Measuring Production and Predicting Outcomes in the National Basketball Association
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Michael Steven Milano, M.S.
Graduate Program in Education
The Ohio State University
2011
Dissertation Committee:
Packianathan Chelladurai, Advisor
Brian Turner
Sarah Fields
Stephen Cosslett
Copyright by
Michael Steven Milano
2011
Abstract
Building on the research of Loeffelholz, Bednar and Bauer (2009), the current study analyzed the relationship between previously compiled team performance measures and the outcome of an “un-played” game. While past studies have relied solely on statistics traditionally found in a box score, this study included scheduling fatigue and team depth. Multiple models were constructed in which the performance statistics of the competing teams were operationalized in different ways. Absolute models consisted of performance measures as unmodified traditional box score statistics. Relative models defined performance measures as a series of ratios, which compared a team‟s statistics to its opponents‟ statistics. Possession models included possessions as an indicator of pace, and offensive rating and defensive rating as composite measures of efficiency. Play models were composed of offensive plays and defensive plays as measures of pace, and offensive points-per-play and defensive points-per-play as indicators of efficiency.
Under each of the above general models, additional models were created to include streak variables, which averaged performance measures only over the previous five games, as well as logarithmic variables. Game outcomes were operationalized and analyzed in two distinct manners - score differential and game winner. Multiple regression analysis was used to explain the relationships between predictors and the “un-played” game‟s score differential, and logistic regression analysis was used when the game winner was the
ii dependent variable. The process of entering each model‟s respective variables into the regression equations was accomplished through simultaneous entry, stepwise entry, and hierarchal entry. Statistical analyses were conducted on both the 2007-2008 and 2008-
2009 National Basketball Association seasons, which served as two populations.
Taking into account goodness-of-fit measures and parsimony, superior models were identified. In regards to explained variance in score differential, the possession model with stepwise entry emerged as the best model for the 2007-2008 and 2008-2009 seasons. For predicting game winner the best models for the 2007-2008 and 2008-2009 seasons were the play stepwise entry model and the possession stepwise entry model respectively. As a whole, non-streak models were substantially more successful at explaining game outcomes than streak models. The increase in explained variance due to the entry of scheduling fatigue variables and team depth contribution factors in the second stage of the hierarchal multiple regression analysis varied among the models as well as the variables that reached statistical significance.
Overall, the findings of the present study indicate little generalizability between the two NBA seasons selected for the study. In general, the variables selected for inclusion into the regressions equations, as well as their relative importance differed from one season to the next. However, the possession models were found to be the best models in terms of predictive capabilities and parsimony, and they were the most stable over the two populations. These findings serve to support the use of composite measures of pace and efficiency in future basketball research as well as decisions made by management and members of the media.
iii Acknowledgements
While my journey through the dissertation process has been both long and arduous, it would have been nearly impossible to complete without the support of a host of other individuals. For all the people who provided assistance along the way, I would like to express my appreciation.
First, I would like to gratefully and sincerely thank Dr. Packianathan Chelladurai,
Dr. Brian Turner, Dr. Sarah Fields, and Dr. Stephen Cosslett, for serving as my dissertation committee members. The feedback and suggestions you all provided at various stages in the process was invaluable, and for that I am thankful. Specifically, I would like to express my gratitude to my advisor, Dr. Chelladurai, for his guidance, patience, and support throughout my graduate studies at The Ohio State University. You are truly a wealth of knowledge, and without your assistance I would have undoubtedly struggled to have success in the program.
I would also like to thank all my good friends, who over the years provided an immeasurable amount of support and encouragement. The underlying inspirations for the current research can be traced back to our countless in-depth conversations on predicting outcomes within the domain of sport. During the dissertation process, the much welcomed distractions you all provided helped me maintain my sanity. In particular, I would like to thank Oleg Mishchenko, for without your assistance in the creation of
iv the two statistic conversion programs, this research endeavor would not have been possible.
Finally, and most importantly, I would like to thank my parents, Robert and
Priscilla, and my brother, Bryan. Words cannot express how important you all were throughout this entire process. Thank you so much for supporting and believing in me.
v Vita
July 4, 1982…………………………………………………..Born – Brooklyn, New York
2004………………………………………………………….B.S. Computer Engineering,
Lehigh University
2005………………………………………………………….M.S. Sport Administration,
Florida State University
2007 – present…………………………….…………………The Ohio State University
Publications
Milano, M., & Chelladurai, P. (2011). Gross domestic sports product: The size of the sport industry in the United States. Journal of Sport Management, 25(1), 24-35.
Fields of Study
Major Field: Education
Focus: Sport Management
Minor: Research Methods in Human Resources Development
vi Table of Contents
Abstract…………………………………………………………………….……………...ii
Acknowledgements…………………………………………………………………….…iv
Vita………………………………………………………………………………………..vi
List of Tables……………………………………………………………………………xiii
List of Figures…………………………………………………………………...... …...xx
Chapter 1: Introduction……………………………………………………………………1
Problem Statement………………………………………………………………...1
Theoretical Foundation……………………………………………………………9
Linear Production Function……………………………………………...13
Cobb-Douglas Production Function……………………………………..16
Constant Elasticity of Substitution Production Function (CES)…………21
Transcendental Logarithmic Production Function…………………….....21
Production Function Applications in Sport………………………………………23
Selection and Justification of Production Function Formulations………….……24
Research Questions………………………………………………………………27
Models of Performance…………………………………………………………..27
Absolute Models…………………………...………………………….…28
vii Relative Models………………………………………………….………31
Possession Models……………..…………………………….…………..34
Play Models…………………………………………………………...…37
Definition of Terms………………………………………………………………38
Assist……………………………………………………………………..40
Blocked Shot……………………………...……..……………………….40
Defensive Rating………………………………………………...……….41
Defensive Rebound………………………………………………..……..41
Depth Contribution Factor……………………...……………….….……41
Free Throw………………………………………………….……………42
Game Location…………………………………………...………………42
Offensive Rating…………………………………………………………43
Offensive Rebound………………………………………………………43
Personal Foul………………………………………….…………………44
“Played” Games………………………………………………………….45
Scheduling Fatigue……………………………………………………….45
Steal……………………………………………………………………....47
Streak………………………………………………………………...…..49
Three Point Field Goal………………………………………...…………49
Turnover………………………………………………….………………50
Two Point Field Goal…………………………………….………………50
“Un-played” Games…………………………………………………..….50
viii Significance of the Problem…………………………………………….………..50
Chapter 2: Review of Literature…………………………………………..……………..53
Individual Performance Measures……………………………………………….53
Team Based Production Research……………………………………………….59
Chapter 3: Methodology…………………………………………………………………75
Research Design…………………………………………………………….……75
Conversion of Statistics……………………………………………….…79
Subject Selection……………………………………………………………..…..81
Outcome Measures……………………………………………………………….83
Data Analysis…………………………………………………………………….85
Chapter 4: Results……………………………………………………………….……….95
Multiple Linear Regression ………………………………………….…………..95
Absolute model 2007-2008………………………….…………...……....95
Absolute model 2008-2009………………………….………………….101
Absolute model (streak variables) 2007-2008……………………...…..106
Absolute model (streak variables) 2008-2009…………………...…..…110
Absolute model (logarithmic variables) 2007-2008……………………114
Absolute model (logarithmic variables) 2008-2009………………....…119
Absolute model (streak logarithmic variables) 2007-2008……………..123
Absolute model (streak logarithmic variables) 2008-2009…………..…127
Relative model 2007-2008………………………………………..….…131
Relative model 2008-2009………………………………………..….…137
ix Relative model (streak variables) 2007-2008…………………………..141
Relative model (streak variables) 2008-2009………………………..…144
Relative model (logarithmic variables) 2007-2008………………….....148
Relative model (logarithmic variables) 2008-2009………………….....152
Relative model (streak logarithmic variables) 2007-2008…………...…155
Relative model (streak logarithmic variables) 2008-2009…………...…159
Possession model 2007-2008…………………………………..……….163
Possession model 2008-2009…………………………………...………167
Possession model (streak variables) 2007-2008………………………..171
Possession model (streak variables) 2008-2009…………………..……174
Possession model (logarithmic variables) 2007-2008…………….....…177
Possession model (logarithmic variables) 2008-2009……………….…180
Possession model (streak logarithmic variables) 2007-2008………...…183
Possession model (streak logarithmic variables) 2008-2009………...…186
Play model 2007-2008………………………………………………….189
Play model 2008-2009………………………………….………………193
Play model (streak variables) 2007-2008………………………...... 197
Play model (streak variables) 2008-2009………………………..…..…200
Play model (logarithmic variables) 2007-2008…………...……….....…203
Play model (logarithmic variables) 2008-2009…………...……….……206
Play model (streak logarithmic variables) 2007-2008……………….…209
Play model (streak logarithmic variables) 2008-2009…………..……...212
x Logistic Regression……………………………………………………………..215
Absolute model 2007-2008……………………………………………..216
Absolute model 2008-2009………………………….………………….220
Absolute model (streak variables) 2007-2008……………………….…224
Absolute model (streak variables) 2008-2009………………….....……227
Relative model 2007-2008……………………………………………...230
Relative model 2008-2009………………………….…………………..233
Relative model (streak variables) 2007-2008………………………..…236
Relative model (streak variables) 2008-2009…………………...... ……239
Possession model 2007-2008…………………………………...………242
Possession model 2008-2009………………………….………..………246
Possession model (streak variables) 2007-2008……………………...... 249
Possession model (streak variables) 2008-2009……………………..…252
Play model 2007-2008……………………………………………….....255
Play model 2008-2009………………………….………………...…….258
Play model (streak variables) 2007-2008……………………...……….261
Play model (streak variables) 2008-2009…………………...……….…264
Chapter 5: Discussion……………………………………………………………….….267
Absolute Models………………………………………………………………..273
Relative Models………………………………………………………………...276
Regression Equation Selection…………………………………………………277
Research Questions……………………………………………………………..277
xi Limitations……………………………………………………………...………302
Conclusion………………………………………………………………...……304
References………………………………………………………………...…………….306
Appendix A: Descriptive statistics for absolute model 2007-2008 NBA season……....313
Appendix B: Descriptive statistics for absolute model 2008-2009 NBA season………316
Appendix C: Descriptive statistics for relative model 2007-2008 NBA season………..319
Appendix D: Descriptive statistics for relative model 2008-2009 NBA season……….321
Appendix E: Descriptive statistics for possession model 2007-2008 NBA season…….323
Appendix F: Descriptive statistics for possession model 2008-2009 NBA season…….324
Appendix G: Descriptive statistics for play model 2007-2008 NBA season……...... 325
Appendix H: Descriptive statistics for play model 2008-2009 NBA season…………...326
Appendix I: Goodness-of-fit measures for null model 2007-2008 NBA season……….327
Appendix J: Goodness-of-fit measures for null model 2008-2009 NBA season……….328
Appendix K: Goodness-of-fit measures for null model 2007-2008 NBA season (streak)………………………………………………………………………………….329
Appendix L: Goodness-of-fit measures for null model 2008-2009 NBA season (streak)……………………………………………………………………………….....330
xii List of Tables
Table 1. Basic model descriptions………………………………………………………...6
Table 2. Independent variables for absolute models……………………………….…….29
Table 3. Independent variables for relative models…………………………..………….33
Table 4. Independent variables for possession models……………………….………….36
Table 5. Independent variables for play models………………………….……………...39
Table 6. Fatigue dummy variable format for previous day……………………...………48
Table 7. Summaries of team production research in basketball………………...……….61
Table 8. Stepwise multiple regression results for the absolute model 2007-2008 ………98
Table 9. Hierarchal multiple regression results for the absolute model 2007-2008…..…99
Table 10. Stepwise multiple regression results for the absolute model 2008-2009….…103
Table 11. Hierarchal multiple regression results for the absolute model 2008-2009…..104
Table 12. Stepwise multiple regression results for the absolute model (streak variables) 2007-2008 ……………………………………………………………………………...107
Table 13. Hierarchal multiple regression results for the absolute model (streak variables) 2007-2008 ……………………………………………………………………………...108
Table 14. Stepwise multiple regression results for the absolute model (streak variables) 2008-2009 …………………………………………………………………….….…….111
Table 15. Hierarchal multiple regression results for the absolute model (streak variables) 2008-2009 ……………………………………………………………………………...112
xiii Table 16. Stepwise multiple regression results for the absolute model (logarithmic variables) 2007-2008 ………………………………………………………..…...…….115
Table 17. Hierarchal multiple regression results for the absolute model (logarithmic variables) 2007-2008 …………………………………………………………..………117
Table 18. Stepwise multiple regression results for the absolute model (logarithmic variables) 2008-2009 ……………………………………………………………..……120
Table 19. Hierarchal multiple regression results for the absolute model (logarithmic variables) 2008-2009……………………………………….…………………………..121
Table 20. Stepwise multiple regression results for the absolute model (streak logarithmic variables) 2007-2008…………………………………………………………..…….....124
Table 21. Hierarchal multiple regression results for the absolute model (streak logarithmic variables) 2007-2008……………………………………………..………..125
Table 22. Stepwise multiple regression results for the absolute model (streak logarithmic variables) 2008-2009…………………………………………………………..……….128
Table 23. Hierarchal multiple regression results for the absolute model (streak logarithmic variables) 2008-2009……………………………………………..….….....129
Table 24. Stepwise multiple regression results for the relative model 2007-2008……..133
Table 25. Hierarchal multiple regression results for the relative model 2007-2008…...135
Table 26. Stepwise multiple regression results for the relative model 2008-2009……..138
Table 27. Hierarchal multiple regression results for the relative model 2008-2009…...140
Table 28. Stepwise multiple regression results for the relative model (streak variables) 2007-2008………………………………………………………………………...... 142
Table 29. Hierarchal multiple regression results for the relative model (streak variables) 2007-2008…………...... 143
Table 30. Stepwise multiple regression results for the relative model (streak variables) 2008- 2009……………………………………………………………………...... ….....145
Table 31. Hierarchal multiple regression results for the relative model (streak variables) 2008-2009…………...... 146
xiv Table 32. Stepwise multiple regression results for the relative model (logarithmic variables) 2007-2008………………………………………………………….………..149
Table 33. Hierarchal multiple regression results for the relative model (logarithmic variables) 2007-2008…………………………………………………….……………..150
Table 34. Stepwise multiple regression results for the relative model (logarithmic variables) 2008-2009…………………………………………………….……………..153
Table 35. Hierarchal multiple regression results for the relative model (logarithmic variables) 2008-2009………………………………………………………………….. 154
Table 36. Stepwise multiple regression results for the relative model (streak logarithmic variables) 2007-2008……………………………………….…………………………..156
Table 37. Hierarchal multiple regression results for the relative model (streak logarithmic variables) 2007-2008…………………………………...…………...………………….157
Table 38. Stepwise multiple regression results for the relative model (streak logarithmic variables) 2008-2009………………………………………………….………………..160
Table 39. Hierarchal multiple regression results for the relative model (streak logarithmic variables) 2008-2009………………………………………..………………………….161
Table 40: Stepwise multiple regression results for the possession model 2007- 2008………………………………………………………………………………..……164
Table 41: Hierarchal multiple regression results for the possession model 2007- 2008………………………………………………………………………………..……166
Table 42: Stepwise multiple regression results for the possession model 2008- 2009……………………………………………………………………………………..168
Table 43: Hierarchal multiple regression results for the possession model 2008- 2009………...... 170
Table 44: Stepwise multiple regression results for the possession model (streak variables) 2007-2008………..…………………………………………………………………..…172
Table 45: Hierarchal multiple regression results for the possession model (streak variables) 2007-2008………………………………………………………………….. 173
Table 46: Stepwise multiple regression results for the possession model (streak variables) 2008-2009………..………………………………………………..……………………175
xv Table 47: Hierarchal multiple regression results for the possession model (streak variables) 2008-2009………...... 176
Table 48: Stepwise multiple regression results for the possession model (logarithmic variables) 2007-2008………..………………………….....……………………………178
Table 49: Hierarchal multiple regression results for the possession model (logarithmic variables) 2007-2008………………………………….………………………...…….. 179
Table 50: Stepwise multiple regression results for the possession model (logarithmic variables) 2008-2009………..………………………………………….………………181
Table 51: Hierarchal multiple regression results for the possession model (logarithmic variables) 2008-2009………...... 182
Table 52: Stepwise multiple regression results for the possession model (streak logarithmic variables) 2007-2008………..…………………………...………………...184
Table 53: Hierarchal multiple regression results for the possession model (streak logarithmic variables) 2007-2008………...... 185
Table 54: Stepwise multiple regression results for the possession model (streak logarithmic variables) 2008-2009………..………………...…………………………...187
Table 55: Hierarchal multiple regression results for the possession model (streak logarithmic variables) 2008-2009………...... 188
Table 56: Stepwise multiple regression results for the play model 2007-2008…...……190
Table 57: Hierarchal multiple regression results for the play model 2007-2008……... 192
Table 58: Stepwise multiple regression results for the play model 2008-2009……...…194
Table 59: Hierarchal multiple regression results for the play model 2008-2009………196
Table 60: Stepwise multiple regression results for the play model (streak variables) 2007- 2008………………………………………………………………….……….….198
Table 61: Hierarchal multiple regression results for the play model (streak variables) 2007-2008……………………………………………………..…………………….….199
Table 62: Stepwise multiple regression results for the play model (streak variables) 2008-2009…………………..……………………………………………………….….201
xvi Table 63: Hierarchal multiple regression results for the play model (streak variables) 2008-2009……………………………………………………………..…………….….202
Table 64: Stepwise multiple regression results for the play model (logarithmic variables) 2007-2008……………………………………………………………………..…….….204
Table 65: Hierarchal multiple regression results for the play model (logarithmic variables) 2007-2008……………………………………………………….……….….205
Table 66: Stepwise multiple regression results for the play model (logarithmic variables) 2008-2009………………………………………………………..………………….….207
Table 67: Hierarchal multiple regression results for the play model (logarithmic variables) 2008-2009…………………………………………….………………….….208
Table 68. Stepwise multiple regression results for the play model (streak logarithmic variables) 2007-2008……………………………………………………...……………210
Table 69. Hierarchal entry multiple regression for the play model (streak logarithmic variables) 2007-2008………………………………………..……………………….....211
Table 70. Stepwise multiple regression results for the play model (streak logarithmic variables) 2008-2009…………………………………………………………………...213
Table 71. Hierarchal entry multiple regression for the play model (streak logarithmic variables) 2008-2009……………………………………………...………………...….214
Table 72. Stepwise logistic regression results for the absolute model 2007-2008……..217
Table 73. Hierarchal logistic regression goodness-of-fit measures for the absolute model 2007-2008…………………………………………………………………………..…..219
Table 74. Stepwise logistic regression results for the absolute model 2008-2009…..…221
Table 75. Hierarchal logistic regression goodness-of-fit measures for the absolute model 2008-2009…………………………………………………………………………..…..223
Table 76. Stepwise logistic regression results for the absolute model (streak variables) 2007-2008………..………………………………………..……………………………225
Table 77. Hierarchal logistic regression goodness-of-fit measures for the absolute model (streak variables) 2007-2008…………………………..………………………………..226
xvii Table 78. Stepwise logistic regression results for the absolute model (streak variables) 2008-2009………..……………………………………………………..………………228
Table 79. Hierarchal logistic regression goodness-of-fit measures for the absolute model (streak variables) 2008-2009…………………………..………………………………..229
Table 80. Stepwise logistic regression results for the relative model 2007-2008…...…231
Table 81. Hierarchal logistic regression goodness-of-fit measures for the relative model 2007-2008…………………………………………………………………………..…..232
Table 82. Stepwise logistic regression results for the relative model 2008-2009…...…234
Table 83. Hierarchal logistic regression goodness-of-fit measures for the relative model 2008-2009…………………………………………………………………………..…..235
Table 84. Stepwise logistic regression results for the relative model (streak variables) 2007-2008………..……………………………………………………..………………237
Table 85. Hierarchal logistic regression goodness-of-fit measures for the relative model (streak variables) 2007-2008…………………………..………………………………..238
Table 86. Stepwise logistic regression results for the relative model (streak variables) 2008-2009………..…………………………………..…………………………………240
Table 87. Hierarchal logistic regression goodness-of-fit measures for the relative model (streak variables) 2008-2009…………………………..………………………………..241
Table 88. Stepwise logistic regression results for the possession model 2007-2008…..243
Table 89. Hierarchal logistic regression goodness-of-fit measures for the possession model 2007-2008………………………………………………………………….…....244
Table 90. Stepwise logistic regression results for the possession model 2008-2009…..247
Table 91. Hierarchal logistic regression goodness-of-fit measures for the possession model 2008-2009…………………………………………………………………..…...248
Table 92. Stepwise logistic regression results for the possession model (streak variables) 2007-2008………..……………………………………………………………..………250
Table 93. Hierarchal logistic regression goodness-of-fit measures for the possession model (streak variables) 2007-2008…………………………..……………….………..251
xviii Table 94. Stepwise logistic regression results for the possession model (streak variables) 2008-2009………..………………………………………..……………………………253
Table 95. Hierarchal logistic regression goodness-of-fit measures for the possession model (streak variables) 2008-2009…………………………..………………….……..254
Table 96. Stepwise logistic regression results for the play model 2007-2008……….…256
Table 97. Hierarchal logistic regression goodness-of-fit measures for the play model 2007-2008…………………………………………………………………………..…..257
Table 98. Stepwise logistic regression results for the play model 2008-2009……….…259
Table 99. Hierarchal logistic regression goodness-of-fit measures for the play model 2008-2009…………………………………………………………………………..…..260
Table 100. Stepwise logistic regression results for the play model (streak variables) 2007- 2008………..……………………………………………………………………………262
Table 101. Hierarchal logistic regression goodness-of-fit measures for the play model (streak variables) 2007-2008…………………………..………………………………..263
Table 102. Stepwise logistic regression results for the play model (streak variables) 2008- 2009………..……………………………………………………………………………265
Table 103. Hierarchal logistic regression goodness-of-fit measures for the play model (streak variables) 2008-2009…………………………..………………………………..266
Table 104. Summary of multiple regression analyses 2007-2008……………….……..278
Table 105. Summary of multiple regression analyses 2008-2009……………………...282
Table 106. Summary of logistic regression analyses 2007-2008………………...…….288
Table 107. Summary of logistic regression analyses 2008-2009………………………290
Table 108. Summary of hierarchal multiple regression analyses…………………...….296
xix List of Figures
Figure 1. Single input and single output production function: A decreasing returns to scale with diseconomies of scale, B increasing returns to scale with economies of scale, C constant returns to scale…………………………………………………………….……14
xx Chapter 1: Introduction
Garnering the attention of the academic community as well as individuals involved in player personnel, revolutionary statistical analyses are pioneering an evolution in the evaluation of sport performances. The influx of statistics that altered professional baseball in the 1990s has inspired a statistical migration of sorts into every major sport (Lewis, 2009). Unfortunately, unlike baseball, evaluations performed within the domain of basketball are inherently complex due to the sport‟s interdependent nature.
