd 0.117 0.109 0.239 0.228 Method 2 PC1 p-value 0.014 0.169 0.689 0.795 sd 0.003 0.025 0.097 0.101 PC2 p-value 0.950 0.865 0.859 0.923 sd 0.036 0.077 0.100 0.057 PC3 p-value 0.398 0.206 0.524 0.575 sd 0.036 0.037 0.114 0.103

349

Table 13 – Results of significance tests of Pearson correlation between stratigraphic positions and the first three PC axes of postorbital shape. Method 1 and 2 show the average of 1000 replicates. Significant values are shown in bold.

Lower All Taxa Centro./Styrac. Centro. Styrac. PC1 p-value 0.041 0.179 0.081 0.556 PC2 p-value 0.290 0.388 0.426 0.324 PC3 p-value 0.000 0.002 0.000 0.720 Method 1 PC1 p-value 0.052 0.225 0.208 0.585 sd 0.028 0.116 0.188 0.209 PC2 p-value 0.319 0.439 0.521 0.376 sd 0.111 0.177 0.259 0.180 PC3 p-value 0.000 0.005 0.008 0.706 sd 0.000 0.006 0.017 0.183 Method 2 PC1 p-value 0.044 0.190 0.145 0.564 sd 0.011 0.046 0.069 0.033 PC2 p-value 0.261 0.427 0.504 0.314 sd 0.032 0.058 0.117 0.046 PC3 p-value 0.000 0.002 0.001 0.759 sd 0.000 0.001 0.001 0.057

350

Table 14 – Results of significance tests of Pearson correlation between stratigraphic positions and the first three PC axes of parietal shape. Method 1 and 2 show the average of 1000 replicates. Significant values are shown in bold.

Lower All Taxa Centro. Styrac. PC1 p-value 0.000 0.759 0.385 PC2 p-value 0.000 0.683 0.093 PC3 p-value 0.177 0.085 0.322 Method 1 PC1 p-value 0.001 0.723 0.444 sd 0.001 0.193 0.196 PC2 p-value 0.000 0.685 0.144 sd 0.000 0.197 0.103 PC3 p-value 0.194 0.125 0.379 sd 0.063 0.082 0.186 Method 2 PC1 p-value 0.001 0.543 0.374 sd 0.000 0.092 0.064 PC2 p-value 0.000 0.703 0.088 sd 0.000 0.095 0.022 PC3 p-value 0.224 0.168 0.308 sd 0.039 0.061 0.051

351

Table 15 – Model-fitting results of time-series analysis of the first three PC axes of the nasal for entire series (Coronosaurus brinkmani, Centrosaurus apertus, and Styracosaurus albertensis).

Simulations of stratigraphic error using methods 1 and 2 show the summary statistics for 1000 replicates. All values shown (with the exception of the percentages) are Akaike weights (based off of delta AIC) illustrating the probability of each model fitting best. The percentages, illustrate the percentages of replicates for which model was most preferred. Analyses performed using 7 m stratigraphic bins.

Lower GRW URW Stasis PC1 0.021 0.185 0.795 PC2 0.033 0.377 0.59 PC3 0.048 0.575 0.377 Method 1 PC1 Mean 0.064 0.484 0.452 Max 0.247 0.822 0.860 Min 0.012 0.128 0.062 St. Dev 0.035 0.179 0.208 Percent 0% 52% 48% PC2 Mean 0.042 0.478 0.480 Max 0.076 0.720 0.900 Min 0.008 0.093 0.204 St. Dev 0.011 0.125 0.136 Percent 0% 55% 45% PC3 Mean 0.044 0.506 0.450 Max 0.087 0.752 0.917 Min 0.006 0.076 0.170 St. Dev 0.011 0.108 0.118 Percent 0% 68% 32% Method 2 PC1 Mean 0.062 0.327 0.611 Max 0.095 0.388 0.757 Min 0.031 0.208 0.518 St. Dev 0.018 0.065 0.082 Percent 0% 0% 100% PC2 Mean 0.032 0.360 0.607 Max 0.038 0.424 0.916 Min 0.006 0.078 0.539 St. Dev 0.007 0.077 0.084 Percent 0% 0% 100% PC3 Mean 0.041 0.490 0.469 Max 0.053 0.501 0.562 Min 0.034 0.400 0.455 St. Dev 0.003 0.015 0.016 Percent 0% 93% 7%

352

Table 16 – Model-fitting results of time series analysis of the first three PC axes of the nasal for

Centrosaurus apertus and Styracosaurus albertensis only. Simulations of stratigraphic error using methods 1 and 2 show the summary statistics for 1000 replicates. All values shown (with the exception of the percentages) are Akaike weights (based off of delta AIC) illustrating the probability of each model fitting best. The percentages, illustrate the percentages of replicates for which model was most preferred. Analyses performed using 7 m stratigraphic bins.

Lower GRW URW Stasis PC1 0.035 0.807 0.158 PC2 0.032 0.816 0.152 PC3 0.024 0.61 0.366 Method 1 PC1 Mean 0.082 0.570 0.348 Max 0.169 0.752 0.784 Min 0.018 0.198 0.140 St. Dev 0.024 0.102 0.117 Percent 0% 83% 17% PC2 Mean 0.044 0.448 0.507 Max 0.106 0.704 0.783 Min 0.019 0.192 0.227 St. Dev 0.014 0.098 0.109 Percent 0% 41% 59% PC3 Mean 0.053 0.566 0.381 Max 0.098 0.688 0.734 Min 0.026 0.236 0.255 St. Dev 0.009 0.096 0.101 Percent 0% 80% 20% Method 3 PC1 Mean 0.066 0.610 0.323 Max 0.167 0.822 0.858 Min 0.012 0.112 0.111 St. Dev 0.018 0.153 0.158 Percent 0% 81% 19% PC2 Mean 0.046 0.422 0.532 Max 0.156 0.717 0.891 Min 0.008 0.102 0.207 St. Dev 0.020 0.135 0.149 Percent 0% 38% 62% PC3 Mean 0.055 0.570 0.375 Max 0.179 0.773 0.882 Min 0.012 0.105 0.159 St. Dev 0.012 0.113 0.118 Percent 0% 83% 17%

353

Table 17 – Model-fitting results of time series analysis of the first three PC axes of the postorbital for the entire series (Coronosaurus brinkmani, Centrosaurus apertus, and

Styracosaurus albertensis). Simulations of stratigraphic error using methods 1 and 2 show the summary statistics for 1000 replicates. All values shown (with the exception of the percentages) are Akaike weights (based off of delta AIC) illustrating the probability of each model fitting best.

The percentages, illustrate the percentages of replicates for which model was most preferred.

Analyses performed using 10 m stratigraphic bins.

Lower GRW URW Stasis PC1 0.085 0.378 0.536 PC2 0.007 0.051 0.942 PC3 0.07 0.567 0.363 Method 1 PC1 Mean 0.073 0.356 0.571 Max 0.246 0.656 0.946 Min 0.006 0.047 0.191 St. Dev 0.041 0.126 0.161 Percent 0% 24% 76% PC2 Mean 0.025 0.188 0.787 Max 0.168 0.496 0.975 Min 0.003 0.021 0.367 St. Dev 0.015 0.096 0.110 Percent 0% 1% 99% PC3 Mean 0.127 0.538 0.335 Max 0.512 0.871 0.788 Min 0.025 0.184 0.008 St. Dev 0.085 0.152 0.215 Percent 0% 66% 34% Method 2 PC1 Mean 0.080 0.401 0.519 Max 0.204 0.589 0.857 Min 0.016 0.127 0.213 St. Dev 0.036 0.093 0.125 Percent 0% 33% 67% PC2 Mean 0.015 0.355 0.631 Max 0.025 0.466 0.790 Min 0.010 0.200 0.517 St. Dev 0.003 0.063 0.064 Percent 0% 0% 100% PC3 Mean 0.138 0.542 0.320 Max 0.426 0.842 0.721 Min 0.031 0.248 0.008 St. Dev 0.105 0.184 0.275 Percent 0% 53% 47%

354

Table 18 – Model-fitting results of time series analysis of the first three PC axes of the postorbital for only Centrosaurus apertus and Styracosaurus albertensis. Simulations of stratigraphic error using methods 1 and 2 show the summary statistics for 1000 replicates. All values shown (with the exception of the percentages) are Akaike weights (based off of delta

AIC) illustrating the probability of each model fitting best. The percentages, illustrate the percentages of replicates for which model was most preferred. Analyses performed using 7 m stratigraphic bins for method 1 and 5 m stratigraphic bins for method 2.

Raw - No error GRW URW Stasis PC1 0.052 0.438 0.51 PC2 0.042 0.352 0.607 PC3 0.045 0.549 0.405 Method 1 PC1 Mean 0.055 0.498 0.447 Max 0.129 0.735 0.916 Min 0.009 0.075 0.192 St. Dev 0.018 0.144 0.154 Percent 0% 62% 38% PC2 Mean 0.045 0.422 0.532 Max 0.138 0.745 0.948 Min 0.004 0.047 0.194 St. Dev 0.012 0.130 0.139 Percent 0% 39% 61% PC3 Mean 0.077 0.606 0.318 Max 0.379 0.909 0.634 Min 0.028 0.338 0.012 St. Dev 0.046 0.137 0.168 Percent 0% 78% 22% Method 2 PC1 Mean 0.058 0.536 0.406 Max 0.122 0.701 0.740 Min 0.020 0.240 0.217 St. Dev 0.019 0.098 0.112 Percent 0% 75% 25% PC2 Mean 0.046 0.434 0.520 Max 0.074 0.597 0.699 Min 0.025 0.258 0.347 St. Dev 0.006 0.061 0.065 Percent 0% 28% 72% PC3 Mean 0.072 0.618 0.310 Max 0.242 0.905 0.610 Min 0.030 0.360 0.012 St. Dev 0.044 0.200 0.239 Percent 0% 58% 42%

355

Table 19 – Model-fitting results of time series analysis of the first three PC axes of the parietal for the entire series (Coronsaurus brinkmani, Centrosaurus apertus, and Styracosaurus albertensis). Simulations of stratigraphic error using methods 1 and 2 show the summary statistics for 1000 replicates. All values shown (with the exception of the percentages) are

Akaike weights (based off of delta AIC) illustrating the probability of each model fitting best.

The percentages, illustrate the percentages of replicates for which model was most preferred.

Analyses performed using 10 m stratigraphic bins.

Lower GRW URW Stasis PC1 0.296 0.901 0.025 PC2 0.051 0.62 0.329 PC3 0.063 0.64 0.296 Method 1 PC1 Mean 0.139 0.681 0.180 Max 0.400 0.860 0.839 Min 0.022 0.140 0.003 St. Dev 0.046 0.126 0.157 Percent 0% 93% 70% PC2 Mean 0.125 0.738 0.136 Max 0.234 0.850 0.833 Min 0.023 0.144 0.017 St. Dev 0.023 0.103 0.121 Percent 0% 97% 30% PC3 Mean 0.036 0.198 0.767 Max 0.132 0.503 1.000 Min 0.000 0.000 0.365 St. Dev 0.017 0.079 0.095 Percent 0% 0% 100% Method 2 PC1 Mean 0.037 0.854 0.109 Max 0.041 0.884 0.137 Min 0.034 0.830 0.077 St. Dev 0.001 0.011 0.012 Percent 0% 100% 0% PC2 Mean 0.034 0.905 0.060 Max 0.041 0.927 0.074 Min 0.033 0.890 0.039 St. Dev 0.002 0.010 0.009 Percent 0% 100% 0% PC3 Mean 0.037 0.743 0.220 Max 0.046 0.805 0.261 Min 0.028 0.705 0.164 St. Dev 0.005 0.025 0.023 Percent 0% 100% 0%

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Table 20 – Model-fitting results of time series analysis of the first three PC axes of the parietal for only Centrosaurus apertus and Styracosaurus albertensis. Simulations of stratigraphic error using methods 1 and 2 show the summary statistics for 1000 replicates. All values shown (with the exception of the percentages) are Akaike weights (based off of delta AIC) illustrating the probability of each model fitting best. The percentages, illustrate the percentages of replicates for which model was most preferred. Analyses performed using 7 m stratigraphic bins for method 1 and 5 m stratigraphic bins for method 2.

