GLS-corrected RLQ analysis: a new multivariate method for incorporating spatial and phylogenetic information into trait-environment analyses by Thomas A. Henry A Thesis presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Integrative Biology Guelph, Ontario, Canada © Thomas A. Henry, May, 2016 ABSTRACT GLS-CORRECTED RLQ ANALYSIS: A NEW MULTIVARIATE METHOD FOR INCOR- PORATING SPATIAL AND PHYLOGENETIC INFORMATION INTO TRAIT- ENVIRONMENT ANALYSES Thomas A. Henry Advisor: University of Guelph, 2016 Dr. Steven G. Newmaster A key goal in community ecology is to understand factors that shape the distribution of species on the landscape. One relevant factor is the relationship between the environment and species traits (environmental filtering). RLQ analysis is a multivariate method that measures the relationship between species traits and the environment. However, it ignores spatial and phy- logenetic patterns in the data. In this thesis, I develop a method for incorporating spatial and phy- logenetic information into RLQ analysis. In the first chapter, I demonstrate that phylogenetic and spatial effects bias multivariate statistical methods, and that methods based on the generalized least squares (GLS) framework can remove this bias. In the second chapter, I develop a new variant of RLQ analysis based on the GLS framework. In the third chapter, I use an existing dataset on herbaceous plant communities in Ontario forests as a case study for the new GLS- based RLQ method. iii Acknowledgements First, I’d like to thank my advisor, Steve Newmaster, for his support and guidance over the past few years, and for encouraging me to pursue my interest in phylogenetics and multivari- ate statistics. Second, I’d like to thank Cort Griswold for his assistance and feedback with this project, and for introducing me to mathematical programming and the wondrous world of eigen- values. To Wayne Bell and Jennifer Dacosta at the Ontario Ministry of Natural Resources and Forestry: thank you for allowing me to use the NEBIE data, and for your assistance in assem- bling the databases. I’d also like to thank everyone in the Newmaster Lab, past and present, for making the lab a fun place to be. Special thanks to Carole Ann Lacroix for assistance with plant identifica- tion, and to Subramanyam Ragupathy (Ragu) and Kawa Cheng for taking care of the PCR and sequencing. Thanks to Jillian Bainard, Royce Steeves, Kevan Berg, Neil Webster, Brian Lacey, Troy McMullin, and Aron Fazekas for many interesting and insightful discussions over the years. Finally, special thanks to Sean Rapai, Lyndsay Schram, and Jose Maloles for your friend- ship and support and all-around good times, and for tolerating my various eccentricities. And lastly, thanks to my family. I couldn’t have done this without you. iv Table of Contents List of Tables ................................................................................................................................ vii List of Figures ............................................................................................................................... vii List of Symbols and Abbreviations.............................................................................................. viii General Introduction ....................................................................................................................... 1 Figures......................................................................................................................................... 7 References ................................................................................................................................... 8 Chapter 1: Non-independence among observations biases multivariate statistical methods ........ 12 1.1 Abstract ............................................................................................................................... 12 1.2 Introduction ......................................................................................................................... 13 1.3 Methods............................................................................................................................... 20 1.3.1 Implementation of multivariate methods and the GLS correction ............................... 20 1.3.1.1 Variances and covariances .................................................................................... 20 1.3.1.2 Principal component analysis ............................................................................... 22 1.3.1.3 Coinertia analysis .................................................................................................. 23 1.3.1.4 Redundancy analysis ............................................................................................. 24 1.3.1.5 Canonical correlation analysis .............................................................................. 24 1.3.2 Analytical approach ..................................................................................................... 25 1.3.2.1 Quadratic forms in random variables .................................................................... 25 1.3.2.2 Sample variance .................................................................................................... 26 1.3.2.3 Sample covariance ................................................................................................ 27 1.3.2.4 One matrix multivariate methods.......................................................................... 30 1.3.2.5 Two-matrix multivariate methods......................................................................... 31 1.3.3 Simulation approach .................................................................................................... 33 1.3.3.1 Generating C matrices........................................................................................... 33 1.3.3.2 Generating multivariate trait data ......................................................................... 34 1.3.3.3 Simulation setup.................................................................................................... 34 1.4 Results ................................................................................................................................. 39 1.4.1 Sample variance ........................................................................................................... 39 1.4.2 Sample covariance and correlation .............................................................................. 40 1.4.3 Principal component analysis ...................................................................................... 41 1.4.4 Coinertia analysis ......................................................................................................... 42 1.4.5 Redundancy analysis .................................................................................................... 42 1.4.6 Canonical correlation analysis ..................................................................................... 42 v 1.5 Discussion ........................................................................................................................... 43 1.6 Tables .................................................................................................................................. 51 1.7 Figures................................................................................................................................. 57 1.8 References ........................................................................................................................... 62 Chapter 2: GLS-corrected RLQ analysis: a new multivariate method for incorporating spatial and phylogenetic information into trait-environment analyses ........................................................... 68 2.1 Abstract ............................................................................................................................... 68 2.2 Introduction ......................................................................................................................... 69 2.3 Methods............................................................................................................................... 78 2.3.1 Generating trait data ..................................................................................................... 78 2.3.2 Generating environment data ....................................................................................... 79 2.3.3 Generating community data ......................................................................................... 79 2.3.4 Simulation setup........................................................................................................... 81 2.4 Results ................................................................................................................................. 82 2.5 Discussion ........................................................................................................................... 83 2.6 Tables .................................................................................................................................. 88 2.7 Figures................................................................................................................................. 89 2.8 References ........................................................................................................................... 90