Designated as interactively dependent under Carron and Chelladurai‟s (1981) sport typology, basketball requires five players to engage in continual interactions and mutual adjustments throughout an individual competition. As a fundamental feature of basketball, seamless interactions between teammates can dramatically increase offensive and defensive success (Oliver, 2004). Simply stated, basketball is a dynamic game that is permeated by instances of mutual dependence and congruent actions.
Problem Statement
While extensive research has been conducted on performance, production, and efficiency measures in basketball, with the exception of Loeffelholz, Bednar and Bauer
(2009), investigations integrating previously compiled statistics with forthcoming game outcomes are sorely lacking. In addition, overreliance on traditional box score statistics has thwarted efforts to explain and predict the underlying factors responsible for on-court
1 success. Finally, although performance measures have been defined through different conceptualizations (i.e., as absolute statistics as well as relative measures), research studies have failed to analyze the relationship between outcomes and composite indicators of pace and efficiency (e.g., possessions and offensive / defensive ratings).
The present study will attempt to fill this dearth in sport performance literature by specifically exploring the aforementioned uncharted regions.
Loeffelholz et al.‟s (2009) research on the predictive capabilities of previously accumulated statistics within the NBA supplied the essential building blocks for the present investigation. With a focus on forecasting accuracy, Loeffelholz et al. used neural networks, which included team averages in a variety of performance categories as inputs, in an effort to predict “un-played” games. These input variables included home and away team statistics in field goal percentage, three point percentage, free throw percentage, offensive rebounds, defensive rebounds, assists, steals, blocks, turnovers, personal fouls, points and a dichotomous variable to indicate game location. Constructed in this manner, the complex relationship between forthcoming competition outcomes and team statistics from previous contests was explored. While incorporating the concepts of
“played” and “un-played” games, the current study expands upon Loeffelholz et al.‟s line of research:
1. Through the inclusion of quantitative variables related to fatigue and depth. More
specifically, each model includes a series of dummy variables to account for the
recent competition schedules of both teams as well as variables that identify the
number of players contributing significant minutes;
2 2. Through the incorporation of composite performance measures, which focus on
team pace and team efficiency;
3. By adopting different statistical approaches (i.e., multiple regression analysis and
logistic regression analysis). Multiple regression analysis is advantageous as it
permits for the exploration of score differential as a dependent variable, which
can provide more information than a single binary game winner variable. Perhaps
more importantly, both multiple regression and logistic regression allow the
individual contributions of independent variables to be identified, which is not
possible for neural networks.
In the domain of sport, where interpreting quantifiable performance measures are of paramount importance, the present conceptualization offers an alternative perspective.
Whereas prior basketball research has generally centered on the relationship between performance statistics accumulated in a given contest and the outcome of that competition (e.g., McGoldrick & Voeks, 2005), or statistics amassed over an entire season and a measure of team success (e.g., Hofler & Payne, 1997), the study at hand adopts the viewpoint of a prognosticator. More specifically, the correlation between team performance measures accumulated prior to entering a given competition and the outcome of the “un-played” contest is under examination. Hence, the present investigation represents an attempt to link known team statistics with an unknown future outcome. Unearthing the relationship between prior performance statistics, which
3 indicate talent levels and overall performance ability, and future outcomes, inevitably reveals the importance of individual performance categories.
With an alternative paradigm, geared toward the prediction of future outcomes, the individual contribution of performance measures can be reexamined. If one were capable of accurately forecasting final game score differentials, solely through knowledge of prior performance measures, the relationship between game statistics and game scores would be further illuminated. Information of this sort not only has implications for determining the value of performance measures, it could potentially alter game strategies as well as personnel decisions. If general managers or coaches were to learn that certain statistics were more pivotal to the production of points, or conversely to the surrendering of points, strategies could be further tailored to increase the likelihood of success against particular opponents.
The numerous models constructed within the present investigation include a host of innovative quantifiable measures, which focus on fatigue arising from scheduling and advantages / disadvantages associated with the average number of contributing players per team. These novel variables include:
1. A series of five dichotomous scheduling fatigue variables for each team, which
detail the team‟s game schedule history. Variables identify whether the “un-
played” game will be the team in questions, fourth game over the previous five
days, third game in four days, second game in three days, or if the team competed
in the an away or home contest the preceding day;
4 2. A contribution factor variable, which accounts for a team‟s roster depth and talent
by recording the number of athletes who play considerable minutes.
While these concepts on their own are hardly revolutionary, previous research has failed to empirically examine their influence on game outcomes. The correlation between traditional box score statistics and competition outcomes is well established, however research including alternative explanatory variables is utterly lacking. A deeper understanding of the impact of these fatigue-related variables would be valuable for coaches and general managers alike. Information pertaining to the number of contributing players could alter player rotation strategies, while quantifying the impact of scheduling fatigue could prompt a change in team traveling protocols.
Two distinct populations (i.e., the first consisting of 2007-2008 NBA regular season contests, the second composed of 2008-2009 NBA regular season games) were analyzed. A total of 24 models were constructed for each population, as shown in Table
1. In essence the populations may be perceived as confirmatory complements. The dual populations facilitate comparisons in regards to prediction preciseness and the individual contributions of independent variables. These 24 models can be segmented into four general classifications, with each model category vastly differing in their operational definitions of performance measures. More specifically, absolute models (i.e., models 1-
6), incorporate absolute performance measures, relative models (i.e., models 7-12) consist of statistics defined in relative terms, possession models (i.e., models 13-18) build on Oliver‟s (2004) conceptualizations of offensive and defensive rating, and play models
(i.e., models 19-24) are composed of points-per-play, an alternative measure of offensive
5 Dependent # Description # IV Variable(s)
1 Absolute model 64 Score differential 2 Absolute model (Streak variables) 64 Score differential 3 Absolute model (Logarithmic variables) 64 Score differential 4 Absolute model (Streak logarithmic variables) 64 Score differential 5 Absolute model 64 Game winner 6 Absolute model (Streak variables) 64 Game winner 7 Relative model 32 Score differential 8 Relative model (Streak variables) 32 Score differential 9 Relative model (Logarithmic variables) 32 Score differential 10 Relative model (Streak logarithmic variables) 32 Score differential 11 Relative model 32 Game winner 12 Relative model (Streak variables) 32 Game winner 13 Possession model 18 Score differential 14 Possession model (Streak variables) 18 Score differential 15 Possession model (Logarithmic variables) 18 Score differential 16 Possession model (Streak logarithmic variables) 18 Score differential 17 Possession model 18 Game winner 18 Possession model (Streak variables) 18 Game winner 19 Play model 20 Score differential 20 Play model (Streak variables) 20 Score differential 21 Play model (Logarithmic variables) 20 Score differential 22 Play model (Streak logarithmic variables) 20 Score differential 23 Play model 20 Game winner 24 Play model (Streak variables) 20 Game winner
Table 1. Basic model descriptions
6 and defensive efficiency. As a whole, each of the 24 models contain a variety of team performance statistics that are amassed and averaged over previously “played” games, and analyzed in relation to the outcome of an “un-played” contest.
Representing the most robust conceptualization, absolute models include performance measures for the competing teams in an unaltered form. These absolute measures mirror statistics conventionally found in a standard box score (e.g., free throws made, defensive rebounds, and steals). Relative models consist of performance measures derived as ratios that compare a team‟s offensive and defensive ability in a given statistical category. The inclusion of performance ratios is beneficial as it inevitably puts a team‟s capabilities into perspective. Examples of the performance measures in the relative models are field goal percentage ratio, offensive rebound ratio and turnover ratio.
Conceived from a different perspective, the possession models along with the play models consist of composite indices of pace (i.e., the speed that a team plays), offensive efficiency and defensive efficiency. Possession models include possessions as a measure of pace, and a team‟s offensive rating and defensive rating as indicators of efficiency.
Alternatively, play models include a team‟s number of offensive plays and defensive plays as indicators of pace, and points per offensive play and points per defensive play as measures of team efficiency. Although similar, these two model categories differ in their classification of offensive rebounds. Gathering an offensive rebound allows a team to maintain control of the ball, yet under the present conceptualization, this situation yields one possession, but two plays.
7 Each subset of models can be further divided into those that focus on team statistics amassed over all previously “played” games and those that include performance measures accumulated only over the previous five games (i.e., streak models). The concept of streak models was adopted from Loeffelholz et al. (2009). Streak models were constructed in order to account for injuries as well as positive / negative fluctuations in play. Over the length of an 82 game season, levels of performance hardly remain steady as teams often undergo hot streaks as well as cold spells. With the intention of accounting for these positive and negative surges in play, as well as the potential impact of injuries, streak models were constructed in the absolute, relative, possession and play based varieties.
While basketball production functions commonly include absolute and relative measures of performance, the possession models and play models offer a fresh outlook.
As mentioned previously, the ability to provide a different perspective to sport-related statistical analyses can be immensely beneficial. Concepts pertaining to pace and efficiency are accepted as valuable tools among the basketball community, yet basic box score statistics are generally referenced by the media when discussing team evaluations.
The present research explores the validity of these composite measures in terms of explaining and predicting competition outcomes. Additional evidence of the value of these efficiency measures could facilitate a shift toward composite team statistics.
The primary purpose of the current investigation entails an extensive analysis of the explanatory and predictive capabilities of the 24 models in relation to the score differential of an “un-played” competition. A secondary intention of the present study
8 parallels the primary purpose, but deviates in its employment of a dichotomous dependent variable, which simply indicates whether the home or away team won the “un- played” game. Lastly, a tertiary purpose for the current study involves an examination of the individual contributions for the independent variables within each model.
Theoretical Foundation
A fundamental tenet of economic theory, a production function delineates the relationship between a series of inputs and outputs for a given entity as a mathematical expression (Bishop, 2004). More specifically, a production function is a heuristic devise used to determine the maximum attainable output produced by an assortment of inputs for a decision making unit (Greene, 1997; Miller, 2008). As noted by Greene, using the multifaceted notion of a decision making unit allows for the inclusion of alternative agents (e.g., service organizations) that transform inputs into outputs through the rearrangement or redistribution of resources, as opposed to an outright transformation of form. In terms of notation, a production function relationship is expressed as a mapping,
+ + + where X is a vector inputs and XK ∈ ℝ , Y is a vector of outputs and YM ∈ ℝ , and f: ℝ K
+ 1 ℝ M . In a similar fashion, the production function is commonly portrayed through the formulation Y = f(X). Simply stated, the production function can be conceptualized as a translucent box which converts a combination of inputs into a combination of outputs.
From a microeconomic perspective, a firm‟s production function reveals the largest possible output achievable for a particular set of inputs, under a given
1 ∈ =An element of, ℝ+ = Positive real number set
9 technological state (Aigner & Chu, 1968). As expected, advances in technology can significantly alter a firm‟s level of production; therefore it is critical to understand that a firm‟s production function over time is hardly static in nature. However, according to
Aigner and Chu, an individual firm‟s production function is conceptually identical to the industry production function, as other firms operating within the industry are similarly bound by predetermined optimal output levels.
While the production function originated as a microeconomic endeavor, macroeconomists soon recognized that the exercise of empirically estimating a production function was applicable at both levels (Miller, 2008). Unlike micro-level production functions, implementing production functions at the macro-level requires the aggregation of resources. The practice of constructing aggregate production functions can be traced back to the innovative and controversial Cobb and Douglas (1928) paper, which examined the aggregate production of the United States manufacturing industry as a function of labor and capital. Fast forwarding to today, the production function has become a cornerstone of econometric theory. Analyses using production functions differ substantially in scope as well as context. Examples of the production function‟s multitude of applications include research focusing on the productivity of the Alabama logging industry (e.g., Duc, Shen, Zhang & Smidt, 2009), Major League Baseball franchises (e.g., Woolway, 1997), Chinese regional economies (e.g., Bairam, 1996),
Jordanian concrete companies (e.g., Bataineh & Bataineh, 2008), and educational institutions in rural Pakistan (e.g., Khan & Kiefer, 2007).
10 In essence the primary intention of all production functions are equivalent (i.e., to express a relationship between inputs and outputs), but differences arising from unique facets in the production process has resulted in numerous mathematical representations.
The presence of production function forms such as the linear production function, the
Cobb-Douglas production function, the Constant Elasticity of Substitution (CES) production function, and the transcendental logarithmic production function allow for variations on critical economic components (e.g., factor substitution, economics of scale and input demand elasticities; Greene, 1997). Prior to discussing in detail the strengths, weaknesses and previous applications for each production function type, a brief synopsis of a few economic concepts will ensue.
Elasticity of substitution is a quantifiable measure relating to the ease of switching between factor inputs (Miller, 2008). For example, the elasticity of substitution for a basic two factor microeconomic production function would measure the firm‟s ability to substitute labor for capital and vice versa. The measure is calculated as the percentage change in factor proportions stemming from a one-unit change in one factor while holding the output level constant (Miller). The values for elasticity of substitution range from zero to infinity, with zero indicating fixed factor proportions with no flexibility, and a value of infinity signifying perfect substitutability (Barnes, Price & Barriel, 2008).
Underdeveloped nations serve as a quintessential example of rigid factor proportions, as these economies tend to be labor intensive with limited opportunities to readily replace labor into capital (McManus, 1988). Economies of this nature support an elasticity of substitution less than unity, as changes in the marginal rate of technical substitution result
11 in minute changes to factor proportions; thus a factor‟s relative share rises when it‟s corresponding quantity declines in relation to other factor quantities (Bronfenbrenner,
1960). At the extreme with fixed factor proportions, the Leontief production function, exhibits an elasticity of substitution equal to zero, indicating perfect complements
(Barnes et al.; Miller). On the opposite end of the spectrum, an elasticity of substitution of infinity (i.e., a linear production function), signifies perfect substitutability with a constant marginal rate of technical substitution (Barnes et al.; Miller).
Economies of scale refer to the benefits gained from firm expansion. In the majority of industries, bigger is better, and consequently increases in output are often accompanied by a decrease in the average cost incurred per unit produced (Bishop,
2004). Economies of scale was and remains the catalyst behind corporate gigantism and can generally be segmented into the internal (e.g., reductions in overhead for research and development or savings arising from specialized labor and machinery) and external (i.e., benefits accrued as a result of industry organization) varieties (Hindle, 2003).
Conversely, diseconomies of scale are negative ramifications associated with organizational growth, such as a loss in productivity as a direct result of failing to efficiently manage the logistics of a larger operation.
Closely aligned with the concept of economies of scale are the economic tertiary terms pertaining to returns to scale. Decreasing returns to scale emerge when a proportional increase in inputs leads to a less than proportional increase in outputs
(Wiens, n.d.). Alternatively, under increasing returns to scale, a proportional increase in inputs results in an output level more than the proportional constant (Weins). Finally,
12 constant returns to scale occur in the production process when inputs are proportionally increased by a constant and outputs increase by the same proportional constant (Miller,
2008).
In order to visually demonstrate the differences between the three concepts a series of rudimentary production functions, which include a single input and single output, are illustrated in Figure 1. As shown in Figure 1A, doubling input values from 15 to 30 results in a disproportionate increase in output, simply stated the proportional increase stemming from ΔI1 is less than the associated proportional increase in output.
Alternatively, Figure 1B depicts increasing returns to scale, as an increase in input leads to a change in output level, ΔQ2, of larger proportion. Lastly, Figure 1C portrays a production function with constant returns to scale, as an increase in input creates an identical proportional increase in output. With a solid grasp of production function basics, the following sections will describe the most common functional forms.
Linear production function. The linear production function is the most simplistic in nature. As the name implies, a linear production function is a production function that is graphically depicted as a straight line. For a firm with two factor inputs, the quantity of production can be empirically estimated with the following equation:
(1) Q = β0 + β1 * L + β2 * K where Q equals the predicted quantity produced, β0 is a constant, β1 represents the coefficient of labor, L equals labor, β2 represents the coefficient of capital and K equals capital. Unfortunately, due to the production function‟s rudimentary form as well as its intrinsic linearity, the linear production function lacks the ability to account for several
13
A
B continued
Figure 1. Single input and single output production function: A decreasing returns to scale with diseconomies of scale, B increasing returns to scale with economies of scale, C constant returns to scale
14 Figure 1 continued
C
15 pivotal economic components (e.g., factor substitutions and economies of scale). The linear production function is the easiest of the four production function formulations to implement and interpret.
The weaknesses associated with the linear production function, along with the strength of alternative formulations, have limited the application of the linear production function in research endeavors. However, despite its disadvantages, in situations where economies of scale, factor substitutions and other fundamental economic tenets are of no concern, use of the linear production function is often deemed appropriate. For example,
Bataineh and Bataineh (2008) in their research investigation of the production for
Jordanian concrete companies incorporated a linear production function including capital, labor and a dummy variable indicating the decreased value of the Jordanian dinar as inputs. In an alternative study, Khan and Kiefer‟s (2007) construction of a linear educational production function for Pakistani institutions, serves as a testament to the versatility of the linear production function. Khan and Kiefer examined the relationship between a vector of student comprehension and mathematic test scores, which were identified as the linear production function‟s output, and a series of vector inputs, which included measures of school quality characteristics (e.g., teacher experience), child and household characteristics (e.g., parent‟s education and wealth), and school inputs under parental control (e.g., student attendance).
Cobb-Douglas production function. Conceived over 80 years ago, the Cobb-
Douglas production function remains “the ubiquitous form in theoretical and empirical analyses of growth and productivity” (Felipe & Adams, 2005, p. 428). Today the Cobb-
16 Douglas production function is an indispensable tool in econometric research and is critical to the estimation of aggregate production function parameters as well as investigations focusing on maximum output levels, technical change and other theoretical economic constructs (Felipe & Adams; McCombie, 1998). The Cobb-Douglas production function emerged as a result of Cobb and Douglas‟s (1928) realization that refinements in physical production measurements opened up the possibility of ascertaining the relationship among labor, capital and quantity produced. Along the same lines, Cobb and Douglas aspired to explore the impact of changes to factor input proportions, which could subsequently be used to deduce marginal products. Armed with an abundance of data relating to the US manufacturing industry from 1899 to 1922, the
Cobb-Douglas aggregate production function took the following form:
(2) Q = 1.01 * L3/4 * K1/4 where Q equals the production level predicted, 1.01 serves as a constant, and L and K represents labor and capital respectively.
While Cobb and Douglas (1928) offered several pieces of evidence in support of the accuracy of their construction (e.g., the close correlations between predicted production and actual production for both when secular trends were included [r = 0.97] and excluded [r=0.94], as well as the adherence to economic principles such as the fact that predicted level of output approaches zero as either L or K approach zero), criticism of the Cobb-Douglas production function has been prevalent. Contentions surrounding the applicability of the Cobb-Douglas production function generally revolve around difficulties pertaining to the aggregation of resources. More specifically, Felipe and
17 Adams (2005) argue that the aggregation of micro-production functions produces macro- production functions that are unrealistic for a real world setting. In addition, detractors of the Cobb-Douglas production function have raised concerns regarding the lack of an exponential time trend component to address technical progress and the alarming similarities between the production function and the accounting identity (McCombie,
1998).
Conversely, supporters of the Cobb-Douglas production function adamantly reference the multitude of cross sectional studies that have rather precisely reproduced the production function with little divergences in the value of the labor and capital exponents (McCombie). Needless to say, there is no general consensus on the appropriateness or accuracy of the Cobb-Douglas production function, or macro-level production functions as a whole.
Since Cobb and Douglas‟s (1928) original inception, applications of their production function have expanded substantially beyond the two factor macro-level conceptualization. In its generalized form, the Cobb-Douglas production function is as follows:
α γ λ (3) Yi = β0 * X1 * X2 * … Xn where Yi is a vector of outputs, β0 serves as a constant, Xn is a vector of inputs with each input accompanied by exponential coefficients (i.e., α, γ and λ). Unlike the linear production function, the Cobb-Douglas production function‟s exponential components permit for variations in economies of scale as well as returns to scale. In the generalized formulation where α + γ + λ < 1, the production function demonstrates diseconomies of
18 scale, as shown in Figure 1a (Farrell, 1957; Weins, n.d.). Alternatively, in Figure 1b, when α + γ + λ > 1, the production function can be categorized by its economies of scale
(Farrell; Weins). As mentioned previously, the ramifications arising from firm growth can be of the positive (i.e., economies of scale) as well as negative (i.e., diseconomies of scale) variety.