Lower GRW URW Stasis PC1 0.003 0.993 0.004 PC2 0.007 0.992 0.001 PC3 0.003 0.977 0.021 Method 1 PC1 Mean 0.148 0.685 0.167 Max 0.508 0.861 0.917 Min 0.012 0.071 0.002 St. Dev 0.055 0.154 0.183 Percent 0% 91% 9% PC2 Mean 0.139 0.765 0.096 Max 0.410 0.863 0.730 Min 0.037 0.233 0.003 St. Dev 0.034 0.101 0.116 Percent 0% 98% 2% PC3 Mean 0.029 0.163 0.808 Max 0.106 0.683 0.993 Min 0.001 0.006 0.211 St. Dev 0.018 0.102 0.120 Percent 0% 1% 99% Method 2 PC1 Mean 0.125 0.841 0.035 Max 0.220 0.885 0.276 Min 0.067 0.654 0.009 St. Dev 0.024 0.030 0.038 Percent 0% 100% 0% PC2 Mean 0.082 0.884 0.034 Max 0.125 0.915 0.272 Min 0.060 0.668 0.004 St. Dev 0.009 0.038 0.040 Percent 0% 100% 0% PC3 Mean 0.037 0.386 0.577 Max 0.057 0.626 0.855 Min 0.012 0.133 0.320 St. Dev 0.011 0.106 0.116 Percent 0% 23% 77%

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Figure 1 – Hypothetical illustrations of the evolutionary modes of anagensis (A), forwarded by

Horner et al (1992), and cladogenesis (B), forwarded by Sampson (1995), for the evolution of

Two Medicine Formation centrosaurine ceratopsids (Modified from Horner et al., 1992), and corresponding modes of anagenesis (C) and cladogensis (D) for Belly River Group centrosaurine ceratopsids.

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Figure 2 – Illustration of the stratigraphic occurrence of centrosaurine bonebeds (left) and a selection of centrosaurine specimens (right) within the Belly River Group, in Dinosaur

Provincial Park, Alberta. Specimens are colour coded based on taxonomy (Coronosaurus brinkmani – green, Centrosaurus apertus – grey, Styracosaurus albertensis – red,

Pachyrhinosaurus-like – blue). Bonebeds with significant collections are indicated with a star.

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Figure 3 – Hypothetical evolutionary patterns of a morphological trait in two successive species through stratigraphy (or time). Both A and B show a correlation between trait score and stratigraphy when both taxa are considered. However, when the species are considered independently, situation A shows no correlation (C), whereas satiation B shows and correlation.

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Figure 4 – Schematic representation of the skull of Centrosaurus apertus showing the majority of the morphometric measurements taken. Not all of these measurements were used in analyses, but they consist of the measurements available in the analyses. See Table 1.

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Figure 5 – Schematic representation of a hypothetical stratigraphic column illustrating all methods of stratigraphic uncertainty simulation used in this study, A) Lower, B) Upper, C)

Mehtod 1, D) Method 2. Method 1 uses the measured height as the mean of a normal probability distribution with a set (and consistent) level of variance. Method 2 uses the measured upper and lower limits of each quarry and host unit as the upper and lower limited of the uniform distribution unique to each quarry. In this case, Method 1 will allow for rare reversal of stratigraphic order, and Method 2 will not.

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Figure 6 – List of morphometric variables and their A) completeness (red), B-C) variation (blue) and D-E) asymmetry (green). Variation and asymmetry are shown for all specimens (B and D) and just C. apertus (C and E). Darker colours represent ornamentation. For values see Table 2.

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Figure 7 – Schematic representation of the skull of Centrosaurus apertus showing areas most consistently preserved in the articulated skulls. Elements shown in darker colours are represented more often than those in lighter colours. For values see Figure 6A, and Table 2. Modified from

Lull (1933)

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Figure 8 – Schematic representation of the skull of Centrosaurus apertus showing areas and regions with the highest variation across the entire sample of articulated skulls. Elements shown in darker colours have higher variance than those in lighter colours. Ornamentation associated with the nasal, postorbital and parietal show higher variances than those of the rest of the skull.

For values see Figure 6B, and Table 2.

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Figure 9 – Schematic representation of the skull of Centrosaurus apertus showing areas and regions with the highest variation across articulated skulls of Centrosaurus apertus. Elements shown in darker colours have higher variance than those in lighter colours. Ornamentation associated with the nasal, postorbital and parietal show higher variances than those of the rest of the skull. For values see Figure 6C, and Table 2.

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Figure 10 – Density plots illustrating the distributions of the variation between skull elements/regions (A), and between measurement types (B). The nasal, postorbital, and frill, as well as measures of length show the highest variance. Vertical dotted lines indicate the means for the samples. For values see Figure 6, and Table 2.

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Figure 11 – Schematic representation of the skull of Centrosaurus apertus showing areas and regions with the highest asymmetry across the entire sample of articulated skulls. Elements shown in darker colours have higher asymmetry than those in lighter colours. Asymmetry of hatched areas could not be quantified. Ornamentation of the parietal show higher variances than those of the rest of the skull. For values see Figure 6D, and Table 2.

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Figure 12 – Schematic representation of the skull of Centrosaurus apertus showing areas and regions with the highest asymmetry across articulated skulls of Centrosaurus apertus. Elements shown in darker colours have higher asymmetry than those in lighter colours. Asymmetry of hatched areas could not be quantified. Ornamentation of the parietal show higher variances than those of the rest of the skull. For values see Figure 6E, and Table 2.

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Figure 13 – Plots illustrating the correlation of variation (variation between specimens) and asymmetry (variation between sides of the same skull) for the measurements of all specimens

(A) and Centrosaurus apertus specimens (B). Both samples are strongly and significantly correlated, with measures of ornamentation being both more variable and more asymmetric.

Solid line represent best fit line (OLS), with dashed 95% confidence intervals.

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Figure 14 – Plots illustrating the correlation of variation (variation between specimens) and asymmetry (variation between sides of the same skull) the cranial elements/regions for all specimens (A) and Centrosaurus apertus specimens (B). Both samples are strongly and significantly correlated, with nasal, postorbital and parietal elements being most variable and often asymmetric. Solid lines represent best fit line (OLS), with dashed 95% confidence intervals.

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Figure 15 – Allometry of the nasal horncore in Centrosaurus apertus. A) Plot of height as a function of anteroposterior length of base. B) Plots of anterior height as a function of posterior height. Solid line indicates best fits line, dashed lines indicate 95% confidence intervals, and dotted lines indicate 1:1 scaling . Nasal are coded as unfused (white), partially fused (grey), fused (black), and unknown (grey with horizontal line). Curves above B indicate distribution of the different stages of fusion. Insets highlight the shapes of specific nasals (i - TMP

1996.012.0288, ii - TMP 1992.036.0442, iii - TMP 1981.026.0003, iv - TMP 1966.033.0017, v -

NMH(UK) R 4859, vi - CMN 347, vii - CMN 348, viii - AMNH 5351: not to scale). Star indicates holotype of Styracosaurus albertensis.

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Figure 16 – Plot of postorbital horncore height as a function of anteroposterior length at base of the horncore for all Belly River Group centrosaurine specimens. A) Coded for fusion of the postorbital to the peripheral elements as unfused (white), partially fused (grey), fused (black), and unknown (grey with horizontal line). B) Coded as presence (black), absence (white), or unknown (grey) of a resorption pit on the postorbital. Solid and dashed lines indicate the best fit and 95% confidence intervals of a logarithmic curve. Dotted line indicates the line of isometry.

Grey line indicated a moving average for the data. Curves above the plot indicate the distribution

(and KS-test results) of fusion across the X axes.

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Figure 17 – Regression of postorbital horncore height on anteroposterior length at base of the horncore for Centrosaurus apertus specimens. A) Coded for fusion of the postorbital to the peripheral elements as unfused (white), partially fused (grey), fused (black), and unknown (grey with horizontal line). B) Coded by taxonomy. Solid and dashed lines indicate the best fit and

95% confidence intervals of a linear fiction. Curves above the plot in A indicate the distribution

(and KS-test results) of fusion across the X axes.

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Figure 18 – Plot of mean (and error bars of standard deviation) of postorbital horncore height binned by intervals of basal horncore length for all Centrosaurus apertus specimens. Grey areas indicate the percentage of specimens in each bin to show fusion to peripheral elements (light grey), and development of resorption pit (drak grey). The initial bins (Phase I) show a large increase in height with no fusion or resorption bin. The latter bins (Phase II) show a cessation of growth of height accompanied by fusion and development of resorption pits.

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Figure 19 – Plot of height of epiparietal ossification as a function of their basal width with P1

(A), P2 (B), P3 (C), P4 (D), P5 (E), P6 (F) and P7 (G). Linear model best fit lines are shown for

Centrosaurus (black) and Styracosaurus (red) (for values see Table 3). H illustrates the relative allometric trend of each horn relative to its basal width for both Centrosaurus (grey) and

Styracosaurus (red). Dotted line in H represents the line of isometry.

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Figure 20 – Principal Component Analyses of nine linear measurements of the nasal showing

PC1 vs. PC2 (A) and PC2 vs. PC3 (B). Size of points incites size of element (anteroposterior length of horncore base) and colours indicate taxonomy. Shaded areas indicate convex hulls of taxonomy. Curves above and to the right of the plot illustrate the distribution of the taxa in that axis. Large overlap in morphospace is seen between the taxa in all axes (for KS test results see

Table 5). For loading see Table 4.

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Figure 21 – Proportion of variance explained by each axes (A) and loadings (B) for the PCA of centrosaurine nasals. PC1-3 are considered significant and plotted in Figure 20. Based on the loadings (B) height variables (HT, AL, PL), width variables (LT, TK, CIR) load similarly.

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Figure 22 – Principal Component Analyses of 10 linear measurements of the postorbital , showing PC 1 vs. PC2 (A) and PC2 vs. PC3 (B). Size of points incites size of element

(anteroposterior length of horncore base) and colours indicate taxonomy. Shaded areas indicate convex hulls of taxonomy. Curves above and to the right of the plot illustrate the distribution of the taxa in that axis. Large overlap in morphospace is seen between the taxa in all axes (for KS test results see Table 7). For loadings see Table 6).

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Figure 23 – Proportion of variance explained by each axes (A) and loadings (B) for the PCA of centrosaurine postorbitals. PC1-3 are considered significant and plotted in Figure 22. Based on the loadings (B) height variables (HT, LM, MH), width variables (LTH, WTH, CIR) load similarly.

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Figure 24 – Principal Component Analyses of 45 linear measurements of the parietal, showing

PC 1 vs. PC2 (A) and PC2 vs. PC3 (B). Size of points incites size of element (parietal sagittal length) and colours indicate taxonomy. Thick black lines indicated known stratigraphy. Shaded areas indicate convex hulls of taxonomy. Curves above and to the right of the plot illustrate the distribution of the taxa in that axis. Taxonomic segregation in morphospace is seen between axes

1 and 2 (for KS test results see Table 9). For loadings see Table 8.

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Figure 25 – Proportion of variance explained by each axes (A) and loadings (B) for the PCA of centrosaurine parietals. PC1-3 are considered significant and plotted in Figure 24.

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Figure 26 – Stratigraphic position (and potential range of error) for of each centrosaurine quarry in the area of Dinosaur Provincial Park. On the right are visual representations of test of stratigraphic overlap between the three successive species, utilizing five different stratigraphic options, and including and excluding BB 156. Generally, overlap is present if BB156 is included and not present if BB156 is excluded.

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Figure 27 – Putative Styracosaurus albertensis parietal fragment TMP 1998.068.0033, from BB

156, in lateral (A), dorsal (B), ventral (C), medial (D), distal/posterior (E) and proximal/anterior

(F) views. G, illustrates reconstruction of the element in the complete frill of Styracosaurus albertensis. Dotted lines indicate extrapolated extent of the broken spike.

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Figure 28 – Figures illustrating the effect of the two stratigraphic uncertainty methods (left –

Method 1, right – Method 2) on both correlation of morphology and stratigraphy (top) and time series analysis (bottom) of PC1 parietal scores. Centrosaurus apertus in black and

Styracosaurus albertensis in red. Correlation plots show 100 replicates (and the extreme slope of the simulations), and time series analysis plots show 3 replicates.

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Figure 29 – Significance tests for Pearson correlation between nasal PC scores and stratigraphic position for: All specimens, Centrosaurus + Styracosaurus, Centrosaurus and Styracosaurus.

‘Raw-No error’ indicates ‘Lower’ stratigraphic postion was used. Values for Method 1 and 2 represent means and standard deviation of 1000 replicates.

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Figure 30 – Significance tests for Pearson correlation between postorbital PC scores and stratigraphic position for: All specimens, Centrosaurus + Styracosaurus, Centrosaurus and

Styracosaurus. ‘Raw-No error’ indicates ‘Lower’ stratigraphic postion was used. Values for

Method 1 and 2 represent means and standard deviation of 1000 replicates.

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Figure 31 – Significance tests for Pearson correlation between parietal PC scores and stratigraphic position for: Centrosaurus + Styracosaurus, Centrosaurus and Styracosaurus.

‘Raw-No error’ indicates ‘Lower’ stratigraphic postion was used. Values for Method 1 and 2 represent means and standard deviation of 1000 replicates.