In a similar fashion the Cobb-Douglas production function is capable of representing entities with decreasing returns to scale, constant returns to scale and increasing returns to scale. While the Cobb-Douglas production function‟s ability to represent varying returns to scale is a commendable upgrade over the linear production function, it nevertheless assumes constant technology and lacks the ability to account for variations in elasticities of substitution (McCombie, 1998). As a result, the elasticity of substitution for the Cobb-Douglas production function is always unitary (Bishop &
Brand, 2003; Wiens, n.d.).
In order to facilitate the use of the Cobb-Douglas production function in statistical analyses, a linear estimation can be obtained by taking the natural logarithm of the variables in the equation (Griffin, Montgomery & Rister, 1987). The result of executing a logarithmic transformation for equation 3 is illustrated in equation 4.
(4) lnY = β0 + β1 * lnX1 + β2 * lnX2 + … βN * lnXn
It is imperative to recognize that the coefficients of equation 4 are not exactly equal to the exponential coefficients of equation 3, as the coefficients of equation 4 are merely estimations of the unknown parameters of equation 3 (i.e., β0 ≠ β0, β1 ≠ α, β2 ≠ γ, and βN
≠ λ). This practice of constructing linear approximations of the Cobb-Douglas
19 production function has been implemented in a copious number of studies, in a myriad of contexts. A few examples of research using the Cobb-Douglas production function will be discussed in the following paragraphs.
Macro-level analyses using the Cobb-Douglas production function have been conducted worldwide. In an effort to explain the economic growth in Poland from 1994 to 2000, the Organization for Economic Co-operation and Development (2001) constructed a linear approximation of the Cobb-Douglas production function with real
GDP serving as the function‟s output and capital stock, as well as labor acting as factor inputs. A second example of a macroeconomic analysis was conducted in Samad‟s
(2009) investigation of the Bangladesh banking industry. Samad used natural logarithms to construct a linear estimation of the Cobb-Douglas production function, which included total loans as the production function‟s output, and total deposits and number of employees as inputs.
Similarly, on the micro-level, the Cobb-Douglas production function has been a vital and versatile tool in production analyses. In the realm of sport, Woolway (1997) defined a Major League Baseball franchise‟s winning percentage as a function of several statistical measures including on-base percentage, slugging percentage, stolen base, earned run average and unearned run average. Like the aforementioned studies,
Woolway represented this relationship through a linear approximation of the Cobb-
Douglas production function. Taken as a whole, as noted by Miller (2008), the widespread use of the Cobb-Douglas production function stems from its two major
20 strengths, the production functions ease of use and its ability to produce solid empirical fits across a multitude of data sets.
Constant elasticity of substitution production function (CES). More advanced than the Cobb-Douglas production function, CES production functions permit for variations in the value of the elasticity of substitution. An example of a generalized three factor CES production function is shown through the following equation (Wiens, n.d.):
-ρ -ρ -ρ -ν/ρ (5) Y = β0 * (β1 * X1 + β2 * X2 + β3 * X3 ) where the ν parameter represents a measure for economies of scale and ρ facilitates the calculation of elasticity of substitution. Like the Cobb-Douglas production function, the
CES‟s exponential components allow for production functions with economies of scale and diseconomies of scale. Furthermore, according to Wiens, differing values of ν permit for production functions with deceasing returns to scale (i.e., ν < 1), constant returns to scale (i.e., ν = 1) and increasing returns to scale (i.e., ν > 1). However, it must be noted, that while the CES production function can represent different elasticities of substitution, its intrinsic complexities convolute the analysis of production. Duc et al.‟s (2009) research on production in the Alabama logging industry serves as an example of calculating the elasticity of substitution between factor inputs with the assistance of a
CES production function.
Transcendental logarithmic production function. Of the four production functions discussed throughout this paper, the transcendental logarithmic production function, or the translog production function for short, is the most comprehensive design.
The introduction of the translog production function was based on Christensen, Jorgenson
21 and Lau‟s (1973) finding that the underlying assumption of commodity-wise additivity was unsatisfactory when constructing a production function with multiple inputs and outputs. The translog production function includes the advantages of the aforementioned production functions (i.e., the ability to represent economies of scale, diseconomies of scale, and all three variations of returns to scale), but additionally permits partial elasticities of substitutions to vary between inputs (Wiens, n.d.). An example of a generalized two factor non-homogeneous translog production function is illustrated by the following (Brand & Brand, 2003):
(6) ln Q = β0 + βL * ln L + βK * ln K + 0.5 * βLL * (ln L)2 + 0.5 * βKK * (ln K)2 +
βLK * ln L * ln K where Q is the function‟s output, and the inputs L and K represent labor and capital respectively. In the time since Christensen et al. created the translog production function, modifications such as Kymn and Hisnanick‟s (2001) derivation, which included a technology parameter, have further increased the functionality of the production function.
Of course additional capabilities are often accompanied by drawbacks, and the translog production function is no different. The inclusion of additional parameters and coefficients unfortunately complicates the interpretation of the production relationship as well as increases the difficulty associated with conducting statistical analyses.
Nevertheless, depending on the context, the translog production function remains a viable and often appropriate option. Evidence of the versatility of the translog production function is demonstrated in Bishop and Brand‟s investigation on the production and efficiency of museums residing in England. For their study, output was operationally
22 defined as the number of physical visits and subscribers, while inputs included running and maintenance costs, and the number of full time equivalent workers.
In concluding this introduction to production functions, their applications, and their functional forms, it is worth briefly noting that while useful in their own right, production functions are commonly a means to an end for efficiency analyses (Greene,
1997). The construct of economic efficiency is composed of two distinct components, technical efficiency, which refers to an economic unit‟s ability to maximize output given a predetermined set of inputs and technologies, and allocative efficiency, which is an economic unit‟s operating capabilities in regards to equating “…its specific marginal value product with its marginal cost” (Kalirajan & Shand, 1999, p. 149). Once a theoretical ideal, known as the production frontier, is obtained from the production function, efficiency can be measured by examining the distance between an actual observation and the production frontier (Greene). Furthermore, the frontier production model assists in comparisons among individual economic actors.
Production Function Applications in Sport
The practice of analyzing performance and efficiency has invaded numerous industries, with professional sport serving as an ideal laboratory for empirical investigations. Equipped with an abundance of observable and quantifiable data, capable of acting as inputs and outputs, sport is uniquely suited for production function and efficiency analyses. While sport at any level is capable of providing a sufficient amount of data in regards to the creation of production functions, the profusion of readily
23 accessible statistics has catapulted professional sport to the forefront of sport related production function research.
Pioneering the trend of conducting production and efficiency research within sport, Scully (1974) sought to explore the degree of monopsonistic exploitation stemming from Major League Baseball‟s restrictive reserve clause. In the process of his analysis,
Scully constructed a pair of linear production functions to explain a team‟s winning percentage and its revenue generated. Along the same lines, with a focus on the National
Football League, Hadley, Poitras, Ruggiero and Knowles‟s (2000) research involved the estimation of team and head coach performance through the creation of a production frontier of offensive and defensive performance measures. Additional research in professional sport includes Kahane‟s (2005) stochastic frontier approach, which estimated National Hockey League franchise production with both Cobb-Douglas production functions and translog production functions, and Haas‟s (2003) evaluation of technical efficiency for Major League Soccer team teams with data envelopment analysis.
Selection and Justification of Production Function Formulations
The present study employs numerous production functions, which are modeled either in the linear or Cobb-Douglas functional form. The rationale for incorporating the
Cobb-Douglas formulation generally revolves around its versatility, high levels of acceptability across a variety of disciplines, as well as its ease of use and interpretation
(McCombie, 1998; Miller, 2008; Woolway, 1997; Zak, Huang & Siegfried, 1979).
Similarly, as the most rudimentary formulation, linear production functions are advantageous due to innate simplicities in terms of use and interpretation. In fact “all
24 previous studies, except those applying the nonparametric method of data envelopment analysis, assume the appropriate functional form for a production function in sports to be either linear or Cobb-Douglas” (Lee & Berri, 2008, p. 57). While general acceptance among the academic community lends credence to appropriateness, the selection of the aforementioned functional forms is additionally grounded in economic and basketball theory.
As mentioned previously, linear production functions contain constant returns to scale and an infinite elasticity of substitution. Under the present conceptualization, constant returns to scale imply that an increase in a team‟s positive performance measures leads to a proportional increase in their score differential. Coinciding with an infinite elasticity of substitution, all inputs for linear production functions are classified as perfect substitutes. Within the context of basketball, perfect substitutability would mean that any performance measure could simply be converted into an alternative performance measure. While the notion of perfect substitutes works conceptually in some instances (e.g., a team could alter their strategy to increase offensive rebounds at the cost of decreasing their field goal percentage defense), it remains problematic for a multitude of performance measures (e.g., substituting free throw percentage for blocked shots). Despite concerns regarding an infinite elasticity of substitution, the linear production function is worth exploring due to its intrinsic simplicities as well as its widespread use in previous research.
Unlike linear production functions, the Cobb-Douglas production function is capable of representing varying returns to scale. Given that the relationships between the
25 inputs and outputs as operationally defined by the current study has yet to be explored, measuring or assessing the potential for differing returns to scale is advantageous. For example, it is entirely possible that an increase in performance inputs leads to a disproportionate increase in outputs (i.e., decreasing returns to scale) due to game time constraints and a finite number of possessions / plays. Of course technically, the absence of a maximum number of overtimes could result in a contest that extends across eternity; however this notion is simply not supported by reality.
With a unitary elasticity of substitution, Cobb-Douglas production functions include inputs which are neither perfect complements nor perfect substitutes (Barnes et al., 2008; Brontenbrenner, 1960). Intuitively, the identification of performance measures in between the spectrum‟s extremes of perfect complements and substitutes is sound.
Although assuming unitary elasticity of substitution with the Cobb-Douglas production function is mildly concerning, assigning an elasticity of substitution for each performance measure is an exercise that lacks practical significance (e.g., determining the elasticity of substitution between ratio of assists and ratio of defensive rebounds). While the CES production function would allow for a single value for the elasticity of substitution measure, as noted by Christensen et al. (1973), this formulation is unsatisfactory for functions with multiple inputs and outputs. Therefore, with a CES production function, it would be assumed that all inputs possess identical elasticities of substitution, hardly a significant improvement over the Cobb-Douglas formulation. Alternatively, while the tranlog production function permits for partial elasticities of substitution between inputs, it convolutes statistical analyses and variable interpretations. With production functions
26 ranging from 18 to 64 inputs, the calculation of partial elasticities of substitution is an arduous task that is both cumbersome and contains little practical implications. As a whole, the Cobb-Douglas production function offers a functional form that is widely accepted and able to depict varying returns to scale with a unitary elasticity of substitution; consequentially, the Cobb Douglas production function is a proper fit for the current investigation.
Research Questions
1) With what accuracy can the absolute model, the relative model, the possession
model, and the play model explain and predict NBA score differentials for the
2007-2008 and 2008-2009 seasons?
2) At what rate of success can the absolute model, the relative model, the possession
model, and the play model classify individual game winners for the 2007-2008
and 2008-2009 NBA seasons?
3) Do streak models predict the dependent variable (i.e., score differential or game
winner) better than their non-streak counterparts?
4) How are NBA game score differentials associated with the competing teams‟
average number of contributing players and fatigue stemming from playing and/or
traveling in the preceding days?
5) In terms of relative importance and variable inclusion, are regression equations
consistent between the 2007-2008 and 2008-2009 NBA seasons?
Models of Performance
27 Absolute models. As shown in Table 1, models 1-6 are of the absolute variety.
Each absolute model contains 64 independent variables, as depicted in Table 2, and includes previously accumulated statistical averages in absolute terms. The use of absolute statistics coincides with previous production function research such as the investigations of Grier and Tollison (1990), Loeffelholz et al. (2009), McGoldrick and
Voeks (2005), Sánchez, Castellanos & Dopico (2007), and Scott, Long and Somppi
(1985). Given that absolute statistical measures of performance are free from manipulation, these figures inevitability encapsulate the greatest amount of information.
From these numerous absolute measures, the independent variables of the other models can be constructed.
Differences among the six absolute models stem from slight variations in the operationalization of the independent and dependent variables (e.g., model 3 is nearly identical to model 1, with the exception that model 3 uses logarithmic variables). Models
1-4 include score differential as the dependent variable, which is simply the away team‟s final score minus the home team‟s final score. Models 5 and 6 include game winner as the dependent variable, which is dummy coded to indicate whether the home or away team was victorious. The independent variables encompassed within the absolute models include home and away team averages upon entering the “un-played” game in a multitude of statistical categories as well as an average depth contribution factor and variables accounting for scheduling fatigue. For both competing teams, offensive and defensive averages per game are computed in two point field goals made, two point field goals attempted, three pointers made, three pointers attempted, free throws made, free
28 # Name Description
1 A2FGMA Away two point field goals made, average per game. 2 A2FGAA Away two point field goals attempted, average per game. 3 A3PMA Away three pointers made, average per game 4 A3PAA Away three pointers attempted, average per game 5 AFTMA Away free throws made, average per game 6 AFTAA Away free throws attempted, average per game 7 AOREBA Away offensive rebounds, average per game 8 ADREBA Away defensive rebounds, average per game 9 AASTA Away assists, average per game 10 ASTLA Away steals, average per game 11 ABLKA Away blocked shots, average per game 12 ATOA Away turnovers, average per game 13 APFA Away personal fouls, average per game 14 A2FGMAA Away two point field goals made allowed, average per game 15 A2FGAAA Away two point field goals attempted allowed, average per game 16 A3PMAA Away three pointers made allowed, average per game 17 A3PAAA Away three pointers attempted allowed, average per game 18 AFTMAA Away free throws made allowed, average per game 19 AFTAAA Away free throws attempted allowed, average per game 20 AOREBAA Away offensive rebounds allowed, average per game 21 ADREBAA Away defensive rebounds allowed, average per game 22 AASTAA Away assists allowed, average per game 23 ASTLAA Away steals allowed, average per game 24 ABLKAA Away blocked shots allowed, average per game 25 ATOAA Away turnovers allowed, average per game 26 APFAA Away personal fouls allowed, average per game 27 ACONA Away team depth contribution factor, average per game 28 A4DP Scheduling Fatigue: Away four days prior 29 A3DP Scheduling Fatigue: Away three days prior 30 A2DP Scheduling Fatigue: Away two days prior 31 A1DPV1 Scheduling Fatigue: Away one day prior variable one 32 A1DPV2 Scheduling Fatigue: Away one day prior variable two
Continued
Table 2. Independent variables for absolute models
29 Table 2 continued
33 H2FGMA Home two point field goals made, average per game. 34 H2FGAA Homes two point field goals attempted, average per game. 35 H3PMA Home three pointers made, average per game 36 H3PAA Home three pointers attempted, average per game 37 HFTMA Home free throws made, average per game 38 HFTAA Home free throws attempted, average per game 39 HOREBA Home offensive rebounds, average per game 40 HDREBA Home defensive rebounds, average per game 41 HASTA Home assists, average per game 42 HSTLA Home steals, average per game 43 HBLKA Home blocked shots, average per game 44 HTOA Home turnovers, average per game 45 HPFA Home personal fouls, average per game 46 H2FGMAA Home two point field goals made allowed, average per game 47 H2FGAAA Home two point field goals attempted allowed, average per game 48 H3PMAA Home three pointers made allowed, average per game 49 H3PAAA Home three pointers attempted allowed, average per game 50 HFTMAA Home free throws made allowed, average per game 51 HFTAAA Home free throws attempted allowed, average per game 52 HOREBAA Home offensive rebounds allowed, average per game 53 HDREBAA Home defensive rebounds allowed, average per game 54 HASTAA Home assists allowed, average per game 55 HSTLAA Home steals allowed, average per game 56 HBLKAA Home blocked shots allowed, average per game 57 HTOAA Home turnovers allowed, average per game 58 HPFAA Home personal fouls allowed, average per game 59 HCONA Home team depth contribution factor, average per game 60 H4DP Scheduling Fatigue: Home four days prior 61 H3DP Scheduling Fatigue: Home three days prior 62 H2DP Scheduling Fatigue: Home two days prior 63 H1DPV1 Scheduling Fatigue: Home one day prior variable one 64 H1DPV2 Scheduling Fatigue: Home one day prior variable two
30 throws attempted, offensive rebounds, defensive rebounds, assists, steals, blocked shots, turnovers and personal fouls. As illustrated in Table 2, variables 1-13 are the away team‟s statistics, variables14-26 are the away team‟s previous opponents‟ performance measures, variables 33-45 are the home team‟s statistics, and variables 46-58 are the home team‟s previous opponents‟ performance measures. Scheduling fatigue indicators for the competing teams include a variable to identify whether the team in question will be playing its fourth game in five days, third game in four days, second game in three days, as well as two variables to determine if the team participated in an away game, home game, or no game, the day immediately preceding the “un-played” contest.
It is worth noting that several traditional box score statistics have been altered or excluded due to redundancies. More specifically, instead of including total field goals made and attempted, which would result in the double counting of three point shots, field goals were separated into two and three point makes and attempts. Additionally, total rebounds were not included as it is simply the combination of offensive rebounds and defensive rebounds, and total points were excluded as points produced represents an alternative measure of two point field goals made, three point field goals made and free throws made.
Relative models. Composed of 32 independent variables, models 7-12 (see Table
1), are classified as the relative models. For these six models, team averages are operationally defined in relative comparison with opponents‟ averages. Previous basketball performance research utilizing relative statistics include Chatterjee, Campbell and Wiseman (1994), Hofler and Payne (1997), McGoldrick and Voeks (2005), Sánchez
31 et al. (2007) and Zak et al. (1979). Descriptions of the independent variables included within the relative models are displayed in Table 3. According to Zak et al. (1979), ratios and differences are beneficial as they provide a comparative measure for each performance statistic. While it may be impressive that team A on average shoots 39% from the three point line, learning that over the season it allows opponents to shoot 41% from three point land significantly reduces the luster associated with team A‟s three point prowess. Alternatively, stating that team A has an average relative three point percentage of 0.95 puts the team‟s statistic in perspective. A ratio hovering around a value of 1 indicates that the team is very similar to its opponents on the specified performance measure (McGoldrick & Voeks). Conversely, ratio values straying further from 1 suggest that a team on average outperforms or underperforms its opposition in regards to that statistical category (McGoldrick & Voeks). Individual independent variables within the relative models are calculated with ratios, which are found by dividing the team‟s statistic by the team‟s allowed statistic (e.g., the away team‟s defensive rebounds divided by the away team‟s defensive rebounds allowed).
The independent variables making up models 7-12 include home and away team average ratios in numerous performance categories as well as a team depth contribution factor and scheduling fatigue measures. While previous studies using relative models
(e.g., McGoldrick and Voeks; Sánchez et al.; Zak et al.) have commonly measured steals and blocked shots as differences, due to the fact that NBA teams in general do not accumulate these statistics in large numbers and therefore it is assumed that differences would result in a more accurate representation than ratios, the current investigation‟s
32 # Name Description
1 A2FGR Away two point field goal ratio 2 A3PR Away three point field goal ratio 3 AFTR Away free throw ratio 4 AOREBR Away offensive rebounds ratio 5 ADREBR Away defensive rebounds ratio 6 AASTR Away assists ratio 7 ASTLR Away steals ratio 8 ABLKR Away blocked shots ratio 9 ATOR Away turnover ratio 10 APFR Away personal foul ratio 11 ACONR Away team depth contribution factor ratio 12 A4DP Scheduling Fatigue: Away four days prior 13 A3DP Scheduling Fatigue: Away three days prior 14 A2DP Scheduling Fatigue: Away two days prior 15 A1DPV1 Scheduling Fatigue: Away one day prior variable one 16 A1DPV2 Scheduling Fatigue: Away one day prior variable two 17 H2FGR Home two point field goal ratio 18 H3PR Home three point field goal ratio 19 HFTR Home free throw ratio 20 HOREBR Home offensive rebound ratio 21 HDREBR Home defensive rebound ratio 22 HASTR Home assist ratio 23 HSTLR Home steal ratio 24 HBLKR Home blocked shots ratio 25 HTOR Home turnover ratio 26 HPFR Home personal foul ratio 27 HCONR Home team depth contribution factor ratio 28 H4DP Scheduling Fatigue: Home four days prior 29 H3DP Scheduling Fatigue: Home three days prior 30 H2DP Scheduling Fatigue: Home two days prior 31 H1DPV1 Scheduling Fatigue: Home one day prior variable one 32 H1DPV2 Scheduling Fatigue: Home one day prior variable two
Table 3. Independent variables for relative models
33 longitudinal approach eliminates any need to incorporate difference measures.
Furthermore, the representation of all performance measures as ratios is beneficial as it maintains consistency throughout all relative model variables.
Possession models. Consisting of 18 independent variables, models 13-18 (see
Table 1), are conceptualized as the possession models. Referring to the number of times a team has acquired the ball, possessions can be estimated with knowledge of field goal attempts, offensive rebounds, defensive rebounds, turnovers and free throws. For the current investigation, the number of possessions employed is estimated with the following equation (Oliver, 2004):
(7) Team's possessions employed = Team's field goals attempted - (Team's offensive
rebounds / (Team's offensive rebounds + Opponent's defensive rebounds)) *
(Team's field goals attempted - Team's field goals made) * 1.07 + Team's
turnovers + 0.4 * Team's free throws attempted
The number of free throws awarded to a player varies depending on the situation. For example, when a player is fouled while missing a three point field goal attempt, three free throws are awarded. In this instance, the first two free throw attempts cannot end the possession. The 0.4 multiplicative constant in equation 7, accounts for the fact that roughly 40% of free throw attempts result in the end of a possession (Oliver).
Additionally, is must be noted that Oliver includes the 1.07 multiplicative constant as a statistical adjustment to address the difference in estimated number of possessions and observed number of possessions.
34 Within a single competition the number of possessions will essentially be equivalent for both teams (Oliver, 2004). Conversely, over a season, the average number of possessions per team can vary significantly as a result of the employment of a diverse array of offensive and defensive schemes and strategies. While it is important that a team is able to set a pace conducive to its offensive and defensive strategies, success on the floor is contingent upon offensive and defensive efficiency, which for the possession models are represented through offensive and defensive ratings. To the naked eye it may appear as if high tempo offenses that tend to score an abundance of points (e.g., the 1990-
1991 Denver Nuggets, 116.9 possessions per game, 119.9 points per game, 102.6 offensive rating) are more efficient than slow plodding offenses (e.g., the 1998-1999
Miami Heat, 86.4 possessions per game, 89 points per game, 103 offensive rating;
Oliver), however, this is hardly always the case.
As the possession models strive to explain and predict the score differential of a competition and the game winner respectively, it is necessary to account for both dimensions of pace and efficiency. Although the importance of including efficiency measures in the exercise of predicting game outcomes is self-explanatory, the inclusion of pace can be equally critical, especially for score differential. Teams employing a lightning pace may not necessarily be more efficient than their deliberate brethren, but unavoidably, teams willing to increase tempo will both score and allow more points per game than slower methodical teams, which can influence score differential. Descriptions of the independent variables within the possession models are displayed in Table 4. For
35 # Name Description
1 APOSA Away possessions employed average per game 2 AOR Away offensive rating 3 ADR Away defensive rating 4 ACONA Away team depth contribution factor, average per game 5 A4DP Scheduling Fatigue: Away four days prior 6 A3DP Scheduling Fatigue: Away three days prior 7 A2DP Scheduling Fatigue: Away two days prior 8 A1DPV1 Scheduling Fatigue: Away one day prior variable one 9 A1DPV2 Scheduling Fatigue: Away one day prior variable two 10 HPOSA Home possessions employed average per game 11 HOR Home offensive rating 12 HDR Home defensive rating 13 HCONA Home team depth contribution factor, average per game 14 H4DP Scheduling Fatigue: Home four days prior 15 H3DP Scheduling Fatigue: Home three days prior 16 H2DP Scheduling Fatigue: Home two days prior 17 H1DPV1 Scheduling Fatigue: Home one day prior variable one 18 H1DPV2 Scheduling Fatigue: Home one day prior variable two
Table 4. Independent variables for possession models
36 each observation, home and away team‟s average possessions, offensive ratings, and defensive ratings, will serve as independent variables. Furthermore, for each team, a depth contribution factor and scheduling fatigue measures will be constructed.
Play models. Constructed as an alteration of the possession models, the play models (i.e., models 19-24, as shown in Table 1) contain a total of 20 independent variables. Unlike possessions, which are more or less equal for the two competing teams in a game, plays can differ substantially within a single contest as the gathering of an offensive rebound represents the initiation of a second play (Oliver, 2004). In the moments following a field goal attempt, neither team truly has control of the ball. From the possession model perspective, when the offensive team regains control of the ball through an offensive rebound, possession is maintained and therefore a single possession is recorded. Alternatively, from the play model‟s viewpoint, gathering an offensive rebound represents the end of the first offensive play and the beginning of a second offensive play. As a result of the differences emerging from the accumulation of offensive rebounds, two separate play measures are created for each team (i.e., average number of offensive plays per game and average number of defensive plays per game), both of which are included as independent variables in the play models. Determining the number offensive and defensive plays can be found through the following respective equations (Oliver):
(8) Total offensive plays = Total field goals attempted + Total turnovers + Total free
throws attempted * 0.4
37 (9) Total defensive plays = Opponent's total field goals attempted + Opponent's total
turnovers + Opponent's total free throws attempted * 0.4
The 0.4 multiplicative constant is included in the above calculations because not every free throw attempt reinitiates live action and is capable of starting a new play (Oliver).
While offensive and defensive plays are measures of pace, just like the possession models, it is imperative that the play models include measures of efficiency. Derived from Berri et al.‟s (2007) points per field goal attempt measure and Oliver‟s (2004) formula for plays, points-per-play is a simple efficiency measure which calculates the number of points on average a team scores / allows on each play. As shown in Table 5, for the home and away teams, offensive points-per-play and defensive points-per-play are independent variables. Finally, as was the case with all of the present investigation‟s models, the play models include independent variables relating to the number of contributing players and measures of scheduling fatigue.
Definition of Terms
For the purposes of increasing one‟s comprehension of the numerous relationships within the present investigation, a list of constitutive and operational definitions of variables will follow. In order to assist individuals unfamiliar with basketball lexicon, definitions for performance measures found in traditional box scores are included. To avoid confusion, it is imperative to understand that operational definitions of performance measures are contingent upon the model and are typically measured in absolute or relative terms. For a detailed description of which independent variables are included
38 # Name Description
1 AOPLA Away offensive plays, average per game 2 ADPLA Away defensive plays, average per game 3 AOPPP Away offensive points-per-play 4 ADPPP Away defensive points-per-play 5 ACONA Away team depth contribution factor, average per game 6 A4DP Scheduling Fatigue: Away four days prior 7 A3DP Scheduling Fatigue: Away three days prior 8 A2DP Scheduling Fatigue: Away two days prior 9 A1DPV1 Scheduling Fatigue: Away one day prior variable one 10 A1DPV2 Scheduling Fatigue: Away one day prior variable two 11 HOPLA Home offensive plays, average per game 12 HDPLA Home defensive plays, average per game 13 HOPPP Home offensive points-per-play 14 HDPPP Home defensive points-per-play 15 HCONA Home team depth contribution factor, average per game 16 H4DP Scheduling Fatigue: Home four days prior 17 H3DP Scheduling Fatigue: Home three days prior 18 H2DP Scheduling Fatigue: Home two days prior 19 H1DPV1 Scheduling Fatigue: Home one day prior variable one 20 H1DPV2 Scheduling Fatigue: Home one day prior variable two
Table 5. Independent variables for play models
39 within each model, reference the previous section (i.e., models of performance) along with the corresponding table.
Assist. An assist refers to a pass that is directly responsible for a successful field goal. Although no restrictions exist in terms of number of permitted dribbles or time elapsed between the pass and shot, the score must represent an immediate reaction
(International Basketball Federation, 2009; National Collegiate Athletic Association,
2009). A pass classified as an assist should ultimately be the catalyst behind the sequence, not merely a routine pass that happened to occur prior to the field goal. Only one assist may be awarded per field goal.
Unlike other traditional box score statistics, the awarding of an assist is subjective. Given the lack of a rigid definition, even the judgments of well respected statisticians may differ in certain situations. Statisticians employed by the home team are responsible for the record books and the divergence in home and away assists serves to empirically support the notion of subjectivity in classifying assists (Biderman, 2009).
Furthermore, as a testament to the influence of perspective, the percentage of field goals in which an assist was credited has increased rather significantly over the years. In the early 1970‟s roughly half of successful field goals were accompanied by an assist, while in the modern era this number has hovered around 60% (Biderman).
Blocked shot. A blocked shot is awarded when a defensive player sufficiently alters the flight path of a field goal attempt by making contact with the basketball and the field goal is unsuccessful (International Basketball Federation, 2009). In order for a blocked shot to be credited, the sequence must be devoid of goaltending and foul calls.
40 While it is not necessary that the ball be detached from the shooter‟s hand, the player must clearly be in the act of shooting. It should be noted that in these instances, it is not always clear whether the defensive player should be awarded with a blocked shot or a steal, determination is ultimately up to the statistician‟s discretion. In the event of a blocked shot, the offensive player is charged with a field goal attempt and the player who gains control of the loose ball is credited with a rebound.
Defensive rating. As a measure of defensive efficiency, a team‟s defensive rating is operationally defined as follows (Oliver, 2004):
(10) Defensive Rating = Team‟s total points allowed / Opponent‟s total
possessions employed * 100
For the concept of defensive rating, lower values signify higher levels of defensive efficiency.
Defensive rebound. Following a failed shot attempt (i.e., field goal or free throw), the ball is considered live as both teams attempt to gain control. A defensive rebound occurs when a player from the non-shooting team secures possession of the ball. Upon acquiring the defensive rebound, the team which was formerly on defense is now on offense. Balls awarded to the defensive team that either, go out-of-bounds before a player had the opportunity to gain control or were never secured as a result of a foul call, are classified as team defensive rebounds.
Depth contribution factor. Even in the world‟s preeminent basketball league, immense discrepancies exist in the talent levels between athletes. The depth contribution factor represents an attempt to assess team depth by focusing specifically on the average
41 number of players who make significant on-court contributions. In order to gain insight into the relationship between team depth and game outcomes, a team‟s depth contribution factor was operationally defined as the number of players who play at least 12 minutes per game. Twelve minutes represents one quarter of a regulation NBA game and is consistent with Berri, Schmidt and Brook‟s (2007) designation of substantial player minutes.
Free throw. As a consequence for certain rule infractions, a referee will award a number of free throws to the victim / victim‟s team. The free throw itself is worth one point and consists of an unguarded shot at the basket, with the game clock stopped, from a distance of 15 feet. The number of free throws awarded can vary from one to three and depends on the circumstances surrounding the foul or infraction. For example, a shooting foul occurring on a three point field goal attempt, that is unsuccessful, results in three free throws; whereas a player fouled in the act of shooting a two point field goal, which goes in, is awarded a single free throw attempt. In addition to shooting fouls, a player or team is granted a trip to the free throw line as a result of technical fouls, flagrant fouls and when a team‟s total fouls exceed four in a regulation period; exceptions often occur when double fouls are issued, during the last two minutes of any quarter and for the entirety of overtime periods (National Basketball Association, 2009b).
Game location. Evidenced by the divergence in home and away NBA winning percentages as well as the results of empirical investigations (e.g., Sánchez et al., 2007), the impact of game location is undeniable. Speculation on the causes of home court advantage includes home team familiarly with the court and the arena, detriments
42 associated with travel fatigue for the visiting team, and a positive impact stemming from crowd noise. Under the present conceptualization, the influence of game location is intrinsically built within the models. The segmentation of performance measures into two distinct categories (i.e., statistics and measures corresponding to the away team or the home team) permit for variations in the individual value of performance measures based on game location. Simply stated, the coefficients associated with each performance measure are not only contingent upon the measure itself, but also the inherent location component (e.g., all possession models include an offensive rating variable for the away team and another offensive rating variable for the home team). Therefore, although game location is not operationally defined explicitly, the separation of statistics and measures based upon location implicitly accounts for the advantages of playing at home as well as the disadvantages of playing on the road.
Offensive rating. As a measure of offensive efficiency, a team‟s offensive rating is operationally defined in the following manner (Oliver, 2004):
(11) Offensive Rating = Team‟s total points scored / Team‟s total possessions
employed * 100
For the concept of offensive rating, high values correspond with high levels of offensive efficiency.
Offensive rebound. In the moments following a missed field goal or free throw attempt, the opposing teams vie for control of the live ball. An offensive rebound is credited when the team responsible for the unsuccessful shot attempt recovers control of the basketball. If no player is able to gain control of the loose ball before it travels out-
43 of-bounds, and the defensive team last touched the basketball, a team offensive rebound is awarded for stat keeping purposes. Additionally, attempts at tipping the ball in order to score a basket are recorded as offensive rebounds.
Personal foul. According to the National Basketball Association (2009b) “a player shall not hold, push, charge into, impede the progress of an opponent by extending a hand, arm, leg or knee or by bending the body into a position that is not normal” (p.
43). When actions of this nature are observed by a referee, a personal foul is assessed, which is followed by an appropriate consequence (e.g., awarding foul shots in the instance of a shooting foul). Infractions identified as personal fouls can occur at the offensive and defensive ends as well as during loose ball situations. Individual players are permitted to commit six personal fouls before fouling out of a contest. Permissible physical contact additionally varies based upon one‟s location on the floor (e.g., defenders are allowed to place a forearm on an offensive player who has his back to the basket when outside of the Lower Defensive Box and below the foul line extended;
National Basketball Association, 2009b). Finally, it is important to recognize that three types of fouls exist within the NBA (i.e., personal fouls, technical fouls and flagrant fouls). While flagrant fouls represent an act of excessive unnecessary physical contact, in which the offender is assessed a personal foul, technical fouls are not accompanied by personal fouls and can be charged for non-physical individual actions (e.g., taunting or hanging on the rim) as well as team violations (e.g., defensive three seconds or a delay of game penalty).
44 “Played” games. The concepts of “played” and “un-played” games were adopted from Loeffelholz et al.‟s (2009) research in order to maintain a chronological sequence of events. As identified on the NBA schedule, all games are slated to be played on a specific date. The concepts of “played” and “un-played” games operate under the pretense that the current date is the date in which the game in question is scheduled to be played; hence, hypothetically, the game in question has not yet occurred. The relationship between the game in question, identified as the “un-played” game, and the performance statistics accumulated in games prior to the game in question, is explored within the current study. The independent variables within each model include previously accumulated statistics and measures for both the home and away teams. For example, in a competition between home team A and away team B, which represents a single observation within the data set, all games prior to the game in question represent
“played” games. Therefore performance measures for team A and team B are compiled in all previously “played” games, which serve as independent variables, and the actual outcome of the contest between A and B, the “un-played” game, represents the dependent variable. It is important to note, that streak models explore the relationship between the
“un-played” game and statistics accumulated in only the previous five “played” games.
Scheduling fatigue. Over the course of the NBA season, teams spend an extraordinary amount of time on the road traveling to arenas all across the country. Even when remaining at home, games occurring in close succession with previous competitions can be taxing on the human body, which consequently negatively affects performance. In situations where an opposing team is forced to play on consecutive days
45 and the second game occurs on the road, the impact of home-court advantage gains additional significance (Bellotti, 1990). For the 1988-1989 NBA season the best road record belonged to the Detroit Pistons (26-15), who were fortunate to play the day preceding an away game only eight times; conversely, the San Antonio Spurs and the
Miami Heat compiled the worst road record in the league (3-38) and played the day before road games a total of 16 and 13 times respectively (Bellotti). Additional support for the negative relationship between fatigue and performance within the context of basketball was revealed in Lyons, Al-Nakeeb and Nevill‟s (2006) investigation of body fatigue and passing accuracy. Outside the realm of basketball, studies such as Royal,
Farrow, Mujika, Halson, Pyne and Abernethy‟s (2006) examination of water polo players as well as Feddock, Hoellein, Wilson, Caudill and Griffith‟s (2007) investigation of nursing interns, have demonstrated the detrimental effect of fatigue on performance.
Expanding on these findings, the present study strived to quantify the frequently overlooked factor of scheduling fatigue.
A brief glance at NBA team schedules reveals that one of the worst case scheduling scenarios for a club is a series of four games in a period of five days (National
Basketball Association, 2009c). Building on this information, scheduling fatigue is operationally defined as a series of dummy variables that chronicle the events of the four preceding days. It is imperative to recognize that these scheduling fatigue variables attempt to ascertain the impact of games occurring in close succession with each other, which represent interdependent events. While the influence of playing a contest four days prior to the game in question would in all likelihood be nonexistent, compounded
46 fatigue stemming from multiple games over the previous four days may adversely affect performance.
For each team, five scheduling fatigue variables are constructed. The first binary variable is used to identify teams playing their fourth game in five days. A second dichotomous variable tracks whether the team is playing its third game in a four day period and a third binary variable denotes two games occurring within the previous three days. While these three scheduling fatigue variables fail to account for variations pertaining to travel (e.g., the difference between four home games in five days and four away games in five days), they nevertheless are capable of accounting for the compounded influence of fatigue. It must be noted that in general, NBA teams do not have games on three straight days.
Unlike the previous three scheduling fatigue variables, the variables that account for the day prior to the “un-played” game include a travel location component. To represent a nominal variable consisting of three categories (i.e., whether the team participated in an away game, a home game or did not have a game at all) two dichotomous variables are required. For example, the combination of H1DPV1 and
H1DPV2 are capable of describing the home team‟s happenings the day prior to the game in question. The dual variable construction for one day prior scheduling fatigue variables are displayed in Table 6.
Steal. A steal is awarded to a defensive player when their positive aggressive action results in an opponent‟s turnover (International Basketball Federation, 2009;
National Collegiate Athletic Association, 2009). Numerous defensive actions qualify
47 Case Variable one Variable two No Game 0 0 Away Game 0 1 Home Game 1 0
Table 6. Fatigue dummy variable format for previous day
48 as a steal and include, but are not limited to, directly taking the ball from a dribbling opponent, intercepting a pass from an opposing player, and deflecting a ball that was in an opponent‟s control to a teammate. As illustrated by these examples, a steal is hardly a passive beneficial action; steals should be awarded when the action is deemed the direct reason for the turnover (International Basketball Federation).
Streak. Streak variables are constructed for a multitude of performance measures.
Operationally defined as each performance measure‟s average over the previous five games, the number of streak variables differ based upon the model. Therefore, streak models solely focus on the relationship between statistics accumulated over the previous five “played” games and the outcome of the “un-played” game. While the majority of investigations have disregarded the potential influence of trend data, Loeffelholz et al.
(2009) analyzed the predictive capabilities of statistics compiled over the previous five games. The incorporation of streak performance measures is advantageous as streak statistics can potentially capture recent surges in play, both of the positive and negative variety, as well injuries or illnesses (Loeffelholz et al.).
Three point field goal. A team is charged with a three point field goal attempt whenever one of its players shoot or tap a live ball in the direction of their opponent‟s basket from outside the three point line. According to National Basketball Association
(2009b) guidelines, the three point area is outside an arc measuring 22 feet from the basket at the corners and 23 feet 9 inches from the apex. However, it is important to note, that in a few instances, a three point field goal attempt can be nullified. For example, if the offensive player was fouled while making the attempt and the attempt was
49 unsuccessful or if a player on the offensive team committed offensive basket inference.
A successful three point field goal attempt results in three points added to the offensive team‟s score.
Turnover. A turnover represents a blunder committed by a player on the offensive team that results in a loss of possession. Furthermore, the loss of control must not be attributed to the expiration of a period, a missed field goal attempt or a missed free throw attempt. Turnovers can be charged to individual players, who directly lose control of the ball (e.g., fumbling the ball into the hands of a defensive player or throwing a pass out-of-bounds), as well as to offensive players away from the ball (e.g., the act of committing a three-second violation or an offensive foul). Regardless of fault, the end result of a turnover is a loss of possession, in which the team formerly identified as the offense is now classified as the defense, and vice versa.
Two point field goal. A team is credited with a two point field goal attempt whenever one of its players shoot or tap a live ball in the direction of their opponent‟s basket from inside the three point line. As was the case with three point field goal attempts, a two point field goal attempt can be nullified in certain situations (e.g., when the offensive team is whistled for goaltending). As the name implies, a two point field goal is worth two points for the offensive team.
“Un-played” games. See “played” games.
Significance of the Problem
By illuminating the complex relationship between performance measures and game outcomes in the National Basketball Association, the present research facilitates in
50 the generation of new knowledge. Formulated as the integration of average performance measures and forthcoming measures of team success, the current investigation explored the exploratory and predicative capabilities of several previously disregarded factors (i.e., indicators of pace, offensive and defensive measures of efficiency, the team depth contribution factor and measures of scheduling fatigue), all of which support practical implications. The emergence of a relationship between measures of efficiency and future game outcomes would provide additional credence for the use of offensive and defensive efficiency measures as composite indicators of team success. Similarly, the discovery of a relationship between the team depth contribution factor or measures of scheduling fatigue, and approaching game outcomes, could alter coaching strategies in the case of the team depth contribution factor or result in the modification of travel arrangements to counter scheduling fatigue.
Perhaps the most significant implication, an examination of the marginal effects of the individual performance measures within the four models will reveal the statistics that have the most and least influence on a game‟s outcome. In the long term, team personnel decisions coinciding with these findings should maximize the likelihood of achieving on-court success. Armed with this information, personnel departments could more efficiently navigate the draft as well as evaluate existing players and potential free agents.
According to Sirmon, Gove and Hitt (2008), resource management consists of the three distinct processes; structuring, which refers the acquisition of resources, bundling, which involves the integration of resources with the intention of contributing to a firm‟s
51 capabilities, and leveraging, which pertains to the deployment of capabilities within the marketplace. A deeper understanding of the marginal effects of individual performance measures would facilitate in these three resource management processes. Through the execution of such practices, teams residing in small markets, inhibited by limited monetary resources, could balance a slanted playing field. Through the 1990s, this exact scenario played out with Major League Baseball. Supporting one of the smallest payrolls, Billy Beane, the general manager of the Oakland Athletics, used sabermetrics to facilitate player evaluations and was ultimately able to regularly field playoff caliber teams despite a disadvantaged fiscal position (Gerrard, 2004).