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Figure 32 – Results of a time-series analysis of three evolutionary modes (trend - GRW, random walk - URW , and stasis) for the multivariate results of the centrosaurine nasal. A, C, and E analyze the complete dataset (all taxa), in bins of 10m, whereas B, D and F only include specimens for Centrosaurus and Styracosaurus, in bins of ~7m. For A and B, no stratigraphic error/uncertainty is simulated and the raw stratigraphic positions (lower) are used. For C and D,

Method 1(consistent variance from a measured mean) for simulating stratigraphic uncertainty was used. For E and F, method 2 (site specific upper and lower bounds) for simulating stratigraphic uncertainty was used. For C-F, means of 1000 replicates are illustrated with error bars representing +/- one standard deviation. “*” indicates the evolutionary mode that best fits the data. Horizontal dashed lines indicates the majority probability mark (50%).

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Figure 33 – Results of a time-series analysis of three evolutionary modes (trend GRW, random walk - URW, and stasis) for the multivariate results of the centrosaurine postorbital. A, C, and E analyze the complete dataset (all taxa), in bins of 10m, whereas B, D and F only include specimens for Centrosaurus and Styracosaurus, in bins of ~7m. For A and B, no stratigraphic error/uncertainty is simulated and the lower stratigraphic positions are used. For C and D,

Method 1 (consistent variance from a measured mean) for simulating stratigraphic uncertainty was used. For E and F, method 2 (site specific upper and lower bounds) for simulating stratigraphic uncertainty was used. For C-F, means of 1000 replicates are illustrated with error bars representing +/- one standard deviation. “*” indicates the evolutionary mode that best fits the data. Horizontal dashed lines indicates the majority probability mark (50%).

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Figure 34 – Results of a time-series analysis of three evolutionary modes (trend - GRW, random walk - URW, and stasis) for the multivariate results of the centrosaurine parietal. A, C, and E analyze the complete dataset (all taxa), whereas B, D and F only include specimens for

Centrosaurus and Styracosaurus. For A and B, no stratigraphic error/uncertainty is simulated and the raw stratigraphic positions (lower) are used. For C and D, Method 1(consistent variance from a measured mean) for simulating stratigraphic uncertainty was used. For E and F, method 2

(site specific upper and lower bounds) for simulating stratigraphic uncertainty was used. For C-

F, means of 1000 replicates are illustrated with error bars representing +/- one standard deviation. “*” indicates the evolutionary mode that best fits the data. Horizontal dashed lines indicates the majority probability mark (50%). Due to different sample sizes and uncertainty simulations, B, used 5 m bins, A, C, and D, used 7 m bins, and E and F used 10 m bins.

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Figure 35 – Stratigraphic section of BB 156 (Quarry 124) from the Dinosaur Park Formation, illustrating the stratigraphic position and lithology. BB 156 is located as part of the basal lag of the lowest rhythm of a multistory stacked channel sequence. Black bars indicate the stratigraphic postion simulated by Method 2, and the potential true stratigraphic position relative to the

Centrosaurus apertus and Styracosaurus albertensis stratigraphic zones.

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Appendix A – Morphological measurements for the articulated centrosaurine skulls examined in this study. For a list of abrevations see Table 1. For paired elements, L= left and R= right.