52 Chapter 2: Review of Literature
Research focusing on production and efficiency within basketball can generally be separated into two types of endeavors, investigations centered on measuring and quantifying the worth of individual participants, and studies interested in calculating productivity from a team perspective. The current investigation falls into the latter classification. With this being the case, the forthcoming review of literature will merely briefly touch on research and measures of the individual variety, before embarking upon a comprehensive discussion of team based production research.
Individual Performance Measures
Quantitative measurements of individual player productivity have occupied the full gamut of possibilities, from simplistic assessments focusing on a particular aspect of the sport (e.g., the accumulation of assists) to immensely complicated analyses representing a composite of numerous contributions. Building on the basic statistics found in a traditional box score, Bellotti (1990) integrated Value of Ball Possession
(VBP), which is determined by dividing points by possessions, into a modified version of assist-to-turnover ratio. Remaining on the simplistic end of the spectrum, Bellotti‟s adjusted field goal percentage evaluates an individual‟s scoring ability by accounting for both field goal percentage and field goals attempted. In essence, adjusted field goal percentage provides an additional reward to individuals who maintain a high field goal
53 percentage while hoisting a large volume of shots. Simply looking at field goal percentage can be deceiving as a player with a limited number of field goal attempts could boast an extraordinary field goal percentage. Hence, the adjusted field goal percentage statistic attempts to address the problematic issue associated with standard field goal percentages.
While the previous measures represent a minor portion of a player‟s on-court contributions, composite performance instruments such as the NBA efficiency measure
(National Basketball Association, 2009a) and Bellotti‟s (1990) points created (PC) measure assimilate a diverse assortment of performance statistics. An individual‟s NBA efficiency measure is calculated through the following:
(12) NBA efficiency = Points + Rebounds + Assists + Steals + Blocked shots –
((Field goals attempted – Field goals made) + (Free throws attempted – Free
throws made) + Turnovers)
In a similar fashion, PC, constructed by Bellotti as an overall measure of a player‟s value, is calculated through the following:
(13) PC = Points + Rebounds * VBP + Assists * (2-VBP) + Steals * VBP +
Blocked shots * VBP – Missed shots * VBP – Turnovers * VBP – (Personal fouls
/ 2) * VBP
Even with the incorporation of VBP, which can vary by season-to-season as well as by team-to-team, PC essentially assigns equal values to the majority of an individual‟s personal statistics and subsequently summates positive contributions while subtracting negative actions. However, as noted by Berri et al. (2007), the NBA efficiency measure
54 as well as Bellotti‟s PC measure, are problematic due to the assumption that each statistical contribution is essentially of equal value. For example, the NBA efficiency formula assigns equivalent penalties for a missed free throw and a missed field goal, a significant flaw when considering the disparity in value between the two shots.
As an alternative individual measure of production, the plus-minus statistic tracks the score differential for the amount of time each player is on the floor. Unfortunately, the plus-minus statistic can produce some rather deceiving results due to its innate reliance on teammate contributions (Lewis, 1999). Conceivably, a marginal player, who consistently plays the majority of his minutes with four of the best players in the world, would produce exceptional plus-minus numbers. In an effort to eliminate the shortcomings associated with the standard plus-minus measure, adjusted plus-minus ratings statistically account for the quality of the surrounding cast for each player as well as the quality of opponents, home court advantage and differences in the relative importance associated with the time of contribution (i.e., clutch minutes versus garbage time; Ilardi, 2007; Rosenbaum, 2004). By controlling for confounding elements, adjusted plus-minus is a significant improvement over the standard plus-minus statistic.
Conceptualized as a model linking individual player statistics to team production,
Berri (1999) developed an econometric model capable of measuring each player‟s marginal product in terms of wins. Berri devised two distinct equations, the first equation broke down points scored into the elements of ball acquisition (i.e., opponent‟s turnovers, defensive rebounds, opponent‟s total scoring), ball handling efficiency (i.e., assist-to- turnover ratio) and scoring success rate (i.e., points-per-shot, free throw attempts, free
55 throw percentage, offensive rebounds); the second equation segmented points surrendered into team tempo (i.e., field goal attempts, free throw attempts, offensive rebounds, turnovers), opponent‟s ball handling efficiency (i.e., opponent‟s assist-to- turnover ratio) and the opponent‟s ability to convert possessions into points (i.e., opponent‟s points-per-shot, opponent‟s free throw percentage, personal fouls, defensive rebounds). Utilizing a sample of the four seasons beginning with the 1994-1995 campaign and concluding with the 1997-1998 season, Berri discovered that the points scored and the points surrendered formulas explained 96.3% of the variance in a team‟s winning percentage and 98.7% of the variance in an opponent‟s total scoring. Since the number of possessions a team employs is essentially equal to the number of possessions its opponents employ, “…wins are solely a function of offensive and defensive efficiency” (Berri et al., 2007, p. 104).
Building on Berri‟s (1999) results, the marginal values for numerous individual player statistics were ascertained. Offensive rebounds were found to have the greatest marginal impact on team wins, which was followed by three point field goals made, turnovers and opponent‟s turnovers. Finally, to determine individual player win production, Berri et al. (2007) adjusted player evaluations on a positional basis, as the five players on the court are undoubtedly complements in the production of wins, as well as adjusted for team defensive statistics, which is illustrated by the following:
(14) Wins produced = (Production per 48 minutes – Average value of
production per 48 minutes for player‟s position + Team defensive adjustment per
48 minutes played + 0.1) / 48 * Minutes played
56 As a testament to the accuracy of the model, a comparison between summated individual player wins and actual team wins revealed a difference of less than one for nine franchises and a difference greater than five for only four organizations (Berri).
In a subsequent analysis, Berri and Schmidt (2002) investigated the construct of instrumental rationality, which states that an economic actor interested in satisfying their own objectives will act in an efficient manner to achieve a desired result. While including draft position to approximate original coach perceptions, an analysis of the correlation between All-Rookie first team votes and two player productivity measures
(i.e., points created and wins produced) was conducted. Results indicated that points created was considerably more congruent with votes received than wins produced (i.e., of the 21 All-Rookie first teamers, 17 were in the top five in points created, whereas only 11 were in the top five in wins produced). Taken as a whole, these findings suggest that coaches fail to correctly use statistics when evaluating players, thus acting contrary to instrumental rationality (Berri & Schmidt).
As an equally complex conceptualization in the measurement of player contributions, Oliver (2004) devised individual offensive ratings and individual defensive ratings. Grounded in Oliver‟s difficult theory for distributing credit in basketball, individual offensive rating refers to the number of points a player produces per hundred individual possessions, while individual defensive rating indicates the number of points allowed per hundred individual possessions. Oliver‟s difficult theory for distributing credit in basketball can best be understood through two fictitious offensive examples. In the situation where a crafty point guard sneaks a pass into a big man underneath the
57 basket, who subsequently dunks the ball, the point guard deserves the majority of the credit. Conversely, if a point guard passes the ball to a teammate with the shot clock winding down and the player scores on a miraculous no-look shot, the point guard does not deserve much, if any, credit for the score. While constraining credit to the actual number of points produced, individual offensive rating consists of three separate formulas
(i.e., individual scoring possessions, individual total possessions and individual points produced), all of which “… apply different weights to different stats for different players based on the difficulty the players face in recording those stats” (Oliver, 2004, p. 152).
On the other hand, individual defensive ratings are comprised of individual defensive stop measures, defensive possessions faced as well as a calculation for the team‟s defensive rating.
Up until this point, the discussion of individual player performance measures have generally revolved around offensive contributions, and while not nearly as common, researchers have constructed individual defensive measures. Boxscore defense, conceptualized by Bellotti (1990), initially matches up each of the five starting players to their positional counterpart on the opposing team (i.e., point guard vs. point guard, center vs. center, etc…). Once the matchups are determined, a player‟s boxscore defense is calculated by the following:
(15) Boxscore defense = (Opponent‟s points * (Minutes played / Total
minutes) / Minutes played) + (Opponent‟s rebounds * Team‟s VBP * (Minutes
played / Total minutes) / Minutes played) + (Opponent‟s assists * Team‟s VBP *
(Minutes played / Total minutes) / Minutes played)
58 As opposed to simply assigning all of an opponent‟s contributions to the defending player in question, boxscore defense incorporates percentages of minutes played. While boxscore defense facilitates in the numerical assignment of defensive contributions, boxscore defense has several flaws including its reliance on pregame positional matchups, which could potentially differ from in-game defensive assignments, as well as an inability to account for reserve players (Bellotti).
Even with access to complex individual defensive measures, such as Oliver‟s
(2004) individual defensive rating or Bellotti‟s (1990) boxscore defense, one must exhibit caution when interpreting results. In addition to the inherent interdependent nature of defense, it must be noted that superior defensive players are often assigned to defend the opponent‟s best offensive player. For these individuals, individual defensive numbers may fail to illustrate the true value of their defensive contribution (Oliver).
Team Based Production Research
Grounded in a production frontier conceptual framework, numerous research investigations have adopted a broader perspective in the evaluation of a team‟s performance. While similarities among these investigations stem from their incorporation of a vector of statistical inputs and an observed output, they differ significantly in the respective samples examined, their operational definitions and the type of statistical methodologies employed. Using either absolute or relative inputs, basketball production frontier research has been conducted with a focus on individual games as well as on team success measured at the conclusion of a season. A summary of
59 team production function research conducted within the context of basketball is provided in Table 7.
For the 1976-1977 season, Zak et al. (1979) explored a sample composed of the five NBA franchises within the Atlantic division (i.e., Boston Celtics, Buffalo Braves,
New York Knicks, New York Nets and the Philadelphia 76ers), to ascertain resource allocation efficiency and determinants of team performance. Given a vector of inputs x, a measure of production efficiency u, and a production function F(x), a firm‟s observed output Y can be represented by Y = F(x) • u. Using the Richmond technique to simplify the production efficiency measure, which restricts the values of u between the boundaries of 0 and 1 (Richmond, 1974), a Cobb-Douglas production function F(x), and a series of in-game statistics that served as the vector of inputs, Zak et al. constructed a production frontier for each NBA contest within the sample. Coinciding with this simplistic version of production efficiency, the value of u equals 1 in situations when a team is 100% efficient and conversely equals a value of 0 in situations where a team is completely inefficient and thus produces no output. Relative input measures included in-game ratios for field goal percentage, free throw percentage, offensive rebounds, defensive rebounds, assists, personal fouls, steals and turnovers, as well as a difference measure for the number of blocked shots, and a binary variable to account for game location (i.e., home =
1 and away = 0). Following along with the relative input measures, Zak et al. operationally defined the dependent variable as a ratio of the finals scores to represent the relative competitiveness in each competition. Overall, the combination of input variables
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Model Sample Independent Variables Dependent Variable
Relative measures of in-game statistics and binary variable for game location [Ratio of field goal percentage, Ratio of free throw percentage, Ratio of offensive rebounds, Ratio of defensive rebounds, Ratio of assists, Ratio of personal fouls, Ratio of steals, Ratio of Zak, Huang & 1976-1977 NBA turnovers, Binary variable for game location, and Difference in Siegfried (1979) Atlantic Division blocked shots] Ratio of final scores
Absolute measures of season performance statistics [Team's field goal 1970-1971 NBA percentage, Opponent's field goal percentage, Team's free throw season through the percentage, Opponent's free throw percentage, Team's rebounds, Scott, Long & 1979-1980 NBA Opponent's rebounds, Team's assists, Opponent's assists, Team's Teams winning Somppi (1985) season personal fouls, and Opponent's personal fouls] percentage
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Absolute measures of season performance statistics along with a coaching efficiency measure. [Coaching efficiency measure, Team's field goal percentage, Opponent's field goal percentage, Team's total rebounds, Opponent's total rebounds, Team's personal fouls, 1974-1975 NBA Opponent's personal fouls, Team's steals, Opponent's steals, Team's season through the blocked shots, Opponent's blocked shots, Team's assists, Opponent's Grier & Tollison 1978-1979 NBA assists, Team's free throw percentage, Opponent's free throw (1990) season percentage, Team's turnovers, and Opponent's turnovers] Total team wins
Season performance statistics measured as per game differentials. [Field goal percentage, Free throw percentage, Offensive rebounds, Chatterjee, Campbell 1991-1992 NBA Defensive rebounds, Assists, Steals, Blocked shots, Turnovers, and Team winning & Wiseman (1994) Season Three point field goal percentage] percentage Continued
Table 7. Summaries of team production research in basketball
Table 7 continued
Relative measures of a team's season performance statistics. [Ratio of field goal percentage, Ratio of free throw percentage, Ratio of Hofler & Payne 1992-1993 NBA offensive rebounds, Ratio of defensive rebounds, Ratio of assists, (1997) Season Ratio of steals, Ratio of turnovers, and Difference in blocked shots] Total team wins
Dichotomous variable to indicate the winner of Relative measures of in-game statistics. [Ratio of two point field goal the contest along with a percentage, Ratio of three point field goal percentage, Ratio of free secondary analysis 2000-2001 NBA throw percentage, Ratio of offensive rebounds, Ratio of defensive utilizing ratio of final
62 McGoldrick & Voeks Season & 2000 rebounds, Ratio of assists, Ratio of personal fouls, Difference in scores as a relative
(2005) WNBA Season steals, Ratio of turnovers, and Difference in blocked shots] outcome.
Model 1 - Relative measures of in-game statistics. [Ratio of field goal percentage, Ratio of free throw percentage, Ratio of defensive rebounds, Ratio of offensive rebounds, Ratio of assists, Ratio of personal fouls, Difference in steals, Ratio of turnovers, and Difference in blocked shots]. Model 2 - Absolute measures of team statistics with dichotomous variable for game location. [Team's field goal Spanish ACB percentage, Team's free throw percentage, Team's defensive rebounds, League Team's offensive rebounds, Team's assists, Team's personal fouls, Championships Opponent's personal fouls, Team's steals, Team's turnovers, Team's Dichotomous variable to Sánchez, Castellanos 2002-2003 & 2003- blocked shots, Opponent's blocked shots, and Binary game location indicate the winner of & Dopico (2007) 2004 variable] the contest.
Continued
Table 7 continued
Series of both teams‟ absolute averages prior to the game in question and a dichotomous variable for game location. [Away team's field goal percentage, Away team's three point field goal percentage, Away team's free throw percentage, Away team's offensive rebounds, Away team's defensive rebounds, Away team's assists, Away team's steals, Away team's blocks, Away team's turnovers, Away team's personal fouls, Away team's points, Home team's field goal percentage, Home team's three point field goal percentage, Home team's free throw percentage, Home team's offensive rebounds, Home team's defensive rebounds, Home team's assists, Home team's steals, Home team's Loeffelholz, Bednar 2007-2008 NBA blocks, Home team's turnovers, Home team's personal fouls, and Outcome of "un-played" & Bauer (2009) Season Home team's points] game 63
produced an adjusted R2 of 87.37%, with all inputs reaching statistical significance at the
5% level except assists, blocked shots and game location. Ratio of field goal percentage was the model‟s most significant contributor with a 1% increase resulting in a 0.61% increase in the ratio of final scores. Lastly, Zak et al. determined that as a whole the five franchises were tremendously efficient, with the New York Knicks registering the lowest level of efficiency at 0.99849.
Along the same lines McGoldrick and Voeks (2005), and Sánchez et al. (2007), constructed production functions from statistics amassed during single games.
McGoldrick and Voeks explored a sample composed of games from the 2000-2001 NBA season and the 2000 WNBA season. In order to examine the influence of relative characteristics of play on the outcome of a respective game (i.e., win or loss), a total of ten independent variables were included in a logit model. As opposed to including a single ratio for field goal percentage a la Zak et al. (1979) or Sánchez et al, McGoldrick and Voeks constructed two ratios (i.e., ratio of two point field goal percentages and ratio of three point field goal percentages). Findings indicated that, for both leagues, two point field goal percentage ratio had the greatest marginal effect on winning. A 1% increase in the two-point field goal percentage ratio resulted in a 4.7% and a 7.1% increase in the probability of winning for the WNBA and the NBA respectively. Focusing on the differences between the two leagues, McGoldrick and Voeks found that turnovers had the second highest marginal impact for the WNBA, whereas defensive rebounds were identified as having the second most influential marginal effect for the NBA.
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As a secondary analysis, McGoldrick and Voeks (2005) used a stochastic frontier approach to estimate the relationship between relative team performance measures and the relative outcome of the competition, which was operationally defined as the ratio of final points. For the WNBA, 82.76% of the variance in the ratio of final scores was explained by the 10 independent variables, which was slightly greater than the 81.18% explained for the NBA. Concentrating on the impact of the individual independent variables revealed that increases in relative shooting percentage, rebounds, assists and steals significantly increased the game outcome ratio. With the ratio of final scores represented as a logarithm, as was the case for several independent variables, the stochastic frontier model could be classified as a linear estimate of a Cobb-Douglas production function.
With a dependent variable mirroring McGoldrick and Voeks‟s (2005) logit analysis (i.e., a dichotomous variable indicating game outcome), Sánchez et al. (2007) created two models from a sample consisting of 2002-2003 and 2003-2004 Spanish ACB
League Championship regular season games. A team‟s ability to achieve success was conceptualized as a combination of four elements including: (a) defensive pressure, which hinders an opponent‟s ability to convert possessions into points; (b) rebounding capacity, which is critical for extending possessions and ending an opponent‟s possession; (c) efficiency in ball handling, which represents contributions to scoring efficiency and maintaining ball possession; and (d) shot effectiveness, which refers to a team‟s ability to convert possessions into points. Model I, composed of a series of ratios and differences in the statistics of the two competing teams, successfully classified
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88.56% of game outcomes. Alternatively, with the purpose of ascertaining the influence of home court advantage, Model II segmented home and away team statistics and correctly classified games at a rate of 83.09% (Sánchez et al.).
Sánchez et al. (2007) deemed that the using the logit probabilistic model to estimate the marginal effect of a play action on the probability that a team emerges victorious was advantageous due to an absence of independent variable distribution restrictions and the non-linear relationship between the probability of winning and the regressors matrix, which mirrors the law of diminishing returns (i.e., decreasing returns to scale). In analyzing the individual contributions of the independent variables, Sánchez et al. found that ratio in field goal percentage and playing at home were the most influential factors in Model I and Model II respectively. While descriptive statistics for Model I indicated that on average a home team shoots 4.5% better than their opponent from the field, the marginal effects of the model revealed that a 1% increase in field goal percentage ratio for the home team increased the probably of winning by 1.36%. On the other hand for Model II, Sánchez et al. discovered that playing a home game increased a team‟s winning probably by 4.54%, and that while holding all other variables constant, a
1% increase in field goal percentage increased a team‟s probability of winning by 3.32%.
Whereas the aforementioned investigations have focused on the relationship between a series of in-game statistics and the outcome of the game in which the statistics were accumulated, a multitude of studies have adopted a broader approach by analyzing team success over the length of an entire season. For a 10 year sample, beginning with the 1970-1971 season and concluding with the 1979-1980 campaign, Scott et al. (1985)
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employed a Cobb-Douglas production function, which quantified team success as a combination of team performance statistics, with the ultimate intention of exploring marginal revenue in professional basketball under both monopsonistic and free market conditions. Assuming a multiplicative error term, team success was operationally defined as a team‟s winning percentage and input variables included season performance statistics in absolute terms (e.g., team and opponent‟s field goal percentage, free throw percentage, rebounds, assists and fouls). Unlike other production function conceptualizations, Scott et al. combined offensive and defensive rebounds into a single measurement. Results indicated that 71% of the variance in a team‟s winning percentage could be explained by the inputted seasonal statistics. Focusing on the impact of the individual performance statistics revealed that opponent‟s field goal percentage was the most significant predicator of winning percentage, followed by field goal percentage and opponent‟s fouls (Scott et al.). More specifically, a 1% increase in an opponent‟s field goal percentage resulted in a 3.45% decrease in a team‟s winning percentage.
Devised in a similar manner, Chatterjee et al. (1994) conducted a series of multiple regression analyses from a sample of 1992 NBA season data. Incorporating numerous season statistics, all measured as differentials, Chatterjee et al. sought to determine the amount of variance explained in win-loss percentage by the input statistics.
Chatterjee et al. identified the use of differentials as natural, with each performance measure calculated as a team‟s per game average minus the team‟s opponents‟ per game average. While the preliminary model explained an impressive 91.8% of the variance in win-loss percentage, only field goal percentage and turnovers were found to be
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statistically significant and additionally multicollinearity among the variables was a concern. After eliminating the statistically insignificant variables of previous renditions, the final linear equation, which included differentials in field goal percentage, free throw percentage, turnovers and a composite measure for rebounds, produced an adjusted R2 of
91.4%.
For the four NBA seasons prior to 1992, Chatterjee et al. (1994) executed another series of analyses. Each campaign produced a regression equation similar to the 1992
NBA season, with the lone exception being the 1990 season, in which the per game differential for three point percentage proved to be statistically significant. Overall the fit for each of the five regression equations, determined by the model‟s adjusted R2, was
2 2 both high and consistent (i.e., 1988: R (adjusted) = 93.7%, 1989: R (adjusted) = 91.1%, 1990:
2 2 2 R (adjusted) = 93.7%, 1991: R (adjusted) = 94.2%, 1992: R (adjusted) = 91.4%; Chatterjee et al.).