Bone Total Total Total Rostral Rostral Rostral Premax Premax Premax Premax Max Max Max Max Nasal Nasal Nasal Nasal Nasal Nasal Nasal Nasal Abreviation SCL TL PRL RSTH RSTL RSTW PM-WD PM-TW PMPL NRSH MAXL MAXH #ALV TRL H1 NH-AL NH-PL NH-CIR LBASE TK1 AFL PL Side L R L R L R L R L R L R L R L R L R L R L R AMNH 5239 Centrosaurus apertus 861 1452 204 195 147 385 358 175 89 AMNH 5351 Centrosaurus apertus 754 1288 523 194 193 174 172 114 28 21 123 309 301 315 329 319 203 289 287 458 447 451 413 167 165 93 196 AMNH 5377 Centrosaurus apertus 267 162 243 AMNH 5429 Centrosaurus apertus AMNH 5442 162 139 CMN 1173 Centrosaurus apertus 320 140 140 80 CMN 11837 Centrosaurus apertus 850 576 21 356 332 132 218 249 366 130 84 297 150 CMN 11839 Centrosaurus apertus CMN 12229 CMN 344 Styracosaurus albertensis 749 1912 573 562 27 28 189 318 313 237 233 334 356 147 149 251 268 534 566 593 438 169 114 224 51 CMN 348 Centrosaurus apertus 793 1541 576 531 207 201 128 129 98 121 314 333 254 266 197 158 204 227 391 387 151 153 80 231 CMN 8790 Monoclonius lowei 822 1630 589 621 20 20 141 294 207 198 312 264 198 229 202 382 154 173 64 220 137 CMN 8795 Centrosaurus apertus 811 1537 632 668 203 189 156 141 22 21 174 257 263 228 227 336 123 274 200 208 226 378 111 124 334 126 CMN 8797 Centrosaurus apertus CMN 8798 Centrosaurus apertus 721 1325 505 504 163 291 304 228 213 237 337 272 334 114 114 87 208 107 CMN 971 Centrosaurus apertus ROM 1427 Centrosaurus apertus ROM 767 Centrosaurus apertus 702 1253 515 520 162 171 129 169 95 24 24 97 301 310 280 283 357 372 152 141 27 26 282 301 293 301 284 352 144 144 51 176 RTMP 86.126.1 Centrosaurus apertus 1323 173 172 147 142 276 291 143 26 240 260 UALVP 11735 Centrosaurus apertus 792 1469 603 607 177 188 172 181 146 32 30 177 267 247 215 204 342 354 124 119 27 28 295 296 208 260 337 398 119 122 81 288 165 UALVP 16248 Centrosaurus apertus 660 1200 175 165 24 USNM 8897 Centrosaurus apertus 718 1264 184 182 157 162 102 17 18 99 294 285 304 291 321 334 204 198 26 268 CMN 12229 TMP 1993.36.117 24 27 118 287 292 291 303 297 292 352 100 79 TMP 1979.10.05 Centrosaurus apertus TMP 1994.12.968 26 84 318 273 140 134 132 297 117 64 131 TMP 1993.70.01 Centrosaurus apertus TMP 1989.97.01 Styracosaurus albertensis 143 27 254 TMP 2009.90.01 Styracosaurus albertensis 827 1424 542 282 234 167 191 167 131 60 228 TMP 2003.12.168 Styracosaurus albertensis TMP 2005.12.58 Styracosaurus albertensis 234 297 239 362 139 86 TMP 1997.85.1 Centrosaurus apertus 868 1586 665 686 193 188 182 183 169 32 32 205 343 227 364 362 147 29 319 221 271 358 432 155 156 112 321 TMP 1982.16.11 TMP 2005.25.1 235 27 305 332 201 215 221 172 NMH R 4859 Centrosaurus apertus 767 1372 588 234 166 33 348 298 148 231 251 232 411 143 144 94 229 NHM R 8648 Spinops sternbergorum 188 194 193 359 126 131 73 214 NHM - Spinops Par 1 Spinops sternbergorum NHM - Spinops Par 2 Spinops sternbergorum YPM 2015 Centrosaurus apertus 757 1256 607 612 186 209 169 173 136 26 26 142 311 352 311 316 313 352 168 182 26 307 318 303 357 352 111 95 247 LAVAL (CMN 347) Centrosaurus apertus 863 1562 660 675 199 202 179 208 116 29 21 123 377 373 286 296 377 192 173 314 334 372 439 153 162 110 349 113 ROM 43214 762 542 30 137 307 281 344 23 273 262 294 356 169 246 UALVP 47979 Centrosaurus apertus TMP 1988.36.20 Styracosaurus albertensis ROM 1426 CMN 34830 53 84 98 321 127 88 227 68 428 Bone Nasal Nasal Nasal Fontenelle Fontenelle Fontenelle Orbit Orbit Total Total PostO PostO PostO PostO PostO PostO PostO PostO Palp Palp Jugal Abreviation ANL VNL 1/2WTH FFL FFMW FD ORBMXW ORBMNW SKH-PO POW HCH ppt OHC-ML OHC-LL HCL HCW OCH-CIR RPIT? PALL PALW MXL Side L R L R L R L R L R L R L R L R L R L R L R L R L R L R L R L R AMNH 5239 136 138 118 107 70 99 96 70 69 N N 318 AMNH 5351 142 143 64 168 11 126 77 407 419 237 55 52 51 59 52 52 79 78 70 75 174 191 Y Y 128 109 45 51 275 AMNH 5377 125 130 108 110 513 262 100 74 104 89 87 89 307 Y 115 58 307 AMNH 5429 145 37 27 78 68 106 97 256 307 114 92 87 95 116 89 89 75 71 273 269 n n 93 96 57 56 261 AMNH 5442 153 54 40 85 70 101 106 294 266 207 64 47 66 58 52 38 73 71 57 53 187 211 Y Y 87 87 46 44 224 CMN 1173 112 120 95 114 51 60 CMN 11837 135 209 65 83 5 131 88 447 164 111 73 86 81 109 96 102 70 249 Y 63 CMN 11839 CMN 12229 CMN 344 139 137 131 138 114 219 65 83 106 86 93 362 364 279 64 61 61 60 53 43 59 49 65 68 52 61 182 191 y y 93 59 59 272 276 CMN 348 142 141 194 147 183 36 119 109 83 103 315 342 368 164 122 210 104 158 119 95 97 74 72 282 270 N Y 104 99 58 59 291 287 CMN 8790 106 114 122 63 244 41 40 256 45 46 44 44 54 50 49 46 68 56 43 50 162 173 n n 94 97 65 42 236 CMN 8795 126 87 74 21 81 93 106 75 69 348 242 85 71 61 63 86 78 97 94 89 100 82 76 273 283 y n 104 93 70 63 245 262 CMN 8797 125 83 142 62 136 93 73 260 n 108 65 276 CMN 8798 142 141 137 126 81 179 57 73 138 138 98 69 256 55 61 48 47 56 56 57 63 64 53 65 66 178 177 n n 94 57 232 CMN 971 60 51 ROM 1427 268 ROM 767 122 149 174 181 36 173 23 29 130 127 84 93 411 408 180 93 100 94 88 89 95 83 93 60 52 222 228 N N 86 89 54 54 264 273 RTMP 86.126.1 157 56 48 128 118 92 103 262 174 43 43 46 44 60 46 147 N N 86 47 220 218 UALVP 11735 163 164 139 149 81 89 23 83 96 97 75 76 346 313 316 145 145 62 64 137 126 141 142 98 100 76 80 307 315 N N 85 85 56 47 257 273 UALVP 16248 USNM 8897 129 18 118 110 93 101 402 248 52 48 51 53 61 58 42 43 181 y 255 268 CMN 12229 TMP 1993.36.117 TMP 1979.10.05 102 119 88 92 42 45 49 47 41 43 57 61 49 50 168 152 Y Y 79 86 43 49 242 TMP 1994.12.968 123 145 54 TMP 1993.70.01 116 25 89 272 107 88 98 86 103 97 82 82 88 83 281 261 Y Y 93 52 TMP 1989.97.01 229 TMP 2009.90.01 138 146 70 126 105 94 102 92 254 52 55 49 51 56 54 68 71 42 37 154 167 N N 84 85 38 38 237 TMP 2003.12.168 118 104 52 47 52 66 44 161 N 96 44 252 TMP 2005.12.58 43 51 48 56 181 ? 90 63 266 TMP 1997.85.1 118 36 71 122 273 292 66 85 59 64 72 89 103 106 76 85 262 268 Y Y 85 67 300 284 TMP 1982.16.11 TMP 2005.25.1 159 174 NMH R 4859 128 142 165 75 112 16 45 124 110 365 261 11 73 86 52 51 212 Y Y 114 63 56 277 NHM R 8648 172 60 127 31 101 116 213 82 72 77 68 91 82 83 99 71 74 248 272 N Y 75 85 57 51 NHM - Spinops Par 1 NHM - Spinops Par 2 YPM 2015 137 142 153 151 71 123 26 113 117 83 64 409 415 276 88 51 81 51 98 62 94 72 61 264 n y 85 95 49 300 319 LAVAL (CMN 347) 179 153 182 174 58 184 27 148 144 90 122 464 417 242 48 53 82 66 75 73 69 54 118 126 75 75 298 321 y y 85 49 60 304 ROM 43214 144 131 92 128 114 UALVP 47979 81 98 86 89 256 52 53 47 45 67 54 74 71 50 38 176 171 N N 76 75 42 45 232 TMP 1988.36.20 ROM 1426 CMN 34830 135 127 88 113 14 46 86 105 271 66 67 53 51 63 66 67 72 67 72 71 75 236 237 n n 93 56 49 429 Bone Jugal Jugal Jugal Jugal Jugal Predent Predent Dentary Dentary Dentary Dentary Dentary Dentary Dentary Dentary Dentary Braincase Braincase Braincase Braincase Abreviation MNW MDPTX EJ? ITFMXW ITFMNW PD-TL PD-WP DNT-L DNT-TRL DNT-H DNT-TRH #ALV DNT DNTW CPH CPL CTK EXOPW BASI OCMXD OCH Side L R L R L R L R L R L R L R L R L R L R L R L R L R L R L R L R AMNH 5239 129 84 52 250 245 155 70 AMNH 5351 Y 54 31 237 249 136 392 362 275 273 126 132 27 26 221 207 205 34 33 198 164 122 66 AMNH 5377 132 94 80 171 77 AMNH 5429 114 21 n 124 AMNH 5442 108 106 19 19 N 55 59 27 36 CMN 1173 58 CMN 11837 n 69 43 CMN 11839 CMN 12229 282 77 CMN 344 123 113 19 20 n y 58 73 28 30 376 391 289 293 118 26 28 189 192 95 95 35 37 251 256 176 72 CMN 348 213 211 78 72 41 45 242 245 112 388 392 126 126 196 196 89 198 75 CMN 8790 104 99 15 n n 50 33 63 CMN 8795 111 116 23 21 n 58 78 23 20 412 279 111 185 90 41 242 238 152 71 CMN 8797 118 n 80 38 CMN 8798 102 105 n 86 89 47 34 CMN 971 ROM 1427 113 n 71 35 227 221 143 66 65 ROM 767 99 104 12 13 Y Y 71 68 46 43 234 241 119 386 404 292 288 98 108 28 29 26 27 182 80 80 30 30 152 145 122 64 60 RTMP 86.126.1 115 116 14 N N 66 214 372 384 272 274 124 27 27 168 30 31 UALVP 11735 117 125 18 20 N Y 73 67 30 27 250 237 179 68 66 UALVP 16248 210 290 93 28 290 127 33 210 175 124 59 62 USNM 8897 100 99 19 15 N N 73 71 47 40 364 371 245 251 103 104 27 108 26 151 151 99 64 62 CMN 12229 TMP 1993.36.117 TMP 1979.10.05 177 9 10 N 79 42 175 62 TMP 1994.12.968 TMP 1993.70.01 TMP 1989.97.01 143 9 N TMP 2009.90.01 168 N N 44 TMP 2003.12.168 N 65 40 TMP 2005.12.58 151 15 N 67 47 TMP 1997.85.1 195 158 32 25 Y N 87 34 154 174 482 473 333 138 135 28 201 36 257 298 174 73 TMP 1982.16.11 TMP 2005.25.1 NMH R 4859 201 18 Y 73 55 107 63 NHM R 8648 NHM - Spinops Par 1 NHM - Spinops Par 2 YPM 2015 121 120 18 n n 65 59 36 43 210 214 159 321 325 271 302 115 112 26 24 24 94 95 33 32 209 200 142 70 LAVAL (CMN 347) y 70 54 75 ROM 43214 151 74 UALVP 47979 109 N 35 TMP 1988.36.20 ROM 1426 246 66 CMN 34830 430 Bone Braincase Braincase Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Abreviation OCW OCG PBMTK PTL PBTL P1/2W PB# PL# PBW PLTK PFW PFL P1-H P1-OL P1-IL P1-L P1-LT P1-TK P1-CIR P1-dist P1-HT P1-OL P1-IL Side L R L R L R L R L R L L L L L L L L R R R AMNH 5239 285 44 775 511 477 6 127 280 AMNH 5351 219 56 678 515 421 7 6 7 138 22 210 161 308 306 207 245 190 98 73 62 252 237 147 AMNH 5377 77 251 50 687 512 242 139 193 133 53 336 102 AMNH 5429 74 651 548 478 5 7 163 17 264 351 188 132 174 112 51 282 166 267 AMNH 5442 596 563 444 6 6 137 19 248 253 95 31 214 112 CMN 1173 182 263 3 CMN 11837 CMN 11839 557 491 369 6 7 111 23 166 217 215 57 55 35 67 33 173 95 CMN 12229 254 713 582 497 504 6 7 159 273 262 317 343 165 CMN 344 73 227 1066 374 407 8 172 34 270 207 22 12 8 64 92 135 105 CMN 348 237 64 747 582 347 564 4 6 6 178 33 33 79 270 312 303 443 242 257 119 79 364 149 337 231 CMN 8790 63 203 719 612 481 5 7 14 46 45 CMN 8795 72 226 692 463 463 456 5 7 7 154 15 14 216 210 275 289 156 124 137 152 52 364 186 156 113 CMN 8797 17 570 495 355 5 110 CMN 8798 587 451 509 481 6 125 20 296 274 297 289 332 336 302 115 55 111 307 297 CMN 971 578 426 498 4 172 18 248 304 31 304 268 ROM 1427 68 202 532 488 483 6.5 150 309 306 28 25 69 97 ROM 767 66 206 584 441 421 5 7 8 148 19 14 100 152 191 194 195 127 142 88 32 246 111 202 137 RTMP 86.126.1 44 552 442 513 489 4 7.5 7.5 146 15 19 214 232 238 244 67 82 32 31 UALVP 11735 68 212 46 662 499 462 471 6 7.5 7.5 173 21 23 221 231 210 199 293 191 242 146 59 377 116 237 159 UALVP 16248 58 382 317 7 102 USNM 8897 66 215 41 615 452 452 423 14 14 194 193 161 194 82 37 222 93 CMN 12229 682 553 498 487 168 264 272 327 352 236 TMP 1993.36.117 TMP 1979.10.05 191 42 4 7 124 17 19 45 40 38 83 25 186 131 56 43 TMP 1994.12.968 TMP 1993.70.01 586 452 412 7.5 152 19 238 295 247 271 121 214 282 223 TMP 1989.97.01 TMP 2009.90.01 637 396 487 8 174 231 274 TMP 2003.12.168 TMP 2005.12.58 714 392 391 6 7 112 166 198 291 36 35 33 69 183 57 43 28 TMP 1997.85.1 221 24 647 523 478 464 4 6 6 142 19 26 271 243 292 286 160 110 143 126 79 291 171 TMP 1982.16.11 TMP 2005.25.1 NMH R 4859 204 36 512 406 428 5 6.5 22 236 184 217 96 42 238 106 NHM R 8648 NHM - Spinops Par 1 93 102 NHM - Spinops Par 2 771 382 118 111 51 275 93 YPM 2015 71 332 553 443 330 423 6 6 7 155 13 15 215 146 294 270 346 299 296 111 41 291 215 352 278 LAVAL (CMN 347) 74 242 597 473 6 156 134 53 298 131 395 397 354 ROM 43214 73 235 UALVP 47979 464 521 411 446 6 7.5 221 207 43 43 20 58 21 123 130 46 42 TMP 1988.36.20 392 ROM 1426 67 212 552 586 468 439 7 7 203 259 321 224 342 42 38 98 123 52 CMN 34830 431 Bone Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Abreviation P1-L P1-LT P1-TK P1-CIR P1-dist P2-HT P2-OL P2-IL P2-L P2-LT P2-TK P2-CIR P2-dist P2-HT P2-OL P2-IL P2-L P2-LT P2-TK P2-CIR P2-dist P3-HT P3-LT P3-TK Side R R R R R L L L L L L L L R R R R R R R R L L L AMNH 5239 138 AMNH 5351 202 114 82 237 67 178 209 108 116 98 26 224 23 189 202 99 128 79 48 204 12 90 124 28 AMNH 5377 AMNH 5429 262 147 40 137 167 86 94 109 50 262 94 97 51 AMNH 5442 54 51 38 87 20 181 228 21 73 16 CMN 1173 CMN 11837 CMN 11839 79 25 69 69 60 73 168 203 CMN 12229 190 88 186 142 63 72 100 48 94 61 89 35 CMN 344 47 38 15 57 24 134 66 551 116 67 CMN 348 236 120 70 348 302 204 92 141 83 51 224 34 252 142 185 93 45 233 17 40 94 26 CMN 8790 46 89 20 186 86 90 11 27 244 199 41 106 18 CMN 8795 136 133 48 315 168 237 162 145 120 51 286 79 212 164 183 118 49 275 73 57 29 CMN 8797 CMN 8798 272 117 76 119 161 251 107 88 94 60 167 224 93 86 94 81 47 85 CMN 971 275 127 37 254 247 213 113 100 98 40 224 51 166 81 85 91 35 214 64 40 70 22 ROM 1427 21 24 67 18 196 36 65 20 ROM 767 151 89 39 244 152 176 121 106 87 36 214 42 168 106 96 86 32 201 71 RTMP 86.126.1 32 63 69 168 73 85 62 65 65 26 140 71 68 22 149 85 83 24 UALVP 11735 214 149 72 358 135 271 165 110 117 57 295 227 280 163 115 104 58 294 230 44 76 30 UALVP 16248 200 146 68 51 74 36 193 95 USNM 8897 59 38 72 63 205 122 108 109 32 168 115 61 64 36 CMN 12229 86 99 117 TMP 1993.36.117 TMP 1979.10.05 40 68 22 147 139 29 79 15 TMP 1994.12.968 TMP 1993.70.01 276 121 159 245 123 143 118 79 194 104 144 99 70 21 70 TMP 1989.97.01 TMP 2009.90.01 TMP 2003.12.168 TMP 2005.12.58 40 75 194 74 27 19 16 59 144 67 34 27 29 74 179 68 109 TMP 1997.85.1 117 71 67 66 122 47 282 137 44 32 33 85 39 206 148 33 122 24 TMP 1982.16.11 TMP 2005.25.1 NMH R 4859 86 27 221 129 22 77 25 NHM R 8648 NHM - Spinops Par 1 95 102 314 265 98 168 233 217 219 84 52 227 202 NHM - Spinops Par 2 112 67 297 112 283 264 258 99 56 186 307 294 293 110 59 265 224 YPM 2015 312 104 51 255 235 196 115 81 80 41 194 49 149 69 71 92 38 225 9 43 59 21 LAVAL (CMN 347) 395 123 53 298 134 189 116 103 97 45 265 222 190 103 93 97 55 257 221 ROM 43214 UALVP 47979 36 59 22 137 106 50 52 50 101 16 183 202 35 67 10 TMP 1988.36.20 12 12 9 40 119 40 16 16 13 43 94 38 182 137 80 ROM 1426 48 85 151 71 59 44 94 212 74 48 41 97 229 13 71 CMN 34830 432 Bone Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Abreviation P3-dist P3-HT P3-LT P3-TK P3-dist P4-HT P4-LT P4-TK P4-dist P4-HT P4-LT P4-TK P4-dist P5-HT P5-LT P5-TK P5-dist P5-HT P5-LT P5-TK P5-dist P6-HT P6-LT P6-TK Side L R R R R L L L L R R R R L L L L R R R R L L L AMNH 5239 AMNH 5351 187 94 125 27 192 67 122 28 192 46 108 18 321 AMNH 5377 20 28 31 22 63 30 AMNH 5429 51 109 27 322 28 90 23 442 21 58 17 492 AMNH 5442 328 7 98 16 422 6 64 16 445 3 42 15 CMN 1173 CMN 11837 CMN 11839 8 60 287 11 70 328 8 68 355 CMN 12229 311 61 108 301 398 24 77 411 12 53 432 23 66 460 13 61 CMN 344 218 512 97 66 678 337 102 67 643 80 65 41 CMN 348 243 117 103 50 255 20 106 29 312 32 105 20 451 48 91 29 352 23 91 23 502 44 82 30 CMN 8790 361 41 89 16 25 79 16 399 37 96 17 14 CMN 8795 297 35 77 26 311 13 44 19 353 45 66 19 415 41 58 21 433 40 64 17 458 32 48 12 CMN 8797 CMN 8798 293 42 56 14 311 34 63 14 431 46 76 14 406 31 54 11 489 35 67 11 471 22 40 CMN 971 281 48 95 27 309 31 66 25 370 28 82 22 410 23 62 16 451 ROM 1427 318 38 56 17 389 54 15 440 25 57 15 ROM 767 39 74 15 294 21 56 10 299 29 60 9 338 14 53 10 336 21 53 9 366 13 64 8 RTMP 86.126.1 226 98 103 40 272 77 87 21 332 75 70 27 364 82 108 29 449 87 67 30 453 84 91 24 UALVP 11735 320 48 82 32 327 23 58 26 372 31 65 26 379 26 68 23 441 30 71 24 463 19 50 21 UALVP 16248 62 85 23 184 49 69 39 54 USNM 8897 207 52 69 31 187 42 57 26 288 57 256 31 57 18 342 27 56 17 327 13 54 13 CMN 12229 TMP 1993.36.117 TMP 1979.10.05 22 68 16 17 65 16 16 81 17 TMP 1994.12.968 TMP 1993.70.01 314 26 45 35 372 24 67 30 408 24 61 27 TMP 1989.97.01 TMP 2009.90.01 151 86 193 143 78 416 93 84 448 TMP 2003.12.168 TMP 2005.12.58 177 263 113 194 133 94 387 93 90 352 42 87 359 40 89 TMP 1997.85.1 313 32 112 26 264 22 96 21 422 5 88 28 389 22 52 17 483 9 77 18 438 12 29 16 TMP 1982.16.11 TMP 2005.25.1 NMH R 4859 268 23 79 24 343 19 62 22 399 22 65 22 NHM R 8648 NHM - Spinops Par 1 NHM - Spinops Par 2 YPM 2015 214 26 54 19 179 28 79 15 283 26 79 19 237 27 83 14 381 31 77 16 307 28 68 14 LAVAL (CMN 347) ROM 43214 UALVP 47979 283 16 56 10 356 15 59 11 394 TMP 1988.36.20 180 116 76 101 64 226 120 75 413 60 77 50 370 ROM 1426 321 23 61 323 14 74 397 17 72 382 32 63 457 21 71 408 52 CMN 34830 433 Bone Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Parietal Squamosal Squamosal Squamosal Abreviation P6-dist P6-HT P6-LT P6-TK P6-dist P7-HT P7-LT P7-TK P7-dist P7-HT P7-LT P7-TK P7-dist SQMTL SQMML SQ# Side L R R R R L L L L R R R R L R L R L R AMNH 5239 475 AMNH 5351 48 110 18 375 44 103 19 412 358 305 5 AMNH 5377 AMNH 5429 6 41 13 458 AMNH 5442 451 288 315 4 CMN 1173 CMN 11837 CMN 11839 3 60 375 4 61 23 376 CMN 12229 471 24 64 501 20 67 498 324 4 CMN 344 433 14 50 36 398 379 292 5 CMN 348 339 41 85 37 567 499 352 349 346 4 5 CMN 8790 475 13 493 449 379 5 5 CMN 8795 466 61 18 450 12 60 12 467 9 57 12 442 413 372 5 CMN 8797 430 350 5 CMN 8798 504 12 55 13 472 427 406 339 372 5 5 CMN 971 29 46 16 502 ROM 1427 487 352 4.5 ROM 767 386 389 309 286 4 4 RTMP 86.126.1 512 78 75 22 482 64 78 14 503 57 72 19 489 312 334 322 321 4.5 4.5 UALVP 11735 462 21 45 21 465 19 49 20 473 22 46 22 469 425 431 342 325 4.5 5.5 UALVP 16248 33 60 40 51 USNM 8897 367 21 61 13 391 10 63 10 397 381 364 CMN 12229 TMP 1993.36.117 TMP 1979.10.05 23 69 16 17 49 18 21 66 17 441 292 5 TMP 1994.12.968 TMP 1993.70.01 414 37 81 25 411 481 322 5.5 TMP 1989.97.01 TMP 2009.90.01 51 69 454 20 43 458 418 321 4.5 TMP 2003.12.168 486 354 4 TMP 2005.12.58 386 20 51 391 423 252 4 TMP 1997.85.1 473 15 62 23 467 443 437 308 284 5 TMP 1982.16.11 TMP 2005.25.1 NMH R 4859 408 360 264 4.5 NHM R 8648 NHM - Spinops Par 1 NHM - Spinops Par 2 YPM 2015 423 30 74 14 335 7 52 13 365 410 357 324 304 5 5 LAVAL (CMN 347) ROM 43214 UALVP 47979 18 62 11 437 19 46 11 468 372 258 4.5 TMP 1988.36.20 14 56 35 349 ROM 1426 455 10 64 431 10 50 472 11 58 428 CMN 34830 434 435

Appendix B – Morphological measurements for the isolated centrosaurine nasals examined in this study. For a list of abrevations see Table 1. A = Lower, B = Upper, C = HT, D = AL, E =

PL, F = LT, G = TK, H =CIR, I = LTH-post, J = LTH-vent, K = LTH-ant, L = WTH1/2.

436

Spec. Number Taxon Site Age Fusion A B C D E F G H I J K L TMP 1966.33.17 Centrosaurus? subad. tip fused 153 192 158 119 46 276 135 145 139 62 TMP 1992.36.224 Centrosaurus? adult fused 243 244 314 168 99 449 133 126 108 91 TMP1987.018.0020 Centrosaurus apertus JR043 adult fused 10.5 15.0 241 247 243 136 88 359 133 112 65 TMP1982.018.0164 Centrosaurus apertus JR043 subad. unfused 10.5 15.0 146 172 143 129 86 310 126 72 TMP1981.018.0053 Centrosaurus apertus JR043 subad. unfused 10.5 15.0 84 98 104 79 52 202 85 47 TMP 1988.18.19 Centrosaurus? adult fused 10.5 15.0 80 94 95 112 72 302 146 111 157 67 TMP 1979.11.166 Centrosaurus? adult fused 10.5 15.0 146 106 399 70 TMP1982.018.0044 Centrosaurus apertus JR043 adult fused 10.5 15.0 301 308 312 126 72 337 141 TMP1979.011.0083 Centrosaurus apertus JR043 subad. partial 10.5 15.0 190 200 184 103 64 278 55 TMP1987.018.0039 Centrosaurus apertus JR043 adult 10.5 15.0 249 252 248 135 72 325 121 135 121 64 TMP1982.018.0067 Centrosaurus apertus JR043 adult fused 10.5 15.0 212 214 274 152 78 364 95 TMP1993.036.0435 Centrosaurus apertus JR043 unfused 10.5 15.0 106 126 113 103 64 266 106 96 53 TMP1982.18.220 Centrosaurus apertus JR043 subad. unfused 10.5 15.0 129 163 139 112 54 276 41 TMP1980.18.310 Centrosaurus apertus JR043 subad. partial 10.5 15.0 240 241 247 116 84 336 62 TMP1980.18.300 Centrosaurus apertus JR043 adult unknown 10.5 15.0 150 100 408 87 TMP 1991.18.90 Centrosaurus apertus JR043 adult fused 10.5 15.0 286 278 314 145 101 417 111 138 TMP2002.068.0007 Coronosaurus brinkmani JR138 subad. partial -14.2 -12.7 194 207 198 110 62 284 112 161 57 TMP2002.068.0009 Coronosaurus brinkmani JR138 juv. unfused -14.2 -12.7 127 138 134 97 46 212 TMP2002.068.0006 Coronosaurus brinkmani JR138 adult fused -14.2 -12.7 204 207 208 92 82 295 79 TMP2002.068.0008 Coronosaurus brinkmani JR138 subad. partial -14.2 -12.7 145 152 155 118 66 284 98 46 TMP2008.78.369 MRR unfused 123 70 318 63 TMP2002.068.0060 Coronosaurus brinkmani JR138 subad. unfused -14.2 -12.7 105 116 124 85 50 202 111 50 TMP2002.068.0062 Coronosaurus brinkmani JR138 subad. unfused -14.2 -12.7 100 104 106 66 44 178 106 53 TMP2002.068.0071 Coronosaurus brinkmani JR138 subad. unfused -14.2 -12.7 126 147 134 100 46 258 98 120 69 53 TMP2002.068.0061 Coronosaurus brinkmani JR138 subad. unfused -14.2 -12.7 128 147 141 112 56 288 110 83 40 TMP2002.068.0063 Coronosaurus brinkmani JR138 subad. unfused -14.2 -12.7 124 156 127 86 52 244 113 53 TMP2002.068.0070 Coronosaurus brinkmani JR138 fused -14.2 -12.7 174 191 173 119 83 337 63 TMP2002.068.0059 Coronosaurus brinkmani JR138 subad. unf. base -14.2 -12.7 122 54 298 103 TMP2002.068.0058 Coronosaurus brinkmani JR138 subad. partial -14.2 -12.7 215 233 224 121 94 328 97 122 62 TMP1997.062.0017 MMR adult fused 141 151 77 96 83 59 TMP1966.20.17 Centrosaurus? unfused 150 164 152 101 52 296 71 TMP1966.35.41 Centrosaurus? unfused 138 196 150 142 62 342 179 113 54 TMP1978.28.7 Centrosaurus? unfused 108 126 122 98 46 234 TMP1982.16.127 Centrosaurus? adult fused 337 347 342 142 133 135 82 TMP2005.49.135 Centrosaurus? 129 176 134 107 98 538 TMP1999.55.165 Centrosaurus? fused 101 142 106 119 68 292 100 98 64 TMP1994.012.0525 Centrosaurus apertus JR091 adult fused 4.0 4.0 343 337 364 145 110 409 86 TMP1995.401.0045 Centrosaurus apertus JR091 adult fused 4.0 4.0 337 335 339 142 81 393 119 183 68 TMP1995.401.0084 Centrosaurus apertus JR091 subad. partial 4.0 4.0 148 162 149 83 60 202 TMP1992.036.0442 Centrosaurus? JR065 juv. unfused 99 112 97 100 34 222 39 TMP1993.036.0587 Centrosaurus apertus JR031 232 247 245 142 84 360 115 112 74 TMP1981.026.0003 Centrosaurus apertus JR052 unfused 133 137 132 85 54 212 104 90 80 TMP1996.012.0286 Centrosaurus? JR005 juv. unfused 53 62 55 59 22 134 TMP1996.012.0288 Centrosaurus apertus JR041 juv. unfused 3.0 56 65 56 54 28 138 TMP2004.666.01 Centrosaurus? adult fused 228 305 476 128 91 365 86 TMP1981.023.0017 Centrosaurus? JR047 adult fused 141 96 367 TMP1964.5.196 Centrosaurus? adult fused 146 93 384 74 TMP1964.5.231 Centrosaurus? adult fused 162 93 405 132 TMP1964.5.228 Centrosaurus? 131 100 365 73 TMP1981.41.110 Centrosaurus? adult fused 137 86 357 147 TMP1984.163.41 Centrosaurus? subad. unf. base 115 102 348 182 153 112 81 TMP 1965.23.19 Centrosaurus? adult fused 314 314 382 138 82 384 72 TMP 1980.16.449 Centrosaurus? adukt fused 160 107 444 136 TMP2003.12.69 Centrosaurus? unfused 99 112 113 77 50 200 52 41 TMP1981.22.10 Centrosaurus? partial 137 161 154 123 62 300 127 124 50 TMP 2008.12.46 Centrosaurus? subad. unfused 151 197 172 122 44 304 114 59 TMP 1986.36.503 Centrosaurus? 149 83 361 143 TMP1995.400.0187 Centrosaurus apertus JR030 subad. partial 6.0 6.3 116 132 112 94 46 236 139 94 56 TMP1995.175.0019 Centrosaurus apertus JR030 partial 6.0 6.3 218 218 58 88 110 44 TMP1995.400.0074 Centrosaurus apertus JR030 juv. unfused 6.0 6.3 49 57 58 49 24 25 TMP1998.093.0163 Styracosaurus albertensis JR050 unfused 69 82 67 49 28 112 25 TMP66.10.19 Styracosaurus albertensis fused 226 261 217 139 73 345 142 TMP66.10.20 Styracosaurus albertensis adult fused 153 73 398 123 73 TMP66.10.21 Styracosaurus albertensis fused 146 67 381 118 74 TMP66.10.23 Styracosaurus albertensis adult fused 124 63 314 61 TMP66.10.22 Styracosaurus albertensis adult fused 178 72 401 TMP66.10.24 Styracosaurus albertensis 142 81 372 65 TMP 2009.31.01 Styracosaurus albertensis Fryberger subad. fused 152 181 169 131 59 313 99 TMP 2009.31.11 Styracosaurus albertensis Fryberger adult fused 167 99 448 162 USNM 12745 Centrosaurus? adult fused 309 319 445 173 71 421 63 131 CMN 190 Centrosaurus? adult fused 218 214 302 123 97 351 CMN 834 Centrosaurus? adult fused 214 211 224 122 77 342 ROM 49861 Centrosaurus? adult fused 252 243 347 108 94 335 ROM 12776 Centrosaurus? adult fused 344 326 339 147 97 364 104 160 114 ROM 49863 Centrosaurus? adult fused 233 257 237 136 79 357 ROM 640 Centrosaurus? adult fused 10.5 15.0 148 83 388 ROM 831 Centrosaurus? adult fused 10.5 15.0 217 222 262 132 80 377 170 156 ROM 728 Centrosaurus? adult fused 255 249 307 126 71 328 ROM 641 Centrosaurus? adult fused 10.5 15.0 269 263 345 142 95 410 113 ROM 636 Centrosaurus? adult fused 404 423 394 134 82 TMP 2009.39.249 Centrosaurus apertus MBB adult fused 224 249 198 164 55 49 TMP 2009.39.365 Centrosaurus apertus MBB subad. unfused 146 167 137 116 66 TMP 2011.12.13 Centrosaurus? adult fused 246 257 264 141 58 UALVP 47967 Centrosaurus? adult fused 322 341 147 67 342 176

437

Appendix C – Morphological measurements for the isolated centrosaurine postorbitals examined in this study. For a list of abrevations see Table 1. A = Resorption Pit, B = Lower, C = Upper, D

= HT, E = LH, F = MH, G = LTH, H = WTH, I = CIR, J = post, K = Orb-lth, L = ant-tk, M = post-tk.