Additionally, the magnitude of the variables, specifically field goal percentage, free throw percentage, turnovers and rebounds were relatively stable over the five year period.
Finally, using the estimated coefficients that emerged from the 1988 dataset, Chatterjee et al. predicted the win-loss percentages of all NBA teams for 1989, 1990, 1991 and 1992.
In each of the four seasons, the winning percentages of the four teams with the highest predicted win-loss percentages were compared with actual win-loss percentages, which revealed a strong correlation between the forecasted and observed percentages.
Representing a slight alteration in the conceptualization of team success, the research conducted by Grier and Tollison (1990) operationally defined the dependent variable in their investigation as the total number of team wins. Focusing on the
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influence of managerial performance on team production, Grier and Tollison examined data from the five NBA seasons prior to the introduction of the three-point shot (i.e.,
1974-75 to 1978-79). In an effort to create a basketball economy, a production function was constructed that defined team wins as a product of 17 input factors, including performance measures operationally defined in absolute terms as well as a measure of coaching efficiency. The construct of coaching efficiency is grounded in the notion that professional basketball suffers from an agency problem, where a player could potentially face a dilemma of either accumulating personal statistics or cooperating with his team‟s strategy. The inherent problem arises from the conflict of interest between the two parties and from the fact that the effort required for player cooperation is difficult to quantify by outsiders, therefore failing to substantially contribute to an individual‟s market value (Grier & Tollison). Guided by a desire to maximize wins, ultimately a coach must efficiently allocate a limited number of field goal attempts. While, the proportion of shots taken by an individual should be related to the player‟s relative skill, a decline in marginal shooting production will result due the opponent‟s defensive strategy
(Grier & Tollison). More specifically, defensive alignments devised to hinder premier scorers should eventually result in open looks for teammates. Using a Spearman rank correlation between each player‟s field goal percentage and his number of attempts, Grier and Tollison operationally defined coach efficiency.
Upon eliminating statistically insignificant variables (i.e., free throw percentage, steals, opponents‟ steals, blocked shots and opponents‟ blocked shots), Grier and Tollison
(1990) found that 84.3% of the variance in team wins was explained by the remaining 12
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independent variables. Furthermore, results indicated that team field goal percentage, team rebounds and team turnovers contained the greatest explanatory power. In terms of the coaching efficiency measure, a one standard deviation increase was found to produce roughly 1.00 additional win per season. Over the entire sample the difference between inferior and superb coaching performances equated to 4.3 wins. In a league where playoff spots and playoff seeding is commonly determined by minuscule margins, a difference of more than four wins can have considerable implications. In analyzing the relationship between coaching efficiency and a coach‟s tenure, Grier and Tollison found that the average coaching efficiency for the six coaches who held their position over the five season sample was almost double that of their coaching brethren.
Facilitated by an advanced stochastic frontier model Hofler and Payne (1997) analyzed potential and efficiency of NBA teams for the 1992-1993 season. Hofler and
Payne adopted a cross-sectional approach and define their dependent variable as the total number of team wins, which mirrors the research conducted by Grier and Tollison
(1990). However, these two research endeavors differed in their operationalization of performance measures; whereas Grier and Tollison used absolute measures, Hofler and
Payne incorporated a series of statistics calculated as ratios and differences. Grounded within stochastic frontier theory, team production functions were conceptualized as Yi =
Xiβ + vi + ui, where Yi is the actual production of team i, Xi represents a team specific production vector, β is a regression coefficient vector, vi is a normally distributed error term and ui represents a team specific error term, which “is a measure of the reduction in output experienced by the ith team due to inefficiency (failure to attain its potential
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number of wins)” (Hofler & Payne, 1997, p. 294). Production functions were estimated through maximum likelihood estimation and a homogeneous Cobb-Douglas formulation.
As a whole, despite suffering from serious issues pertaining to collinearity, the adjusted R2 indicated that the model explained 88% of the variance in wins (Hofler &
Payne, 1997). Of the input statistics, defensive rebounds was identified as the most significant contributor to team wins, followed by turnovers, steals and field goal percentage. In terms of efficiency, NBA franchises were deemed highly efficient with the league average equaling roughly 89%.
Adopting an alternative perspective and expanding upon previous findings in regard to the wins produced performance measure, Lee and Berri (2008) analyzed NBA teams‟ level of productivity based upon three position classifications (i.e., guards, small forwards and big men). With an interest in assessing technical efficiency, Lee and Berri explored several production function formulations, as well as a stochastic frontier approach, in order to determine levels of efficiency. Results of a litany of econometric hypothesis tests revealed that a Cobb-Douglas production function with constant returns to scale, estimated as a time invariant model with fixed effects, was most appropriate.
Using a production function of this form, efficiency was examined as well as relative importance based on player position. Findings indicated that big men have the greatest impact on team wins, as big man contributions nearly doubled that of guards.
With a similar focus on positional importance, Page, Fellingham and Reese
(2007) examined the relationship between skill performance by position and game outcome. Essentially the primary purpose of Page et al. was to determine optimal skill
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sets by position. Employing a hierarchal Bayesian approach, team point differentials were delineated as a function of ten individual player performance categories. Additional model effects addressed dependency issues (e.g., player‟s team and opponent‟s team) and game location. Individual marginal effects by position generally followed expectations, although several surprising findings emerged. These unforeseen findings included the importance of centers who record more steals than their opposition, the relative positive contribution of assists for small forwards a well as the magnitude of negative effect for small forward turnovers, and the fact that defensive rebound differentials were only significant for guards (Page et al.).
Representing almost a synthesis of production function research executed for individual games and of cross-sectional investigations of seasonal performance,
Loeffelholz et al. (2009) explored neural networks as a tool for predicting game outcomes. Representing roughly half of each team‟s games, the first 650 games of the
2007-2008 NBA season were explored. Segmenting the sample into a 620 game training set and a 30 game validation set, which represented “un-played games”, Loeffelholz et al. used in-game statistics as well as a dichotomous variable to indicate home or away to construct a series of neural networks. Season averages were used to predict “un-played games”, in which averages were compiled in numerous statistical categories, including field goal percentage, three point percentage, free throw percentage, offensive rebounds, defensive rebounds, assists, steals, blocked shots, turnovers, personal fouls and points.
Additionally, two alternative schemes were explored, but each failed to produce results superior to that of the aforementioned current season average model. The first alternative
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technique separated each team‟s games into home and away contests, and computed two sets of averages based on game location. The second alternative technique attempted to incorporate streaks and injuries by solely analyzing the previous five games for each team.
Four neural networks were analyzed in Loeffelholz et al.‟s (2009) investigation.
A total of ten training / validation sets were created, in which the first set maintained the actual sequence of events (i.e., the 620 game training set represented the first 620 games of the NBA season) and the other nine sets were randomly constructed from the 650 game sample. When including all 22 variables, the feed forward network, the probabilistic neural network fusion and the Bayesian belief network, achieved the highest averages over the ten sets, predicting outcomes with an accuracy of 71.67%. Utilizing the Signal-to-Noise Ratio method to “… reduce the dimensionality of the data set”
(Loeffelholz et al., 2009, p. 10), a secondary analysis was performed with only turnovers and points, in which the aforementioned three networks achieved a maximum average accuracy of 70.67% over the ten validation sets. An alternative reduction, which focused on shooting percentages (i.e., field goal percentage, three point percentage and free throw percentage), yielded an improvement over the previously discussed models, as the same three networks produced an average accuracy of 72.67% for the ten validation sets.
Finally, a fourth reduction was administered by incorporating only the home and away team‟s field goal percentage and free throw percentage. Using these four variables, the feed forward network correctly predicted on average 74.33% of the games included within the ten validation sets. Furthermore, for validation set 1, the feed forward
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network, the probabilistic neural network and the generalized neural network accurately predicted the outcomes of 25 of the 30 “un-played” games, resulting in a forecasting percentage of 83.33% (Loeffelholz et al).
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Chapter 3: Methodology
Research Design
The current research investigates the complex relationship between performance measures and future outcomes. Conceived as a series of production functions, the association between a multitude of inputs (i.e., previously compiled performance statistics, fatigue measures and team depth contribution factors) and the outcome of an
“un-played” game is explored. It must be noted that the inclusion of previously compiled performance measures, as operationalized in the present study, differs conceptually from traditional production function research in that inputs are not directly transformed into outputs. Conventionally, a production function expresses the explicit transformation of a vector of inputs into a series of outputs; whereas the current investigation explores a relationship that is more indirect in nature. While contentions that prior accumulated performance statistics have no impact on future outcomes have some validity, it would be rather shortsighted to presume that performance measures are not indicative of team talent or overall playing ability. Given that the present study estimates output through numerous relevant and legitimate inputs, the fundamental tenets of production function theory are satisfied.
Using NBA box scores from the 2007-2008 and 2008-2009 seasons, compiled with the assistance of ESPN (2009), a primary data set of team statistics was constructed.
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The NBA is composed of 30 teams, which are segmented into six divisions (i.e., atlantic, central, southeast, northwest, pacific and southwest). Each NBA franchise participated in a total of 82 regular season games, which resulted in the creation of an initial data set of
1,230 unique observations per season. Constructed as a standard text file, this primary data set contained information on each NBA regular season game including the date of the competition, three letter symbols for both competitors and the final box score statistics of both teams.
The current investigation incorporated averages for a variety of previously amassed performance measures, thus a myriad of calculations were performed. When time progresses within the data sets, team statistics inevitably change as games once identified as “un-played” join the “played” categorization. For example as the Oklahoma
City Thunder prepare to play their 50th game of the season, statistics from their previous
49 games are used as predictors; however, during the Thunder‟s following game, all performance measures are constructed from the first 50 games of the season.
The independent variables for each absolute model, as shown in Table 2, are the competing team‟s simple performance measure averages entering the “un-played” game.
Away and home team statistics going into the “un-played” game are calculated in two point field goals made, two point field goals attempted, three point field goals made, three point field goals attempted, free throws made, free throws attempted, offensive rebounds, defensive rebounds, assists, steals, blocked shots, turnovers and personal fouls.
As displayed in Table 3, the independent variables for the relative models are computed as ratios. In order to execute the necessary calculations, team statistics are
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summated (e.g., offensive rebounds) and are subsequently divided by the summation of the team statistic allowed (e.g., opponents‟ offensive rebounds). Ratios for the two competing teams are calculated in two point field goal percentage, three point field goal percentage, free throw percentage, offensive rebounds, defensive rebounds, assists, steals, blocked shots, turnovers and personal fouls.
Using the aggregate intermediate variables from “played” games, the pace and efficiency measures of the possession models (see Table 4) are calculated. Possessions employed is calculated by equation 7, and incorporates field goals attempted, field goals made, free throws attempted, offensive rebounds, defensive rebounds and turnovers.
Efficiency is measured via offensive rating, which is calculated by equation 11 and includes total points scored and possessions employed, and defensive rating, which is calculated by equation 10 and includes total points allowed and opponents‟ total possessions employed.
In a similar fashion, play models (see Table 5) include composite measures of pace (i.e., offensive plays and defensive plays) and efficiency (i.e., offensive points-per- play and defensive points-per-play). Offensive plays, calculated by equation 8, incorporate total field goals attempted, total turnovers and total free throws attempted.
While defensive plays, calculated by equation 9, incorporate opponents‟ total field goals attempted, opponents‟ total turnovers and opponents‟ total free throws attempted. The efficiency measures are computed by simply dividing the number of total points scored by the number of offensive plays or the number of points allowed by the number of defensive plays, for offensive points-per-play and defensive points-per-play respectively.
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While the current data collection procedures deviate from traditional survey research, the design includes longitudinal elements, which specifically resemble a panel study as the data is gathered on numerous occurrences from the same subjects over an extended period of time (Ary, Jacobs, Razavieh & Sorensen, 2006). Subsets of descriptive research, associational studies are conducted in order to explain and predict.
More specifically, the present investigation is categorized as predictive correlational, as it attempts to predict dependent variables based on independent variables that occurred at earlier points in time (Vogt, 2005). By analyzing the underlying relationships among the variables, explanatory relationships are established and predictive capabilities are investigated. However, it is imperative to note that while correlational research results can bear evidence for cause and effect, inferences of causality require experimental research (Ary et al.).
While the four model categories (as shown in Tables 2 through 5) operationally define performance measures in different ways, each ultimately is concerned with explaining and predicting the outcome of an “un-played” game. Each model includes home and away team averages of previously accumulated statistics as independent variables. For example, in a competition between home team A and away team B, which represents a single observation within the data set, team A‟s averages prior to the game and team B‟s averages prior to the game serve as the independent variables, and the outcome of the contest between A and B is the dependent variable. Furthermore, with a focus on team averages, it is crucial to account for aberrations resulting from small samples. For example, as team C prepares to play their second game of the season, any
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attempt at predicting the outcome solely based on performance measures from the first game of the season would be a monumental blunder. In the opening game of the season if team C simply had an off-night or played the league‟s premier squad, team C‟s corresponding performance measures would inadequately represent their skill level; conversely, if team C put together an abnormally exceptional performance on opening night or played one of the league‟s bottom-rung clubs, team C‟s performance measures would over represent team C‟s abilities.
In order to stabilize performance measures and reduce issues associated with small samples, the first 300 games of the season did not undergo statistical analysis for non-streak models. Equaling roughly a quarter of the NBA schedule, the first 300 games amount to each team playing approximately 20 games, which at minimum should be semi-representative of future performances. Therefore for each of the two populations, the data set for non-streak models consisted of 930 observations. Alternatively, streak models focus exclusively on team performance over the previous five games, thus concerns regarding stability are not as relevant. For all streak models the first 125 games of the season will be removed from statistical analysis to ensure that each team has completed the five game prerequisite. As a result, for each population, all streak models are composed of 1105 unique observations.
Conversion of statistics. Given the enormity of time and effort required to construct the data files for each model, a Java computer program, Statistic Converter, was created to complete the process of converting the primary collected data into new variables. From the command line, Statistic Converter, which requires an accompanying
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text file in a predetermined format, is executed. Using the information contained in the preliminary data set, the program performs a multitude of computations and upon completion creates 16 text files, one for each non-logarithmic model. A second program written in the computer language Ruby, Logarithmic Statistic Converter, created the appropriate data files for logarithmic models. Logarithmic Statistic Converter takes the natural logarithm for specified variables and creates a text file for each of the eight logarithmic based models. It should be noted that the natural logarithm is not taken for scheduling fatigue variables or score differential, based on the fact that it is not possible to take the natural logarithm of 0 or negative numbers. These dichotomous independent variables as well as the score differential dependent variable remain in their original form throughout all models.
As a whole the development of Statistic Converter and Logarithmic Statistic
Converter reduces the incredibly laborious and repetitive process of constructing variables. With computations originally standing in the hundreds of thousands, the combination of these two programs makes the present study into a more feasible endeavor.
It is important to note that ensuring the accuracy of the numerous text files created by the two computer programs was a critical step in the research process. After all, these constructed data sets were the objects that underwent statistical analysis. Consequently, it was pivotal that all variables were calculated and outputted precisely in the predetermined formats. In order to complete the testing process, several test cases with all corresponding variables were calculated by hand, which were subsequently compared
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with the program‟s output files. Each case included unique situational factors to test the accuracy of the program as a whole. For all models, performance measures for game number 301, Charlotte Bobcats at Miami Heat, game number 395 Utah Jazz at Chicago
Bulls and game number 486 New Orleans Hornets at Portland Trailblazers, were tested.
As the number of “played” games per team fluctuates, it was imperative to test different possible circumstances. For game 301, the Charlotte Bobcats and the Miami Heat each competed in 20 contests prior to their “un-played” game; game 395 focused on the situation when the home team participated in more previous games than the away team
(i.e., the Utah Jazz had 28 “played” games compared with the 26 “played” games of the
Chicago Bulls); and game 486 presented the case when the away team (i.e., Portland
Trailblazers, 32 “played” games) played more games than the home team (i.e., New
Orleans Hornets, 28 “played” games). Furthermore, all streak models were tested with game number 132, Golden State Warriors at Los Angeles Clippers, to ensure correct computations for cases occurring before game 300. Following the completion of these rigorous tests, an independent party with in-depth knowledge of basketball performance measures confirmed the programs‟ accuracy by similarly computing all variable values for the four test cases. Based on these tests the researcher is supremely confident in the computer programs‟ abilities to correctly produce the numerous data sets.
Subject Selection
Utilizing two populations, the first composed of the entire 2007-2008 NBA regular season, the second consisting of the 2008-2009 NBA season, the current study ascertained the overall predictive capabilities and relative importance of numerous
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independent variables. With each regular season contest representing a single element within the population, the present study circumvented the limitations associated with sampling by implementing a census. Whereas sampling procedures provide only an estimate, a result of not including information on all experimental units, a census is a study conducted on the entire population. While sport offers countless opportunities to explore both performance and the relationship between previously accumulated measures and future outcomes, National Basketball Association regular season games were selected due to several factors.
With an abundance of accessible quantifiable measures, the NBA represents an ideal candidate for research investigations. Of the major professional leagues within the
United States, the NBA supports the highest degree of competitive imbalance. In order to determine the competitive balance of a league, the Noll-Scully statistic “simply takes the standard deviation of winning percentage and divides by the idealized standard deviation” (Berri et al., 2007, p. 49), in which the idealized standard deviation represents the distribution of equally talented teams. For the twenty year period beginning with the
1987-1988 season, the Noll-Scully measure of competitive balance for the NBA was 2.84 or nearly three standard deviations from the ideal (Berri et al., 2007). In contests featuring teams of relatively similar ability levels, one would anticipate an intense competition with an uncertain outcome. In addition, leagues composed of highly competitive teams should have little variation in individual team winning percentages.
From the perceptive of a prognosticator, the NBA‟s lack of competitive balance significantly reduces uncertainty, which consequently leads to more accurate forecasts.
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Finally, playoff competitions are excluded from analysis, as a result of their unpredictable nature. Once the postseason rolls around, offensive performance generally suffers as teams‟ employ a slower pace, hence limiting possessions and opportunities to score (Bellotti, 1990). Playoff series involve numerous games between the same teams and as a result, coaches regularly implement strategic adjustments that further limit offensive productivity. In addition to the strategic alterations within postseason play, incorporating playoff success is problematic due to small sample sizes. For example, in a seven game series between a superior team and an inferior team, in which the superior team would on average win two out of three encounters, the inferior will nevertheless emerge victorious in the seven game series roughly 20% of the time (Mlodinow, 2008).
Outcome Measures
The importance of establishing a valid, reliable and suitable instrument cannot be understated. Validity, which pertains to systematic errors in measurement, is comprised of several distinct facets, including content validity, criterion-related validity, construct validity, face validity, internal validity and external validity (Vogt, 2005). As a whole, only a valid instrument is capable of measuring the full extent of what it purports to measure (Ary et al., 2006). Equally important, reliability addresses occurrences of random error in measurement, as only a reliable instrument is capable of producing consistent results (Ary et al.). Beyond the psychometric properties of an instrument, research investigations must both acknowledge and limit the presence of internal and external validity threats. While internal validity threats (e.g., self-selection effect, mortality, regression effect) are concerned with ensuring that the treatment caused the
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outcome, external validity threats pertain to the generalizability of the research findings
(e.g., interaction of personalogical variables and treatment, multiple treatment interference and the Hawthorne effect; Vogt).
Correlational in nature, the current study used NBA box scores compiled by the
Elias Sports Bureau (2007), the official statistician of the NBA, which are readily available and accessible via ESPN‟s website. The numerous statistics within these box scores served as an initial dataset for the present investigation. In conceptualizing the box score as the instrument, the utilization of a previously established reputable and accepted measure significantly reduces questions and concerns regarding psychometric properties. Although psychometrically sound, it should be noted that box scores are hardly infallible, for they are subject to human error (e.g., data entry) as well as inconsistencies due to the inherent subjectivity in the assignment of several measures
(e.g., assists). Furthermore, as the study lacks a treatment in line with the traditional experimental definition, issues pertaining to internal validity are not pertinent. Finally, as discussed previously, the investigation will incorporate data from all NBA games throughout the 2007-2008 and 2008-2009 seasons. The execution of a census is advantageous due to the fact that all elements of the population are included, thus concerns related to external validity are nonexistent.
Alternatively, the perspective that the 2007-2008 and 2008-2009 seasons represent purposive non-probability samples, constructed from the population of all professional basketball games throughout history, is intuitively problematic. Over the years, NBA rules as well as overall talent levels have changed immensely.
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Consequently, attempts at generalizing findings stemming from the period of modern professional basketball to previous eras, and vice versa, would be a significant miscalculation. For example, in the period spanning from 1974 to 2002, possessions per game have reached as high as 107.2 in 1974 to as low as 89.6 in 1999; similarly average efficiency ratings peaked at 108.4 in 1987 and hit a low of 98.6 in 1974 and 1975 (Oliver,
2004).
Data Analysis
In a research investigation, it is vital that the selection of statistical procedures is appropriate given the study‟s research objectives. Upon gathering and organizing data in a manner consistent with the four general model categories, the analysis of the relationships between independent and dependent variables were accomplished through both multiple regression analysis and logistic regression analysis. A detailed discussion on the implementation and interpretation of these statistical analyses will ensue.
Multiple regression analysis is used to predict or explain a dependent variable from a set of independent variables, in which the independent variable set is permitted to include an assortment of metric and non-metric variables (Licht, 1995). With score differential serving as the dependent variable for each case in the population, multiple regression analysis assessed the relationship between the independent variables within each respective model and the difference in the “un-played” competition‟s final score. A coefficient of determination (R2) was calculated for each regression equation to determine the amount of variance explained in the dependent variable by the independent variable set (Hair, Black, Babin, Anderson & Tatham, 2006). Additionally, beta
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coefficients were interpreted to determine the relative importance of the independent variables within each regression equation (Hair et al., 2006).
For each multiple regression analysis, the underlying assumptions of the statistical procedure were tested. More specifically, to ensure a linear relationship existed between the independent variables and the dependent variable, the residual plot was examined
(Osborne & Waters, 2002). A null plot, for the scatter plot of standardized residuals as a function of standardized predicted values, is evidence of a linear relationship (Osborne &
Waters). Furthermore, this null plot illustrates that the variance of errors is equal for all levels of the predictors (i.e., homoscedasticity), another assumption for multiple regression (Licht, 1995). Finally, to test for normality of the error term distribution, the histogram of residuals and the normal probability plot were visually examined. A distribution that is approximately normal for the histogram of residuals, as well as a residual line that follows a straight diagonal line for the normal probability plot, is evidence that the assumption of normality is not violated (Hair et al., 2006). Unless otherwise noted, all multiple regression based models satisfied these aforementioned assumptions.
Selection of the independent variables for the multiple regression analyses were accomplished through several techniques. Initially, in order to assess the overall predictive accuracy of each model, all respective independent variables were entered simultaneously. While including all independent variables inevitably resulted in the largest amounts of variance explained, parsimony was sacrificed, as variables that failed to significantly contribute were included (Hair et al., 2006).
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As a second entry technique for multiple regression analyses, a stepwise estimation was employed for each model. Sequential search methods, such as stepwise estimation, are advantageous as they objectively select variables that maximize predictive capabilities (Hair et al., 2006). Regression equations arising from sequential search methods employ the fewest number of variables. Given the sheer number of variables within the models of the current study (e.g., 64 independent variables for absolute models), a stepwise approach was valuable, for it facilitated the analysis and interpretation of the independent variables. Furthermore, unlike the simultaneous entry method, stepwise estimation ensures that independent variables within the regression equation are statistically significant. Statistically significant regression coefficients reject the null hypothesis, therefore indicating that their value is indeed different from zero
(Hair et al.). The determination of individual contributions for independent variables was accomplished via stepwise estimation.
Stepwise estimation is conducted through multiple stages. At each step, the independent variable that is statistically significant and capable of explaining the largest portion of unexplained variance is added to the regression equation (Hair et al., 2006).
Subsequently, the regression equation is recalculated with the new independent variable included. Once completed, all independent variables within the regression equation are examined to determine if they all should be kept (Hair et al.). This methodical process continues until none of the remaining independent variable candidates would significantly contribute to the prediction of the dependent variable (i.e., remaining partial regression coefficients are non-significant; Hair et al.).
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Finally, hierarchal multiple regression analysis was used to ascertain the contribution of the present study‟s unique independent variables (i.e., scheduling fatigue measures and team depth contribution factors). Independent variables for each model were entered through two stages. In the first stage, a stepwise estimation was conducted over a pool of the model‟s respective performance measures. For the first stage, scheduling fatigue measures and team depth contribution factors were not included among the independent variable candidates. In the second stage, all unique variables were simultaneously entered into the regression equation. The amount of additional variance explained, as a result of the second stage of entry, was used to determine the influence of the scheduling fatigue measures and team depth contribution factors.
In a secondary analysis, logistic regression was used to determine each model‟s proficiency at predicting the winner of “un-played‟ games. Logistic regression analysis attempts to predict membership, represented as a dichotomous dependent variable, through a set of independent variables (Wright, 1995). For the current investigation, a value of 0 for the dependent variable indicated an away team victory and a value of 1 represented a win by the home team. For each model, the classification matrix was examined to determine overall fit. The classification matrix reveals the hit ratio, which is the percentage of correctly classified cases (Hair et al., 2006). Therefore, for the present investigation, the hit ratio identified the percentage of correctly predicted games.
Additionally, to further facilitate model comparisons, likelihood values, expressed as -2 log likelihood (-2LL), are reported as a second indication of model fit (Wright). Model fitness is negatively correlated with -2LL (i.e., a lower value of -2LL indicates a better
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fitting model). Baseline goodness-of-fit measures (i.e., the hit ratio and the -2LL) were established by the null model, which includes a constant and zero independent variables.
Under the present conceptualization the basic conditions for logistic regression were satisfied. More specifically, the dependent variable (i.e., game winner) is a dichotomous variable, which is statistically independent. Statistical independence requires that each observation in the data set only appears once (Wright, 1995).
Furthermore, the analyses employ a dependent variable that is mutually exclusive and collectively exhaustive (Wright). Away team victory and home team victory are mutually exclusive and collective exhaustive, because an individual contest cannot conclude simultaneously as an away team win and a home team win. As a whole, logistic regression is a preferred statistical technique as it does not face the stringent assumptions
(e.g., multivariate normality and equal variance) that limit alternative analyses (Hair et al., 2006).
Logistic regression analysis assumes that the relationship between the independent variable(s) and the dependent variable is nonlinear in nature (Wright, 1995).
By employing a logit transformation, logistic regression predicts the probability of the dependent variable occurring. The logistic curve of predicted values is fitted to the observed data, in which predicted probabilities that exceed 0.50 coincide with a prediction that the event has occurred (i.e., the dependent variable equals one), and predicted probabilities less than 0.50 are associated with a prediction that the event has not occurred (i.e., the dependent variable equals zero; Hair et al., 2006). Due to the
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incorporation of the logistic curve, which is S-shaped, the interpretation of logistic regression coefficients differs from that of multiple regression coefficients.
Logistic regression coefficients in their raw form are difficult to interpret, as they represent the change in the natural logarithm of the odds ratio (Wright, 1995). As a whole, the exponentiated logistic coefficient, which is calculated by raising the logistic regression coefficient to the exponent of the mathematical constant e, is used to determine the influence of the predictor variables. Exponentiated coefficients greater than 1.0 indicate a positive relationship between the independent variable and the dependent variable, whereas exponentiated coefficients less than 1.0 represent a negative relationship between the independent variable and the dependent variable (Hair et al.,
2006). Within the current study, in which a dependent variable valued at zero coincides with an away team victory and a value of one represents a home team win, exponentiated coefficients greater than one are positively correlated with a home team victory and exponentiated coefficients less than one are negatively correlated with a home team win, and therefore positively correlated with an away team victory. The magnitude of the relationship between the independent variables and the dependent variables are calculated as a percent change in odds. The percentage change in odds is found by subtracting 1.0 from the exponentiated coefficient and then by multiplying this figure by 100 (Hair et al.). The influence of an independent variable is expressed in terms of the percentage change in the odds of a home team victory, stemming from a one unit increase in the respective independent variable.
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Paralleling the aforementioned multiple regression analyses, the process of entering the independent variables into the logistic regression equations was accomplished through several techniques. In an effort to determine the overall forecasting ability of each model, all respective independent variables were entered simultaneously. Although the inclusion of all independent variables has the potential to produce regression equations that include lower -2LL values, parsimony is sacrificed, as variables that fail to significantly contribute are included within the regression equation
(Hair et al., 2006). Minimizing the number of independent variables is beneficial as the resultant regression equation is more likely to be generalizable and numerically stable
(Homer & Lemeshow, 2000). In addition, the simultaneous entry of all independent variables for the more robust models (e.g., the absolute models), can be problematic due to a violation of the rule of ten.
The rule of ten for logistic regression states that in order to construct a stable regression equation, a minimum of 10 events are required per predictor variable (Homer
& Lemeshow, 2000). In the present study, an event can be defined as either an away team win or a home team win, thus in order to satisfy the rule of ten, the number of independent variables multiplied by ten should be less than the number of events in the smaller of the two categories (i.e., in the present study, the number of away team victories). Thus, for the 930 observations for the 2007-2008 NBA season, which included 364 away team wins and 566 home team wins, the maximum number of predictor variables in the logistic model would be 36. While violating the rule of ten is a concern, it must be noted that according to Vittinghoff and McCulloch (2006), this rule of
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thumb may ultimately be too conservative. Based on a series of simulations, Vittinghoff and McCulloch analyzed confidence interval coverage, type I error and relative biases, and found that problems were uncommon with models consisting of five to nine events- per-variable and that even increasing the number to 10 to 16 events-per-variable didn‟t entirely erase problematic elements.
As a second method of variable entry, stepwise estimation was employed for the logistic regression analyses. While there has been a shift away from deterministic model building, stepwise procedures, which select and delete variables based on importance determined by a statistical algorithm, can be rather useful when the influence of the individual independent variables is an unknown (Homer & Lemeshow, 2000). Given the large number of variables within the models of the current study, a stepwise approach is advantageous as it facilitates the construction of regression equations that maximize predictive capabilities while simultaneously minimizing the number of predictors (Sarkar,
Midi & Rana, 2010). Logistic regression analysis offers a variety of stepwise procedures including forward entry based on a reduction in the -2LL value, forward entry based on the greatest Wald coefficient, forward entry based on the highest conditional probability, backward elimination based on a reduction of the -2LL value, backward elimination based on greatest Wald coefficients, and backward elimination based on the highest conditional probability (Hair et al., 2006).
In order to increase the likelihood of creating a model that adheres to the rule of ten, backward elimination stepwise procedures were removed from consideration. The rationale from not using backward elimination procedures was two-fold. First, the
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process of entering variables in a forward manner was consistent with the stepwise procedures implemented in the current investigation‟s multiple regression analyses.
Second, backward elimination is problematic as several of the initial models in the present study (e.g., the absolute models) fail to satisfy the rule of ten prior to the removal of the independent variables.
All three forward stepwise procedures initially involve the evaluation of the score statistic. At each step, based on a significance cut-off point for inclusion, which was set at a level of 0.15, the variable containing the most significant score statistic is added to the regression equation (Homer & Lemeshow, 2000). Previous research has shown that the default level of 0.05 is too stringent and often results in the exclusion of important variables (Homer & Lemeshow). Upon adding the predictor to the regression equation, the computer examines the variables to determine if any of the included predictors should be removed. This removal process is where the three forward stepwise procedures differ.
According to Field (2009), the likelihood ratio method is the superior removal criterion, because the conditional statistic is merely a less intensive arithmetic version of the likelihood ratio method and the Wald statistic can produce unstable results. Forward entry based on the reduction of the -2LL statistic, compares the current equation‟s -2LL value to a secondary equation‟s -2LL value with the variable in question removed. In the case that the removed variable makes a significant difference in the prediction of the observed data, the variable is retained (Field). The significance level for removal was set at 0.20, which is consistent with the recommendations of Homer and Lemeshow. To avoid the entry and exit of a variable in successive steps, it was critical that the
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significance level for removal (0.20) was set at a larger value than the significance level for entry (0.15).
Lastly, to examine the influence of scheduling fatigue measures and team depth contribution factors, hierarchal logistic regression analysis was conducted. During the first stage, performance statistics entered the regression equation through forward stepwise entry based a reduction in the -2LL value. The unique variables of the present study were not included among the pool of candidates for the first stage of entry. In the second stage, scheduling fatigue measures and team depth contribution factors were simultaneously entered into the equation. A comparison of the overall model fit (i.e., the hit ratio and -2LL values for logistic regression based models), indicated the influence of the unique variables in regards to the explanation and prediction of the “un-played” game‟s winner.
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Chapter 4: Results
Descriptive statistics for the independent and dependent variables of the four major model classifications (i.e., absolute model, relative model, possession-based model and play model), for each population, are provided in Appendixes A to H. It must be noted that descriptive statistics for streak models, logarithmic models and logarithmic streak models are excluded, as these descriptive statistics are essentially redundant and fail to provide additional insight. Additionally, multiple linear regression results for models utilizing score differential as the dependent variable are reported in Tables 8-71.
Finally, logistic regression results are documented for models including a dichotomous game winner variable as the dependent variable in Tables 72-103.
Multiple Linear Regression
Absolute model 2007-2008. Basic descriptive statistics, including means, standard deviations, minimums and maximums, for the 2007-2008 absolute model are presented in Appendix A. As all NBA teams compete in an equal number of away and home competitions, team performance measures averaged across the entire league, segmented under the present conceptualization, are approximately equivalent. For example, as team D enters an away competition, their previous performance measures are included as variables classified under the away categorization. However, when team D
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prepares for an upcoming home contest the following night, its updated performance measures are identified under the heading of home statistics. Thus the averages of each individual statistical measure, regardless of its classification as an offensive or defensive statistic, regardless of location indicator, are essentially equal (e.g., A2FGMA,
A2FGMAA, H2FGMA and H2FGMAA). Differences among these statistics are merely due to rounding as well as slight divergences in the number of home or away games each team has played over the 930 game observation set. Alternatively, a comparison of final score averages, separated by team origin (i.e., away or home), reveals a differential in favor of the home team of 3.563 points per game for the 2007-2008 season. This difference in average final scores empirically supports the notion of home court advantage.
An examination of scheduling fatigue variables reveals information pertaining to the logistics of league scheduling. Over the final 930 games of the 2007-2008 NBA season, the away team entering the competition in question was playing their fourth game in five days 4.8% of the time, third game in four days 37.8% of the time, second game in the previous three days 83.3% of the time, was coming off a home game the previous day
13.2% of the time and had an away contest the previous day 19.8% of the time.
Frequency percentages for the home team were four games in five days 1.3%, three games in four days 27.0%, two games in three days 74.0%, a home game the previous day 0.8% and an away game the previous day 13.9%.
The entire model with all 64 independent variables was statistically significant
(F64, 865 = 6.384; p < 0.001) and explained 32.1% of the variance in score differential (i.e.,
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away team final score minus home team final score). In the stepwise regression (see
Table 8), 14 independent variables entered the equation to explain 28.5% of the variance in score differential and was statistically significant (F14, 915 = 26.050; p < 0.001). Of these 14 variables, home defensive rebounds allowed (β = 0.221), home turnovers (β =
0.210), and away three pointers made (β = 0.191) had the greatest influence on score differential. Every one additional home defensive rebound allowed resulted in a 1.768 change in score differential. The positive regression coefficient represents an increase in the away team‟s favor, thus increasing the likelihood of an away team victory. Intuitively this is sound, as an allowed defensive rebound corresponds to a failed offensive rebound opportunity, which is advantageous to the opponent.
In the hierarchal regression analysis (Table 9), 14 absolute performance variables entered the equation through a stepwise technique, which was followed by simultaneous entry of scheduling fatigue measures and team depth contribution factors. The addition of the present study‟s unique variables increased the variance explained by 0.9% for a total of 29.4%. Away one day prior (variable one) was the sole unique variable to reach statistical significance (p = 0.019).
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Variable Coefficient Std. Error β t
Constant -1.459 16.061 - -0.091 Home assists -0.747** 0.221 -0.114 -3.376 Away three pointers made 1.708** 0.327 0.191 5.217 Away defensive rebounds allowed -1.484** 0.280 -0.187 -5.302 Away steals 2.396** 0.407 0.171 5.889 Home defensive rebounds allowed 1.768** 0.274 0.221 6.464 Home defensive rebounds -1.200** 0.341 -0.130 -3.522 Away defensive rebounds 1.447** 0.321 0.156 4.510 Away turnovers -1.292** 0.354 -0.123 -3.644 Home steals -1.375** 0.491 -0.098 -2.800 Home turnovers 2.246** 0.358 0.210 6.280 Home three pointers made -1.061** 0.331 -0.121 -3.205 Away three pointers made allowed -1.216* 0.558 -0.071 -2.178 Home offensive rebounds -0.904* 0.359 -0.086 -2.518 Home free throws made -0.496* 0.227 -0.076 -2.190
R 0.534 R² 0.285 Std. Error 11.628 * Significant at 0.05 level ** Significant at 0.01 level
Table 8. Stepwise multiple regression results for the absolute model 2007-2008
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Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .217a .047 .046 13.331 .047 45.733 1 928 .000
2 .298b .089 .087 13.043 .042 42.346 1 927 .000
3 .361c .130 .127 12.750 .041 44.121 1 926 .000
4 .412d .170 .166 12.464 .039 43.982 1 925 .000
5 .436e .190 .186 12.316 .020 23.340 1 924 .000
6 .455f .207 .202 12.195 .017 19.415 1 923 .000
7 .474g .224 .218 12.066 .018 20.818 1 922 .000
8 .489h .239 .232 11.960 .015 17.544 1 921 .000
i 99 9 .503 .253 .245 11.857 .014 17.029 1 920 .000
10 .512j .262 .254 11.784 .010 12.345 1 919 .000
11 .520k .270 .261 11.731 .008 9.455 1 918 .002
12 .520l .270 .262 11.724 .000 .029 1 918 .864
13 .524m .275 .266 11.693 .005 5.898 1 918 .015
14 .527n .278 .269 11.671 .003 4.424 1 917 .036
16 .534p .285 .274 11.628 .004 4.797 1 915 .029
17 .542q .294 .274 11.631 .009 .959 12 903 .486 Continued
Table 9. Hierarchal multiple regression results for the absolute model 2007-2008
Table 9 continued
a. Predictors: (Constant), HASTA
b. Predictors: (Constant), HASTA, A3PMA
c. Predictors: (Constant), HASTA, A3PMA, ADREBAA
d. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA
e. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA
f. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA, HDREBAA
g. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA, HDREBAA, HDREBA
h. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA, HDREBAA, HDREBA, ADREBA
i. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA
100 j. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA
k. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HSTLAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA
l. Predictors: (Constant), HASTA, A3PMA, ADREBAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA
m. Predictors: (Constant), HASTA, A3PMA, ADREBAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA, H3PMA
n. Predictors: (Constant), HASTA, A3PMA, ADREBAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA, H3PMA, A3PMAA
o. Predictors: (Constant), HASTA, A3PMA, ADREBAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA, H3PMA, A3PMAA, HOREBA
p. Predictors: (Constant), HASTA, A3PMA, ADREBAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA, H3PMA, A3PMAA, HOREBA, HFTMA
q. Predictors: (Constant), HASTA, A3PMA, ADREBAA, ASTLA, HDREBAA, HDREBA, ADREBA, ATOA, HSTLA, HTOA, H3PMA, A3PMAA, HOREBA, HFTMA, A2DP, H1DPV1,
H1DPV2, A4DP, A1DPV1, H2DP, H4DP, A1DPV2, H3DP, ACONA, A3DP, HCONA
r. Dependent Variable: Score_Differential
Absolute model 2008-2009. Descriptive statistics of the absolute model for the
2008-2009 NBA campaign are provided within Appendix B. Under the present conceptualization, as was the case for the 2007-2008 absolute model, league wide performance measure means are essentially equivalent for both away and home competitors, as well as for statistics accumulated on both the offensive and defensive ends. For example, the average offensive rebounds for the away team going into the game in question was 11.078, whereas away offensive rebounds allowed, home offensive rebounds and home offensive rebounds allowed equaled 11.088, 11.080 and 11.085 respectively. As stated previously, the underlying reason for the similarities in these values is due to the fact that teams throughout the season appear on both sides of the ledger (i.e., as an away team and as a home team), contingent upon on the individual game in question.
The scheduling fatigue measures provide some rather fascinating insight into the frequencies of certain scheduling events occurring. The away team over the 930 cases in the 2008-2009 season, played four games within a five day span 4.2% of the time, three games in four days 34.2% of the time, two games in three days 81.9% of the time, a home competition the previous day 14.0% of the time and an away game the previous day
17.2% of the time. Similarly for the home team, the frequency percentage of four games in five days was 1.6%, three games in four days 26.6%, two games in three days 72.8%, a home game the previous day 1.1% and an away game the previous day 14.2%. Finally, the difference between final scores of the away team and the home team for the 2008-
101
2009 season equaled 3.504 points in favor of the home team, once again signifying the importance of playing on one‟s home floor.
Simultaneous entry of all 64 variables resulted in a statistically significant equation (F64, 865 = 5.009; p < 0.001), which predicted 27.0% of the variance in score differential (i.e., away team final score subtracted by home team final score). As shown in Table 10, in the stepwise regression, ten independent variables entered the equation, which explained 22.2% of the variance in score differential and was statistically significant (F10, 919 = 26.195; p < 0.001). Home assists allowed (β = 0.323) emerged as the most important relative contributor, as every home team assist allowed resulted in a
2.463 increase differential. An increase in the home team average assists allowed is an indictment on their defense, thus the change in score differential represents movement in favor of the away team. Following home assists allowed, home personal fouls allowed (β
= -0.242) and home personal fouls (β = 0.239), were the most important respective independent variables.
As displayed in Table 11, hierarchal entry of the independent variables initially entered nine absolute measures of performance through the stepwise technique. Entering the 12 unique variables in the second stage of the entry process resulted in a R2 change of
0.023, as the 21 variable statistically significant equation (F21, 908 = 13.053; p < 0.001) explained 23.2% of the variance in score differential. In terms of the individual independent variables, away four days prior (β = -6.243; p = 0.002) and home three days prior (β = 2.204; p = 0.027) were only unique variables that reached the 0.05 level of statistical significance.