438

Specimen Number Taxon Site Age Side Fusion A B C D E F G H H I J K L TMP1980.16.1694 v. young right isolated no 10.5 15.0 8 13 9 50 TMP1979.011.0157 Centrosaurus apertus JR043 v. young left isolated no 10.5 15.0 21 21 21 26 12 65 71 14 TMP1982.018.0139 Centrosaurus apertus JR043 v. young left isolated no 10.5 15.0 24 25 22 30 17 72 88 15 TMP1979.011.0020 Centrosaurus apertus JR043 sub-adult left isolated 10.5 15.0 62 62 51 52 30 126 119 31 TMP1980.018.0221 Centrosaurus apertus JR043 sub-adult right partial 10.5 15.0 58 61 47 72 47 174 90 37 47 TMP1980.018.0083 Centrosaurus apertus JR043 adult left fused 10.5 15.0 68 72 74 77 59 235 120 101 42 48 TMP1979.011.0040 Centrosaurus apertus JR043 adult right fused 10.5 15.0 68 78 57 93 62 261 134 103 30 59 TMP1980.018.0295 Centrosaurus apertus JR043 left partial no 10.5 15.0 70 70 56 72 45 186 93 TMP1980.018.0350 Centrosaurus apertus JR043 left partial no 10.5 15.0 66 68 57 77 50 181 105 TMP1989.018.0040 Centrosaurus apertus JR043 left fused yes 10.5 15.0 75 73 56 82 55 227 158 97 35 62 TMP1989.018.0090 Centrosaurus apertus JR043 left partial yes 10.5 15.0 57 63 64 79 65 221 178 94 49 TMP1986.018.0101 Centrosaurus apertus JR043 left partial yes 10.5 15.0 58 57 57 70 50 204 109 84 28 39 TMP1986.018.0058 Centrosaurus apertus JR043 left isolated no 10.5 15.0 68 65 62 50 43 145 124 32 TMP1979.011.0100 Centrosaurus apertus JR043 left isolated no 10.5 15.0 57 56 51 34 35 TMP1982.018.0262 Centrosaurus apertus JR043 left isolated no 10.5 15.0 63 66 59 54 39 169 110 30 TMP1979.011.0084 Centrosaurus apertus JR043 left isolated ? 10.5 15.0 69 67 64 67 55 196 30 42 TMP 1983.18.37 Centrosaurus apertus JR043 left isolated yes 10.5 15.0 74 79 54 78 57 216 39 57 TMP1979.011.0128 Centrosaurus apertus JR043 old adult left no 10.5 15.0 74 79 74 74 61 224 33 50 TMP1979.011.0041 Centrosaurus apertus JR043 old adult left yes 10.5 15.0 59 73 59 83 56 226 36 47 TMP1982.018.0017 Centrosaurus apertus JR043 subadult left isolated no 10.5 15.0 64 58 57 59 46 171 129 24 36 TMP1982.018.0002 Centrosaurus apertus JR043 adult left partial no 10.5 15.0 85 81 54 83 55 201 118 28 52 TMP1980.018.0309 Centrosaurus apertus JR043 left partial no 10.5 15.0 55 59 59 62 43 181 149 95 TMP1979.011.0129 Centrosaurus apertus JR043 adult left no 10.5 15.0 67 68 62 78 58 219 TMP1979.011.0066 Centrosaurus apertus JR043 subadult right isolated no 10.5 15.0 67 66 63 53 41 160 34 33 TMP1979.011.0120 Centrosaurus apertus JR043 adult right no 10.5 15.0 98 96 81 76 61 230 55 TMP1989.018.0009 Centrosaurus apertus JR043 no 10.5 15.0 62 59 62 63 46 172 74 26 38 TMP1982.018.0104 Centrosaurus apertus JR043 adult left fused no 10.5 15.0 85 84 86 82 63 224 35 55 TMP1980.018.0303 Centrosaurus apertus JR043 old right fused yes 10.5 15.0 81 94 78 92 66 246 124 96 35 60 TMP1989.018.0024 Centrosaurus apertus JR043 adult left fused no 10.5 15.0 71 71 66 87 62 251 129 36 54 TMP1989.018.0004 Centrosaurus apertus JR043 adult left fused? ? 10.5 15.0 75 49 122 33 38 TMP1979.011.0085 Centrosaurus apertus JR043 old left fused yes 10.5 15.0 74 68 57 75 66 222 97 41 56 TMP1980.018.0349 Centrosaurus apertus JR043 adult right fused yes 10.5 15.0 89 88 68 81 66 235 61 TMP1986.018.0050 Centrosaurus apertus JR043 adult? left yes 10.5 15.0 55 58 49 67 49 185 37 TMP1979.011.0089 Centrosaurus apertus JR043 adult right fused 10.5 15.0 96 99 86 86 68 252 89 37 63 TMP1979.011.0080 Centrosaurus apertus JR043 adutl/old right fused 10.5 15.0 67 72 45 63 53 173 86 39 54 TMP1979.011.0031 Centrosaurus apertus JR043 adult left fused 10.5 15.0 77 71 59 71 45 185 156 42 TMP1979.011.0163 Centrosaurus apertus JR043 adult ? fused? 10.5 15.0 91 92 79 46 224 31 TMP1980.018.0322 Centrosaurus apertus JR043 old adult right fused 10.5 15.0 61 67 46 73 54 196 49 TMP1980.018.0016 Centrosaurus apertus JR043 v. young right unfused 10.5 15.0 22 23 19 26 13 76 15 TMP1980.018.0315 Centrosaurus apertus JR043 subadult left unfused 10.5 15.0 71 68 64 66 41 186 26 33 TMP1979.011.0081 Centrosaurus apertus JR043 adult right fused 10.5 15.0 93 99 92 83 62 247 92 39 58 TMP1989.018.0064 Centrosaurus apertus JR043 subadult right isolated no 10.5 15.0 67 63 59 54 34 151 26 31 TMP1980.018.0108 Centrosaurus apertus JR043 subadult right isolated no 10.5 15.0 75 75 64 59 42 161 36 46 TMP1979.011.0117 Centrosaurus apertus JR043 subadult right isolated no 10.5 15.0 52 47 45 45 29 121 TMP1981.018.0029 Centrosaurus apertus JR043 subadult right isolated no 10.5 15.0 89 88 85 71 45 189 39 50 TMP1980.018.0236 Centrosaurus apertus JR043 adult right fused yes 10.5 15.0 53 62 47 77 59 225 50 44 TMP1979.011.0042 Centrosaurus apertus JR043 adult right fused yes 10.5 15.0 55 62 54 67 62 225 32 TMP1982.018.0085 Centrosaurus apertus JR043 subadult left isolated? no 10.5 15.0 89 88 83 60 50 174 49 TMP1986.018.0043 Centrosaurus apertus JR043 adult? left isolated? no 10.5 15.0 90 92 82 80 50 208 TMP1980.018.0002 Centrosaurus apertus JR043 subadult right isolated no 10.5 15.0 52 39 141 TMP1994.012.0524 Centrosaurus apertus JR091 subadult right partial no 4.0 4.0 94 92 82 71 59 224 37 46 TMP1988.036.0033 Centrosaurus apertus JR091 adult/sub right isolated no 4.0 4.0 85 83 83 68 56 202 51 TMP1992.036.1017 Centrosaurus apertus JR091 subadult left isolated no 4.0 4.0 70 67 77 55 43 183 24 37 TMP1995.401.0107 Centrosaurus apertus JR091 subadult right isolated no 4.0 4.0 54 51 56 47 33 128 27 TMP1995.401.0044 Centrosaurus apertus JR091 adult left fused no 4.0 4.0 82 79 74 85 69 256 TMP1995.401.0004 Centrosaurus apertus JR091 adult right fused ? 4.0 4.0 84 62 236 191 94 35 57 TMP1995.401.0002 Centrosaurus apertus JR091 old adult right fused yes 4.0 4.0 59 58 54 78 63 251 104 33 57 TMP1997.012.0213 Centrosaurus? JR060 adult left partial no 95 99 73 75 60 238 34 50 TMP1994.012.0154 Centrosaurus? JR060 adult? left isolated? no 91 96 99 58 53 175 TMP1994.012.0942 Centrosaurus? JR028 right isolated no 55 51 49 53 37 143 149 20 30 TMP1994.012.0940 Centrosaurus? JR028 left partial no 78 78 66 73 54 212 82 26 46 TMP1994.012.0838 Centrosaurus? JR028 adult right fused no 85 81 81 71 67 204 34 50 TMP1988.036.0269 Centrosaurus? JR090 adult right fused no 112 113 90 83 67 247 89 37 55 TMP1965.23.27 old adult right fused yes 51 61 47 73 61 203 34 51 TMP1965.23.27 old adult right fused yes 54 66 46 64 59 193 108 41 52 TMP1965.12.5 old adult left fused yes 59 57 54 71 59 131 81 33 49 TMP1965.023.0010 adult left fused no 58 48 49 51 46 154 52 TMP1982.016.0178 Centrosaurus? JR076 adult right fused no 98 99 71 75 70 253 57 TMP1981.022.0013 Centrosaurus? JR046 subadult right isolated no 52 51 50 43 31 122 26 TMP1982.019.0244 Centrosaurus apertus JR061 old adult right fused yes 48 54 41 61 45 192 44 25 TMP1992.036.0398 Centrosaurus? JR050 subadult right isolated 64 61 57 49 29 134 25 TMP1992.036.0650 Centrosaurus apertus JR130 adult left fused? no 71 71 97 70 58 211 24 43 TMP1990.057.0026 Centrosaurus apertus JR030 adult right fused no 6.0 6.3 54 56 61 71 49 219 29 47 TMP1995.12.145 Centrosaurus apertus BB30? adult right fused yes 6.0 6.3 72 73 93 97 60 258 153 103 30 50 TMP1995.400.0256 Centrosaurus apertus JR030 juvenile right isolated no 6.0 6.3 68 62 64 49 29 126 30 TMP1995.400.0114 Centrosaurus apertus JR030 juv. (tiny) left isolated no 6.0 6.3 38 36 35 32 21 93 18 TMP1995.400.0164 Centrosaurus apertus JR030 juv. (tiny) left isolated 6.0 6.3 37 36 35 31 21 96 21 TMP1994.12.387 Centrosaurus apertus JR30 Adult left fused no 6.0 6.3 76 81 72 77 58 220 34 54 TMP1981.019.0024 Centrosaurus? JR016 adult/sub right isolated? no 67 64 56 62 41 174 39 TMP2008.079.0047 Centrosaurus? JR168 subadult right isolated no 1.0 4.0 66 67 68 52 36 157 33 TMP1998.93.34 left partial no 65 63 55 64 45 174 126 85 20 35 TMP2005.9.7 left isolated no 68 73 61 63 48 196 32 43 TMP1993.666.44 left partial no 77 76 75 77 58 211 95 29 57 TMP 1987.78.15 subadult left partial no 72 73 57 41 27 117 36 TMP2004.72.06 old adult left yes 37 41 37 73 50 191 49 TMP2002.12.77 subadult left isolated no 69 66 64 60 39 170 33 TMP2006.12.240 old adult left fused yes 50 68 45 74 61 246 38 54 TMP2007.12.59 old adult left fused no 62 64 47 67 56 178 34 TMP1994.12.472 subadult ? fused no 65 66 58 63 46 194 42 439

Specimen Number Taxon Site Age Side Fusion A B C D E F G H H I J K L TMP 1994.12.158 subadult right isolated no 64 62 59 51 42 146 38 TMP1999.55.260 adult ? fused yes 103 108 98 80 60 234 TMP1980.16.1347 old adult right fused yes 59 64 52 78 59 225 42 59 TMP1967.20.234 adult right fused yes 60 61 63 66 51 177 127 87 23 47 TMP1967.14.2 old adult right fused yes 58 59 38 80 56 202 54 35 TMP1967.8.13 adult right fused no 83 81 61 79 59 235 37 TMP1979.14.10 old adult right fused yes 67 72 71 83 72 254 59 TMP1980.16.1043 subadult right isolated no 61 61 58 48 32 135 31 22 TMP1967.20.23 subadult right isolated no 70 66 62 49 39 152 32 TMP1980.16.1677 subadult right isolated no 81 78 79 58 43 35 TMP1987.48.8 subadult right isolated no 65 72 49 66 42 185 22 TMP1991.50.124 subadult right isolated no 75 76 74 63 45 181 35 TMP1979.8.742 adult? right isolated yes 66 66 64 63 45 186 32 39 TMP1990.36.434 old adult left fused no 50 47 42 60 48 165 35 52 TMP.1992.36.856 old adult left fused ? 69 71 67 68 47 202 32 42 TMP1986.36.315 old adult right fused no 64 65 43 68 50 197 39 58 TMP1992.36.786 adult/sub left isolated no 79 79 83 71 57 202 46 TMP1982.16.12 old adult left fused yes 70 72 53 79 68 236 42 61 TMP1981.19.6 old adult left fused yes 36 54 43 69 60 201 27 40 TMP1980.16.673 adult/old left fused yes 77 82 65 72 69 233 32 59 TMP1979.14.180 subadult left isolated no 78 77 74 50 37 135 TMP1981.16.80 adult/old left fused no 98 92 88 85 67 246 33 52 TMP1980.16.515 adult/old left fused yes 91 94 79 78 69 231 41 TMP1983.126.6 old adult left fused ? 81 87 64 89 81 268 56 TMP1997.12.192 subadult right isolated no 85 84 81 53 40 154 32 TMP2007.20.36 subadult right isolated no 54 57 57 55 37 147 31 TMP2007.20.78 subadult right isolated no 40 37 38 40 34 115 24 TMP2007.20.38 subadult left isolated no 70 71 66 48 32 129 28 TMP2007.29.40 adult/old right isolated yes 60 64 48 69 50 181 24 46 TMP2007.20.119 adult left isolated yes 80 76 75 80 70 245 31 47 TMP1998.68.8 adult left fused? yes 78 79 67 84 71 256 35 59 TMP1994.172.101 adult right fused yes 56 61 53 61 60 174 35 TMP1985.47.4 adult left fused yes 86 89 82 85 73 284 TMP1980.16.1047 subadult right isolated no 68 67 65 55 41 152 37 TMP1983.56.30 adult right fused ? 54 58 42 57 52 184 37 52 TMp1985.56.33 subadult left no 66 68 56 61 46 185 48 TMP1986.36.217 subadult left isolated no 71 71 63 58 41 181 37 TMP 2003.12.168 S. albertensis Adult left fused no 29.0 30.0 53 52 54 70 54 169 156 108 18 48 TMP2005.12.67 S. albertensis adult left fused no 29.0 30.0 56 61 57 75 61 214 83 25 40 TMP1998.93.64 S. albertensis subadult right partial no 29.0 30.0 53 54 43 54 41 148 131 99 17 39 TMP2002.70.1 S. albertensis BB42 old adult left fused yes 29.0 30.0 45 51 48 61 63 212 115 25 48 TMP2009.031.0012 S. albertensis JR301 subadult right isolated no 29.0 30.0 58 58 52 55 40 164 41 TMP2002.068.0011 Coronosaurus brinkmani JR138 subadult left isolated -14.2 -12.7 64 68 65 51 36 138 143 34 TMP2002.068.0010 Coronosaurus brinkmani JR138 subadult right isolated -14.2 -12.7 81 84 73 60 35 164 176 39 TMP2002.068.0005 Coronosaurus brinkmani JR138 adult right fused -14.2 -12.7 78 79 114 68 64 220 146 89 25 51 TMP2002.068.0012 Coronosaurus brinkmani JR138 adult right fused -14.2 -12.7 87 84 91 66 61 202 160 88 37 52 TMP2002.068.0013 Coronosaurus brinkmani JR138 adult right fused -14.2 -12.7 83 79 99 63 61 210 144 28 45 TMP1999.82.0002 Coronosaurus brinkmani adult left fused -14.2 -12.7 90 113 165 73 77 243 27 45 TMP2002.068.0028 Coronosaurus brinkmani JR138 sub right isolated? no -14.2 -12.7 65 63 71 49 41 140 30 TMP2002.068.0031 Coronosaurus brinkmani JR138 adult/sub left fused no -14.2 -12.7 68 69 81 63 53 183 137 77 17 42 TMP2002.068.0021 Coronosaurus brinkmani JR138 adult/sub left fused no -14.2 -12.7 73 71 92 69 41 181 84 21 39 TMP2002.068.0032 Coronosaurus brinkmani JR138 adult/sub right partial no -14.2 -12.7 69 72 81 76 52 191 125 90 26 42 TMP2002.068.0030 Coronosaurus brinkmani JR138 adult left fused yes -14.2 -12.7 62 62 58 67 62 208 146 33 46 TMP2002.068.0018 Coronosaurus brinkmani JR138 adult right fused no -14.2 -12.7 98 102 106 73 59 196 161 97 31 48 TMP2002.068.0033 Coronosaurus brinkmani JR138 old left fused yes -14.2 -12.7 58 71 57 81 58 226 164 116 27 49 TMP2001.12.230(20) Coronosaurus brinkmani JR138 subadult right isolated no -14.2 -12.7 69 51 78 68 54 171 147 37 TMP1979.010.0005 old adult right fused yes 46 46 41 69 49 182 92 25 44 TMP1999.82.02 MRR old adult left fused 66 78 62 67 69 219 TMP2008.78.176 MRR subadult left isolated no 69 72 67 47 30 127 102 33 TMP1997.62.39 MRR subadult left isolated no 65 59 48 52 31 141 120 29 TMP1997.129.6 MRR subadult right isolated no 57 58 51 35 25 107 26 TMP2008.78.314 MRR adult right isolated no 61 43 175 120 22 42 TMP2008.78.327 MRR subadult left fused no 62 64 73 60 32 165 146 40 TMP2008.78.315 MRR juv right isolated no 59 58 52 35 33 107 22 TMP2008.78.285 MRR no 60 65 53 42 26 107 25 TMP.2008.78.286 MRR no 52 51 49 43 24 114 27 TMP 1992.78.5 MRR subadult left isolated 86 82 84 44 41 134 TMP 2009.39.281 Centrosaurus apertus MBB adult right fused 86 82 48 99 32 53 TMP2002.69.10 Albertaceratops nesmoi subadult right isolated 121 124 108 52 58 171 UALVP 49475 Centrosaurus apertus BB41 subadult right isolated 3.0 6.0 64 57 55 63 42 185 UALVP 52648 Centrosaurus apertus BB41 juv right isolated 3.0 6.0 51 43 48 51 34 142 56 23 UALVP 53702 subadult left 85 84 78 60 54 188 USNM 12742 adult left fused 57 53 48 65 41 153 178 56 USNM 12742 adult right fused 59 60 55 63 44 168 174 41 53 ROM 12787 Centro adult left fused yes 77 77 75 83 75 237 85 35 56 440

Appendix D – Morphological measurements for the isolated and articulated centrosaurine parietals examined in this study. For a list of abrevations see Table 1.

441

Spec. Number Taxon Site Lr. Upr. SCL TL PBMTK PTL PBTL P1/2W PB# PL# PBW PLTK PFW PFL P1-OL AMNH 5239 Centrosaurus apertus 6.0 8.0 861 1452 44 775 511 477 6 127 AMNH 5351 Centrosaurus apertus 26.7 28.7 754 1288 56 678 515 421 7 6.5 138 22 185.5 307 241 AMNH 5377 Centrosaurus apertus 50 687 512 242 AMNH 5429 Centrosaurus apertus 651 548 478 5 7 163 17 264 351 227.5 AMNH 5442 Monoclonius sp. 596 563 444 6 6 137 19 248 253 CMN 11839 Centrosaurus apertus 557 491 369 6 7 111 23 166 216 57 CMN 12229 Centrosaurus apertus 713 582 500.5 6.5 159 267.5 330 165 CMN 344 Styracosaurus albertensis 43.5 47.5 749 1912 1066 374 407 8 172 34 270 207 22 CMN 348 Centrosaurus apertus 793 1541 64 747 582 455.5 4 6 178 33 174.5 307.5 390 CMN 8790 Monoclonius lowei 822 1630 719 612 481 5 7 14 46 CMN 8795 Centrosaurus apertus 1.5 3.5 811 1537 692 463 459.5 5 7 154 14.5 213 282 156 CMN 8798 Centrosaurus apertus 10.5 17.5 721 1325 587 451 495 6 125 20 285 293 316.5 CMN 971 Centrosaurus apertus 578 426 498 4 172 18 248 304 304 ROM 1427 Centrosaurus apertus 8.0 8.0 532 488 483 6.5 150 309 306 28 ROM 767 Centrosaurus apertus 4.5 10.5 702 1253 584 441 421 5 7.5 148 16.5 126 192.5 198.5 RTMP 86.126.1 Styracosaurus albertensis 42.3 45.3 1323 44 552 442 501 4 7.5 146 17 223 241 32 UALVP 11735 Centrosaurus apertus 2.0 11.3 792 1469 46 662 499 466.5 6 7.5 173 22 226 204.5 265 UALVP 16248 Centrosaurus apertus 25.0 27.0 660 1200 382 317 7 102 USNM 8897 Centrosaurus apertus 22.0 25.0 718 1264 41 615 452 437.5 14 193 TMP 1979.10.05 Centrosaurus apertus 42 4 7 124 18 50.5 TMP 1993.70.01 Centrosaurus apertus 586 452 412 7.5 152 19 238 288.5 TMP 2009.90.01 Styracosaurus albertensis 48.0 52.4 827 1424 637 396 487 8 174 231 274 TMP 2005.12.58 Styracosaurus albertensis 37.0 41.8 714 392 391 6 7 112 182 291 39.5 TMP 1997.85.1 Centrosaurus apertus 6.5 6.5 868 1586 24 647 523 471 4 6 142 22.5 257 289 160 TMP 1982.16.11 Centrosaurus apertus 5.5 6.8 18 204 166 6 31 18 36 6 NMH R 4859 Centrosaurus apertus 25.0 30.3 767 1372 36 512 406 428 5 6.5 22 236 NHM - Spinops Par 1 Spinops sternbergorum NHM - Spinops Par 2 Spinops sternbergorum 771 382 118 YPM 2015 Centrosaurus apertus 11.0 11.0 757 1256 553 443 376.5 6 6.5 155 14 180.5 282 349 LAVAL (CMN 347) Centrosaurus apertus 0.0 7.0 863 1562 597 473 6 156 397 UALVP 47979 Centrosaurus apertus 464 521 428.5 6 7.5 221 207 44.5 TMP 1988.36.20 Styracosaurus albertensis 45.0 48.0 392 ROM 1426 Centrosaurus apertus 5.0 9.5 552 586 453.5 7 203 290 283 47 TMP 1994.182.01 Centrosaurus apertus 643 513 6 248 337 279 TMP 1992.82.01 Centrosaurus apertus 1228 495 412 374 7 6 122 167 247 82.5 TMP 1980.54.01 Centrosaurus apertus 574 521 456 7 122 212 232 30 TMP 1987.52.1 35.0 38.0 475 8 141 222 AMNH 3998 Monoclonius crassus 538 476 418 4 6 120 14 241 322 26 TMP 1988.36.20 Styracosaurus albertensis 392 TMP 1984.93.1 Styracosaurus albertensis 47 552 232 4 TMP 1978.6.01 Centrosaurus apertus 727 514 506 5 7 136 17 276 342 245 TMP 1995.666.36 Centrosaurus apertus 6 125 TMP 2002.68.02 Coronosaurus brinkmani BB 138 -14.2 -12.7 482 426 5 140 43 TMP 2008.79.23 Centrosaurus apertus BB 168 1.0 4.0 150 TMP 2008.79.15 Centrosaurus apertus BB 168 1.0 4.0 TMP 2008.79.184 Centrosaurus apertus BB 168 1.0 4.0 198 TMP 1966.10.6 Styracosaurus albertensis TMP 1981.19.222 Styracosaurus albertensis BB 42 29.0 30.0 TMP 1966.10.28 Styracosaurus albertensis 32 TMP 1991.36.254 Styracosaurus albertensis 37 101 33 TMP 1966.10.9 Styracosaurus albertensis 18 TMP 1981.19.157 Styracosaurus albertensis 30 TMP 1981.19.160 Styracosaurus albertensis 32 TMP 1966.10.4 Styracosaurus albertensis 29 46 TMP 2001.12.4 Styracosaurus albertensis BB 42 29.0 30.0 26 101 TMP 1999.55.2 Styracosaurus albertensis BB 42 29.0 30.0 31 87 74 TMP 1999.55.5 Styracosaurus albertensis 31 110 TMP 1981.18.213 Centrosaurus apertus 25 151 38 TMP 1981.16.115 Centrosaurus apertus TMP 2005.09.69 Styracosaurus albertensis 45 234 3 94 52 TMP 2005.09.74 Centrosaurus apertus BB5 TMP 1966.32.34 Centrosaurus apertus Hilda TMP 1995.401.0036 Centrosaurus apertus BB91 1.5 3.5 TMP 1995.400.160 Centrosaurus apertus BB30 6.0 6.3 TMP 1995.400.54 Centrosaurus apertus BB31 6.0 6.3 462 413 4 101 44 TMP 1995.175.21 Centrosaurus apertus BB32 6.0 6.3 529 518 7 104 41 TMP 1979.11.76 Centrosaurus apertus BB43 10.5 15.0 26 92 37 TMP 1982.18.277 Centrosaurus apertus BB43 10.5 15.0 23 125 213 TMP 1982.18.79 Centrosaurus apertus BB43 10.5 15.0 41 148 232.5 TMP 2002.68.003 Coronosaurus brinkmani BB 138 -14.2 -12.7 96 TMP 2002.68.014 Coronosaurus brinkmani BB 138 -14.2 -12.7 93 TMP 1999.82.01 Coronosaurus brinkmani BB 138 -14.2 -12.7 40 124 101 TMP 1992.30.41 MRR 31 TMP 1984.93.05 Coronosaurus brinkmani BB42 29.0 30.0 57 TMP 1994.12.607 Centrosaurus apertus Hilda 25 127 109 TMP 1979.11.19 Centrosaurus apertus BB43 10.5 15.0 40 160 USNM 7950 Brachceratops 21 299 293 5 40 USNM 11869 Styracosaurus ovatus 41 3 442

Spec. Number P1-IL P1-L P1-LT P1-TK P1-CIR P1-dist P2-OL P2-IL P2-L P2-LT P2-TK P2-CIR P2-dist P3-HT P3-LT P3-TK AMNH 5239 280 138 AMNH 5351 147 196 106 77.5 237 64.5 205.5 103.5 122 88.5 37 214 17.5 92 124.5 27.5 AMNH 5377 139 193 133 53 336 102 AMNH 5429 132 218 129.5 45.5 282 151.5 167 86 94 103 50.5 262 94 51 109 27 AMNH 5442 95 31 214 112 54 51 38 87 20 181 228 21 73 16 CMN 11839 55 35 67 33 173 95 69 69 60 76 25 168 203 8 60 CMN 12229 190 88 186 142 63 72 100 48 94 61 98.5 35 CMN 344 12 8 64 92 135 105 47 38 15 57 24 134 66 551 116 67 CMN 348 236.5 246.5 119.5 74.5 356 225.5 228 117 163 88 48 228.5 25.5 78.5 98.5 38 CMN 8790 45 46 89 20 186 86 90 11 27 244 199 41 97.5 17 CMN 8795 118.5 136.5 142.5 50 339.5 177 224.5 163 164 119 50 280.5 76 35 67 27.5 CMN 8798 287 116 65.5 115 237.5 100 87 94 70.5 44.5 70.5 14 CMN 971 268 275 127 37 254 247 189.5 97 92.5 94.5 37.5 219 57.5 44 82.5 24.5 ROM 1427 25 69 97 21 24 67 18 196 36 65 20 ROM 767 132 146.5 88.5 35.5 245 131.5 172 113.5 101 86.5 34 207.5 56.5 39 74 15 RTMP 86.126.1 31 32 65 69 168 77.5 85 62 65 66.5 24 144.5 78 98 93 32 UALVP 11735 175 228 147.5 65.5 367.5 125.5 275.5 164 112.5 110.5 57.5 294.5 228.5 46 79 31 UALVP 16248 200 146 68 51 74 36 193 95 62 85 23 USNM 8897 161 194 70.5 37.5 222 93 122 108 90.5 47.5 186.5 115 56.5 66.5 33.5 TMP 1979.10.05 41.5 39 75.5 23.5 166.5 135 29 79 15 TMP 1993.70.01 235 273.5 121 186.5 219.5 113.5 143.5 108.5 74.5 21 70 TMP 2009.90.01 151 86 TMP 2005.12.58 31.5 36.5 72 188.5 65.5 30.5 23 22.5 66.5 161.5 67.5 263 111 TMP 1997.85.1 110 143 121.5 79 291 171 57.5 49.5 49.5 103.5 43 244 142.5 32.5 117 25 TMP 1982.16.11 6 6 16 7 42 6 6 6 22 5 62 5 20 6 NMH R 4859 184 217 96 42 238 106 86 27 221 129 22 77 25 NHM - Spinops Par 1 94 102 273.5 217 242 91 52 227 185 NHM - Spinops Par 2 111.5 59 286 102.5 295 279 275.5 104.5 57.5 265 205 YPM 2015 288.5 304 107.5 46 273 225 172.5 92 76 86 39.5 209.5 29 34.5 56.5 20 LAVAL (CMN 347) 354 395 128.5 53 298 132.5 189.5 109.5 98 97 50 261 221.5 UALVP 47979 42.5 28 58.5 21.5 130 118 50 52 50 101 16 183 202 35 67 10 TMP 1988.36.20 14 14 11 41.5 106.5 39 182 126.5 78 ROM 1426 43 91.5 137 72.5 53.5 42.5 95.5 220.5 18 66 TMP 1994.182.01 229 125 156 312 179 114 104 62 TMP 1992.82.01 64 59 78.5 28 123 69 55.5 47.5 69.5 24.5 131 36 71 TMP 1980.54.01 31 28.5 84.5 19.5 166 115.5 45.5 47 31 67.5 12 132 209.5 29 56.5 8 TMP 1987.52.1 12 45 61 17 49 47 356 122 AMNH 3998 24 22 83 24 134 33 32 31 66 16 132 213 15 59 12 TMP 1988.36.20 14 14 11 41.5 106.5 39 182 126.5 78 TMP 1984.93.1 58 92 35 203 135 73.5 36 201.5 187 98 57 TMP 1978.6.01 175 199 114 50 285 151 233 157.5 165 105 54 21 84 107 27 TMP 1995.666.36 110.5 44.5 265 114 174 83 93 107.5 37.5 248.5 80 TMP 2002.68.02 35 31 103.5 27.5 209.5 182.5 TMP 2008.79.23 99 119 80 41 196 158 124 71 87 72 46 181 66 TMP 2008.79.15 95 31 234 129 95 53 48 78 46 211 152 TMP 2008.79.184 139 172 103 37 228 138 TMP 1966.10.6 TMP 1981.19.222 TMP 1966.10.28 50 31 151 95 57 27 141 114 40 53 16 TMP 1991.36.254 25 18 48.5 28 122.5 87 48 23.5 152 107.5 37 TMP 1966.10.9 21 10 53 32 151 29 20 41 27 106 182 67 43 TMP 1981.19.157 73 30 172 95 59 29 135 119 TMP 1981.19.160 84.5 26 184 101.5 54 27 22 46 34 124 76.5 TMP 1966.10.4 35.5 27 70 38 165.5 103.5 47.5 30 25 50 27.5 129.5 69.5 TMP 2001.12.4 55 39.5 167 93.5 73 63.5 60 62 26 154 72 65 92 28 TMP 1999.55.2 44 38 71 31 161 116 67 28 156 51 TMP 1999.55.5 60.5 27 145 77 57 54 58 26.5 135 60 95 78.5 30 TMP 1981.18.213 44.5 33 68.5 29 156.5 140 36 36 33 76 15 165 233 28 76 10 TMP 1981.16.115 37 75 TMP 2005.09.69 45 41 118 30 213 170 TMP 2005.09.74 62.5 36 139 124 59 62 82 43 196 96 36 TMP 1966.32.34 42 61 25 TMP 1995.401.0036 110 42.5 254 213 128 103 111 83 TMP 1995.400.160 TMP 1995.400.54 38 32 78 26 183 131 TMP 1995.175.21 32 18.5 72.5 36.5 177 103 TMP 1979.11.76 29.5 21 66.5 25 147 119 23.5 17.5 12.5 50 10.5 101.5 196 TMP 1982.18.277 152 176 113 42 263 132 153 87 91 98 41.5 229 108 TMP 1982.18.79 159 181 110 49 265.5 127 179 65 73.5 107 46 249 83 TMP 2002.68.003 72 69 68 39 151 168 69 59 61 54 35 108 80 TMP 2002.68.014 69 76 80 44 192 68 63 58 34 28 96 49 64 35 TMP 1999.82.01 67 36.5 113 103.5 326.5 144.5 108 97.5 82 107.5 59 271.5 46 TMP 1992.30.41 109 59 294 118 43 31 145 61 85 54 TMP 1984.93.05 34 21 77 37 191 165 154 159 85 49 230 153 45 TMP 1994.12.607 67 72 38 128 TMP 1979.11.19 105 109 92 54.5 239.5 194 156 88 88.5 85.5 54.5 208.5 148.5 USNM 7950 9 10 USNM 11869 3 3 23 49 29 21 11 36 25.5 112 29.5 290 95.5 47 443

Spec. Number P3-dist P4-HT P4-LT P4-TK P4-dist P5-HT P5-LT P5-TK P5-dist P6-HT P6-LT P6-TK P6-dist P7-HT P7-LT P7-TK P7-dist AMNH 5239 AMNH 5351 189.5 67 122 28 192 46 108 18 321 48 110 18 375 44 103 19 412 AMNH 5377 20 28 31 22 63 30 AMNH 5429 322 28 90 23 442 21 58 17 492 6 41 13 458 AMNH 5442 328 7 98 16 422 6 64 16 445 3 42 15 451 CMN 11839 287 11 70 328 8 68 355 3 60 375 4 61 23 376 CMN 12229 306 24 77 404.5 17.5 59.5 446 18.5 62.5 486 20 67 498 CMN 344 218 512 97 66 678 337 102 67 643 80 65 41 433 14 50 36 398 CMN 348 249 26 105.5 24.5 381.5 35.5 91 26 427 42.5 83.5 33.5 453 CMN 8790 361 31 87.5 16.5 399 14 475 13 493 CMN 8795 304 29 55 19 384 40.5 61 19 445.5 32 54.5 15 458 10.5 58.5 12 454.5 CMN 8798 302 40 69.5 14 418.5 33 60.5 11 480 17 47.5 13 488 CMN 971 295 29.5 74 23.5 390 23 62 16 451 29 46 16 502 ROM 1427 318 38 56 17 389 54 15 440 25 57 15 487 ROM 767 294 25 58 9.5 318.5 17.5 53 9.5 351 13 64 8 RTMP 86.126.1 249 76 78.5 24 348 84.5 87.5 29.5 451 81 83 23 497 60.5 75 16.5 496 UALVP 11735 323.5 27 61.5 26 375.5 28 69.5 23.5 452 20 47.5 21 463.5 20.5 47.5 21 471 UALVP 16248 184 49 69 39 54 33 60 40 51 USNM 8897 197 42 57 26 272 29 56.5 17.5 334.5 17 57.5 13 379 10 63 10 397 TMP 1979.10.05 22 68 16 17 65 16 19.5 75 16.5 19 57.5 17.5 TMP 1993.70.01 314 26 45 35 372 24 67 30 408 24 61 27 414 37 81 25 411 TMP 2009.90.01 193 143 78 416 93 84 448 51 69 454 20 43 458 TMP 2005.12.58 185.5 113 92 369.5 42 87 359 40 89 386 20 51 391 TMP 1997.85.1 288.5 13.5 92 24.5 405.5 15.5 64.5 17.5 460.5 13.5 45.5 19.5 470 TMP 1982.16.11 87 5 22 5 102 4 21 5 113 4 16 8 120 NMH R 4859 268 23 79 24 343 19 62 22 399 22 65 22 408 NHM - Spinops Par 1 NHM - Spinops Par 2 YPM 2015 196.5 27 79 17 260 29 80 15 344 29 71 14 379 7 52 13 365 LAVAL (CMN 347) UALVP 47979 283 16 56 10 356 15 59 11 394 18 62 11 437 19 46 11 468 TMP 1988.36.20 180 226 110.5 69.5 413 60 77 50 370 14 56 35 349 ROM 1426 322 15.5 73 389.5 26.5 67 432.5 10 58 443 10.5 54 450 TMP 1994.182.01 TMP 1992.82.01 272 26 75 320 22 82 14 84 366 TMP 1980.54.01 285 20.5 61.5 7.5 337.5 10 59.5 7.5 15 66 6 428 11 8 449 TMP 1987.52.1 396 109 52 362 27 42.5 16 396 17.5 42.5 14 434 AMNH 3998 304 14 65 16 376 5 68 13 3 80 13 409 TMP 1988.36.20 180 226 110.5 69.5 413 60 77 50 14 56 35 349 TMP 1984.93.1 209 TMP 1978.6.01 327 24 74 18 413 18 59 18 12 54 14 481 19 51 16 513 TMP 1995.666.36 TMP 2002.68.02 TMP 2008.79.23 TMP 2008.79.15 TMP 2008.79.184 TMP 1966.10.6 144 72 39 65 72 28 28 44 21 TMP 1981.19.222 50 24 36 62 18 22 51 12 TMP 1966.10.28 201 TMP 1991.36.254 TMP 1966.10.9 TMP 1981.19.157 TMP 1981.19.160 TMP 1966.10.4 TMP 2001.12.4 222 TMP 1999.55.2 TMP 1999.55.5 202 TMP 1981.18.213 340 25 8 426 TMP 1981.16.115 29 70 29 67 32 72 TMP 2005.09.69 TMP 2005.09.74 TMP 1966.32.34 33 60 18 33 58 15 2 56 11 16 55 17 TMP 1995.401.0036 TMP 1995.400.160 32 71 19 26 70 14 26 42 14 TMP 1995.400.54 TMP 1995.175.21 TMP 1979.11.76 TMP 1982.18.277 TMP 1982.18.79 TMP 2002.68.003 TMP 2002.68.014 TMP 1999.82.01 TMP 1992.30.41 202 TMP 1984.93.05 TMP 1994.12.607 TMP 1979.11.19 USNM 7950 USNM 11869 44.5 281.5 99.5 42.5 42 103 75 36