102
Variable Coefficient Std. Error β t
Constant -37.185** 12.654 -2.939 Home assists allowed 2.463** 0.313 .323 7.868 Home personal fouls allowed -2.257** 0.325 -.242 -6.934 Home personal fouls 2.065** 0.297 .239 6.952 Away defensive rebounds 1.374** 0.300 .152 4.583 Away four days prior -6.271** 1.884 -.097 -3.329 Away defensive rebounds allowed -1.230** 0.218 -.173 -5.641 Away three pointers made 0.982** 0.289 .117 3.399 Home three days prior 2.433** 0.854 .083 2.849 Home offensive rebounds -1.032** 0.376 -.090 -2.744 Home three pointers attempted 0.291 -.103 -2.583 allowed -0.753**
R 0.471 R² 0.222 Std. Error 11.448 * Significant at 0.05 level ** Significant at 0.01 level
Table 10. Stepwise multiple regression results for the absolute model 2008-2009
103
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .268a .072 .071 12.441 .072 71.955 1 928 .000
2 .327b .107 .105 12.212 .035 36.102 1 927 .000
3 .367c .135 .132 12.027 .028 29.789 1 926 .000
4 .413d .170 .167 11.783 .036 39.696 1 925 .000
5 .429e .184 .179 11.692 .014 15.361 1 924 .000
6 .437f .191 .186 11.648 .007 8.023 1 923 .005
7 .446g .199 .192 11.599 .008 8.901 1 922 .003
8 .450h .203 .196 11.575 .004 4.775 1 921 .029
104 9 .457i .209 .201 11.537 .006 7.131 1 920 .008
10 .482j .232 .214 11.442 .023 2.273 12 908 .008 Continued
Table 11. Hierarchal multiple regression results for the absolute model 2008-2009
Table 11 continued
a. Predictors: (Constant), HASTAA
b. Predictors: (Constant), HASTAA, AASTAA
c. Predictors: (Constant), HASTAA, AASTAA, HPFAA
d. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA
e. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA
f. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA
g. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA
h. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HOREBA
i. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HOREBA, H3PAAA
j. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HOREBA, H3PAAA, H4DP, A1DPV1, H2DP, H1DPV1, A1DPV2, HCONA, A4DP, A2DP,
105
H1DPV2, ACONA, H3DP, A3DP
k. Dependent Variable: Score_Differential
Absolute model (streak variables) 2007-2008. Simultaneous entry of all 64 independent variables yielded an equation that was statistically significant (F64, 1040 =
4.345; p < 0.001) and predicted 21.1% of the variance in score differential (i.e., away team final score minus home team final score). Results of employing a stepwise entry technique are presented in Table 12. The constructed equation contained 14 independent variables, was statistically significant (F14, 1090 = 15.806; p < 0.001), and explained 16.9% of the variance in score differential. Home defensive rebounds allowed (β = 0.173) emerged as the most influential independent variable, as each additional home defensive rebound allowed resulted in a 0.881 increase in score differential. The positive regression coefficient signifies that the practice of allowing defensive rebounds hinders a team‟s ability to be successful. Therefore, an increase in the number of defensive rebounds allowed for the home squad corresponds to an increase in score differential in favor of the away team. Away three pointers attempted (β = 0.169) and away defensive rebounds allowed (β = -0.165) were the second and third most important contributors to the model respectively.
Hierarchal multiple regression analysis results for the absolute model (streak variables) are shown in Table 13. The 14 absolute streak performance measures entered in the first stage explained 16.8% of the variance in score differential. Simultaneous entry of the current investigation‟s 12 unique variables in the second stage increased the variance explained to 18.2%. However, of the 12 unique variables, only away one day prior (variable one; β = -3.187; p = 0.010) and home three days prior (β = 2.388; p =
0.015) were statistically significant at the 0.05 level.
106
Variable Coefficient Std. Error β t
Constant 10.267 9.861 - 1.041 Home assists -0.446** 0.139 -.098 -3.211 Away defensive rebounds allowed -0.840** 0.157 -.165 -5.335 Home turnovers 1.031** 0.190 .157 5.422 Away assists allowed -0.404** 0.137 -.088 -2.942 Away steals 1.037** 0.251 .120 4.126 Home defensive rebounds allowed 0.881** 0.154 .173 5.729 Home three pointers attempted -0.258** 0.091 -.085 -2.839 Home defensive rebounds -0.684** 0.160 -.130 -4.275 Home steals -0.878** 0.247 -.102 -3.550 Away turnovers -0.658** 0.189 -.102 -3.482 Away three pointers attempted 0.517** 0.091 .169 5.700 Away two point field goals 0.098 .111 3.173 attempted allowed 0.310** Away one day prior (variable one) -2.480* 1.089 -.063 -2.278 Away offensive rebounds allowed -0.518* 0.235 -.074 -2.202
R 0.411 R² 0.169 Std. Error 12.520 * Significant at 0.05 level ** Significant at 0.01 level
Table 12. Stepwise multiple regression results for the absolute model (streak variables) 2007-2008
107
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .188a .035 .035 13.407 .035 40.522 1 1103 .000
2 .244b .060 .058 13.243 .024 28.461 1 1102 .000
3 .278c .077 .075 13.123 .018 21.210 1 1101 .000
4 .302d .091 .088 13.029 .014 16.943 1 1100 .000
5 .317e .101 .097 12.968 .009 11.371 1 1099 .001
6 .329f .108 .104 12.919 .008 9.452 1 1098 .002
7 .340g .116 .110 12.871 .007 9.171 1 1097 .003
8 .350h .122 .116 12.829 .007 8.176 1 1096 .004
108 9 .364i .133 .125 12.760 .010 12.893 1 1095 .000
10 .373j .139 .131 12.720 .006 7.928 1 1094 .005
11 .382k .146 .137 12.676 .007 8.684 1 1093 .003
12 .380l .144 .136 12.680 -.001 1.798 1 1093 .180
13 .389m .151 .143 12.635 .007 8.804 1 1093 .003
14 .396n .157 .148 12.598 .006 7.505 1 1092 .006
15 .396o .157 .148 12.592 .000 .009 1 1092 .922
16 .401p .161 .152 12.566 .004 5.616 1 1092 .018
17 .406q .165 .155 12.544 .004 4.727 1 1091 .030
18 .410r .168 .157 12.527 .003 3.983 1 1090 .046
19 .426s .182 .162 12.490 .014 1.544 12 1078 .103 Continued
Table 13. Hierarchal multiple regression results for the absolute model (streak variables) 2007-2008
Table 13 continued
a. Predictors: (Constant), HASTA
b. Predictors: (Constant), HASTA, A3PMA
c. Predictors: (Constant), HASTA, A3PMA, ADREBAA
d. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA
e. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA
f. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, AASTAA
g. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, AASTAA, ASTLA
h. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, AASTAA, ASTLA, HDREBAA
i. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA
j. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA
109 k. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA
l. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA
m. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA
n. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA, A3PAA
o. Predictors: (Constant), HASTA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA, A3PAA
p. Predictors: (Constant), HASTA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA, A3PAA, A2FGAAA
q. Predictors: (Constant), HASTA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA, A3PAA, A2FGAAA, AOREBAA
r. Predictors: (Constant), HASTA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA, A3PAA, A2FGAAA, AOREBAA, HPFAA
s. Predictors: (Constant), HASTA, ADREBAA, HTOA, AASTAA, ASTLA, HDREBAA, H3PAA, HDREBA, HSTLA, ATOA, A3PAA, A2FGAAA, AOREBAA, HPFAA, A2DP,
H1DPV2, H1DPV1, A4DP, ACONA, HCONA, H4DP, A1DPV1, H2DP, A1DPV2, H3DP, A3DP
t. Dependent Variable: Score_Differential
Absolute model (streak variables) 2008-2009. Entry of all 64 independent variables produced an equation that was statistically significant (F64, 1040 = 3.959; p <
0.001) and explained 19.6% of the variance in score differential (i.e., away team final score subtracted by home team final score). Results from employing a stepwise entry technique are shown in Table 14. The 14 variable stepwise equation was statistically significant (F14, 1090 = 13.704; p < 0.001) and predicted 15.0% of the variance in score differential. In terms of relative importance, away free throws attempted allowed (β = -
0.190) emerged as the largest individual contributor. Each free throw attempted allowed by the away team corresponded with a 0.626 decrease in score differential (i.e., a shift in score differential in favor of the home team). Following away free throws attempted allowed, home three pointers attempted (β = -0.169) and home assists allowed emerged as the equation‟s most important independent variables.
As displayed in Table 15, hierarchal multiple regression analysis was conducted in two stages, in which stepwise entry for the pool of absolute steak performance measures was followed by the simultaneous entry of the present study‟s unique variables.
The addition of the 12 unique variables, in the second stage, increased the variance explained by 2.8% for a total of 16.0%. However, of the unique variables, only away team depth contribution factor (β = 2.292; p = 0.001), away four days prior (β = -4.673; p
= 0.014) and home three days prior (β = 2.285; p = 0.018), were statistical significant at the 0.05 level.
110
Variable Coefficient Std. Error β t
Constant -25.562* 12.373 - -2.066 Home assists allowed 0.824** .146 .165 5.636 Home defensive rebounds allowed 0.533** .144 .115 3.710 Away offensive rebounds allowed -0.376 .205 -.054 -1.833 Home three pointers attempted -0.525** .101 -.169 -5.186 Away assists allowed -0.457** .147 -.094 -3.099 Away team depth contribution factor 2.379** .659 .103 3.611 Home personal fouls allowed -0.740** .171 -.132 -4.327 Home free throws attempted allowed 0.387** .098 .123 3.943 Home two point field goals made -0.466** .144 -.108 -3.242 Home three days prior 2.458** .831 .084 2.957 Away free throws attempted allowed -0.626** .174 -.190 -3.592 Away personal fouls 0.826** .309 .141 2.676 Away four days prior -4.371* 1.756 -.070 -2.490 Away defensive rebounds 0.339* .148 .068 2.290
R 0.387 R² 0.150 Std. Error 12.154 * Significant at 0.05 level ** Significant at 0.01 level
Table 14. Stepwise multiple regression results for the absolute model (streak variables) 2008-2009
111
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .174a .030 .029 12.903 .030 34.443 1 1103 .000
2 .221b .049 .047 12.785 .018 21.303 1 1102 .000
3 .256c .065 .063 12.678 .017 19.703 1 1101 .000
4 .278d .078 .074 12.602 .012 14.452 1 1100 .000
5 .299e .089 .085 12.526 .012 14.311 1 1099 .000
6 .311f .096 .092 12.483 .007 8.589 1 1098 .003
7 .322g .104 .098 12.439 .007 8.769 1 1097 .003
8 .336h .113 .106 12.382 .009 11.208 1 1096 .001
112 9 .343i .118 .111 12.351 .005 6.455 1 1095 .011
10 .353j .125 .117 12.310 .007 8.312 1 1094 .004
11 .358k .128 .120 12.288 .004 4.939 1 1093 .026
12 .363l .132 .122 12.270 .003 4.178 1 1092 .041
13 .400m .160 .141 12.135 .028 3.035 12 1080 .000 Continued
Table 15. Hierarchal multiple regression results for the absolute model (streak variables) 2008-2009
Table 15 continued
a. Predictors: (Constant), HASTAA
b. Predictors: (Constant), HASTAA, HDREBAA
c. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA
d. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA
e. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA
f. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA
g. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA
h. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA, H2FGMA
113 i. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA, H2FGMA, AFTAAA
j. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA
k. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, ADREBA
l. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, ADREBA, ASTLA
m. Predictors: (Constant), HASTAA, HDREBAA, AOREBAA, H3PAA, AASTAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, ADREBA, ASTLA, A1DPV2, H2DP, H1DPV1, HCONA,
H4DP, ACONA, A1DPV1, A4DP, A2DP, H3DP, H1DPV2, A3DP
n. Dependent Variable: Score_Differential
Absolute model (logarithmic variables) 2007-2008. Simultaneous entry of all 64 independent variables produced a statistically significant equation (F64, 865 = 6.340; p <
0.001) that predicted 31.9% of the variance in score differential (i.e., away team final score minus the home team final score). As presented in Table 16, the equation resulting from stepwise entry consisted of 13 independent variables, explained 27.3% of the variance in score differential, and was statistically significant (F64, 865 = 6.340; p < 0.001).
Home defensive rebounds allowed (β = 0.237), away steals (β = 0.192), and away assists allowed (β = -0.190) emerged as the most significant contributors respectively. More specifically, a one unit increase in the natural logarithm of home defensive rebounds allowed results in a 58.418 increase in score differential. One must exercise caution when attempting to interpret the regression coefficients of logarithmic variables in this manner. While it may appear that a substantial increase or decrease in score differential is incorrect, it must be borne in mind that natural logarithms operate on an exponential scale. Therefore, for example, increasing the natural logarithm of home defensive rebounds allowed by one unit actually requires an extensive increase in the number of home defensive rebounds allowed. As noted previously, coefficients corresponding to a positive increase in score differential imply movement in favor of the away team.
Alternatively, regression coefficients can be interpreted as a percentage change of the independent variable, which is invaluable for determining the corresponding influence on the dependent variable within logarithmic based models. For the 2007-2008
114
Variable Coefficient Std. Error β t
Constant 17.819 53.147 - .335 Home assists -20.949** 4.790 -.145 -4.373 Away assists allowed -32.464** 4.944 -.190 -6.566 Home steals allowed 19.869** 3.413 .179 5.821 Away three pointers made 10.214** 1.632 .184 6.258 Away steals 19.906** 3.153 .192 6.314 Home defensive rebounds allowed 58.418** 8.309 .237 7.031 Home defensive rebounds -44.837** 10.351 -.157 -4.332 Away free throws made allowed -15.960** 3.602 -.135 -4.430 Home steals -8.630** 3.307 -.084 -2.609 Away free throws made 14.548** 3.948 .119 3.685 Home three pointers attempted -7.058** 2.257 -.111 -3.127 Away team depth contribution factor 29.123** 10.215 .090 2.851 Home offensive rebounds -9.008* 3.748 -.078 -2.404
R 0.523 R² 0.273 Std. Error 11.715 * Significant at 0.05 level ** Significant at 0.01 level
Table 16. Stepwise multiple regression results for the absolute model (logarithmic variables) 2007-2008
115
absolute logarithmic model with stepwise entry, a 10% increase in the number of home defensive rebounds allowed corresponds to a 4.901 increase in score differential.2
The first stage of the hierarchal regression analysis produced an equation through stepwise entry, which was composed of 12 absolute logarithmic performance variables and predicted 26.7% of the variance in score differential (see Table 17). The simultaneous entry of scheduling fatigue and team depth contribution variables in stage two yielded an R2 change of 0.014, meaning the variables explained an additional 1.4% of the variance in score differential. Of the 12 unique independent variables, only away team depth contribution factor (β = 27.300; p = 0.008) and away one day prior (variable one; β = -2.529; p = 0.050), reached statistical significance at the 0.05 level.
2 In order to accurately calculate the percentage change associated with the independent variables encapsulated within the logarithmic based models, it is imperative that compounding is taken into account. Compounding is a consequence of the exponential nature of the logarithmic based models. First the percent change must be converted into decimal form (e.g., 1% = 0.01). Subsequently, this figure is either added or subtracted to 1.00, contingent upon whether the impact of an increase or a decrease in the independent variable is sought. Finally, the natural logarithm of this figure is multiplied by the regression coefficient.
For example, in 2007-2008 absolute logarithm model with stepwise entry, the impact of a 10% increase in home defensive rebounds allowed is found through the following: ln (1.10) * 51.418 = 4.901. Therefore a 10% increase in home defensive rebounds allowed corresponds to a 4.901 change in score differential in favor of the away team.
116
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .217a .047 .046 13.329 .047 46.003 1 928 .000
2 .299b .090 .088 13.036 .042 43.182 1 927 .000
3 .356c .127 .124 12.775 .037 39.155 1 926 .000
4 .396d .157 .153 12.557 .031 33.515 1 925 .000
5 .433e .188 .183 12.334 .031 34.754 1 924 .000
6 .455f .207 .202 12.190 .020 22.902 1 923 .000
7 .475g .226 .220 12.052 .019 22.359 1 922 .000
8 .488h .238 .232 11.963 .012 14.750 1 921 .000
117 9 .499i .249 .242 11.885 .011 13.069 1 920 .000
10 .506j .256 .248 11.837 .007 8.506 1 919 .004
11 .512k .262 .253 11.792 .006 8.014 1 918 .005
12 .517l .267 .257 11.760 .005 5.939 1 917 .015
13 .530m .281 .262 11.722 .014 1.505 12 905 .116 Continued
Table 17. Hierarchal multiple regression results for the absolute model (logarithmic variables) 2007-2008
Table 17 continued
a. Predictors: (Constant), HASTA
b. Predictors: (Constant), HASTA, AASTAA
c. Predictors: (Constant), HASTA, AASTAA, HSTLAA
d. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA
e. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA
f. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA
g. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA
h. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA, AFTMAA
118 i. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA, AFTMAA, HSTLA
j. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA, AFTMAA, HSTLA, AFTMA
k. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA, AFTMAA, HSTLA, AFTMA, H3PAA
l. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA, AFTMAA, HSTLA, AFTMA, H3PAA, HOREBA
m. Predictors: (Constant), HASTA, AASTAA, HSTLAA, A3PMA, ASTLA, HDREBAA, HDREBA, AFTMAA, HSTLA, AFTMA, H3PAA, HOREBA, A2DP, H1DPV1, H1DPV2, A4DP,
A1DPV1,H2DP, H4DP, ACONA, A1DPV2, H3DP, A3DP, HCONA
n. Dependent Variable: Score_Differential
Absolute model (logarithmic variables) 2008-2009. Entry of the 64 independent variables resulted in a statistically significant equation (F64, 865 = 5.032; p < 0.001), which predicted 27.1% of the variance in score differential (i.e., away team final score subtracted by home team final score). Results of employing a stepwise entry technique are shown in Table 18. The 18 independent variable equation was statistically significant
(F13, 916 = 21.214; p < 0.001) and explained 23.1% of the variance in score differential.
Home personal fouls allowed (β = -0.246) had the greatest influence on the dependent variable (i.e., score differential). A 10% increase in the number of home personal fouls allowed results in a 4.678 decrease in score differential. The negative regression coefficient signifies movement in favor of the home team, which was as expected. The home personal fouls allowed statistic represents the average number of personal fouls the home team‟s numerous opponents accumulate. Following home personal fouls allowed, home personal fouls (β = 0.242) and home assists allowed (β = 0.194) were the most important independent variables respectively.
Hierarchal multiple regression analysis results are shown in Table 19. The first stage of the process entered 11 absolute logarithmic measures of performance into the equation. The simultaneous entry of the current investigations unique variables, in the second stage, increased the variance explained in score differential by 2.2%, for a total of
23.9%. Of these 12 unique independent variables, only away four days prior (β = -6.117; p = 0.003) and home three days prior (β = 2.128; p = 0.032) reached statistical significance at the 0.05 level.
119
Variable Coefficient Std. Error β t
Constant -81.717 51.418 - -1.589 Home assists allowed 31.039** 5.463 .194 5.681 Away assists allowed -10.660 5.733 -.068 -1.859 Home personal fouls allowed -49.083** 6.926 -.246 -7.087 Home personal fouls 44.700** 6.641 .242 6.731 Away defensive rebounds 36.589** 9.699 .133 3.773 Away four days prior -5.940** 1.880 -.092 -3.160 Away defensive rebounds allowed -28.794** 7.644 -.133 -3.767 Away three pointers made 5.070** 1.874 .092 2.705 Home three days prior 2.368** .852 .081 2.779 Home assists -11.110* 5.562 -.061 -1.997 Home two point field goals made 6.537 .092 2.746 allowed 17.948** Home offensive rebounds -10.853** 4.120 -.087 -2.634 Home three pointers made -4.387* 1.789 -.079 -2.452
R 0.481 R² 0.231 Std. Error 11.396 * Significant at 0.05 level ** Significant at 0.01 level
Table 18. Stepwise multiple regression results for the absolute model (logarithmic variables) 2008-2009
120
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .271a .073 .072 12.432 .073 73.352 1 928 .000
2 .330b .109 .107 12.197 .036 37.170 1 927 .000
3 .369c .136 .133 12.015 .027 29.198 1 926 .000
4 .414d .171 .167 11.777 .035 38.836 1 925 .000
5 .429e .184 .180 11.690 .013 14.861 1 924 .000
6 .437f .191 .186 11.645 .007 8.189 1 923 .004
7 .445g .198 .192 11.603 .007 7.675 1 922 .006
8 .450h .203 .196 11.576 .005 5.334 1 921 .021
121 9 .456i .208 .200 11.545 .005 5.835 1 920 .016
10 .460j .211 .203 11.524 .004 4.445 1 919 .035
11 .466k .217 .208 11.490 .005 6.420 1 918 .011
12 .489l .239 .219 11.403 .022 2.166 12 906 .012 Continued
Table 19. Hierarchal multiple regression results for the absolute model (logarithmic variables) 2008-2009
Table 19 continued
a. Predictors: (Constant), HASTAA
b. Predictors: (Constant), HASTAA, AASTAA
c. Predictors: (Constant), HASTAA, AASTAA, HPFAA
d. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA
e. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA
f. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA
g. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA
h. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HASTA
i. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HASTA, H2FGMAA
j. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HASTA, H2FGMAA, HOREBA
122 k. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HASTA, H2FGMAA, HOREBA, H3PMA
l. Predictors: (Constant), HASTAA, AASTAA, HPFAA, HPFA, ADREBA, ADREBAA, A3PMA, HASTA, H2FGMAA, HOREBA, H3PMA, A1DPV2, H2DP, H1DPV1, H4DP, A1DPV1,
A4DP, A2DP, HCONA, ACONA, H3DP, H1DPV2, A3DP
m. Dependent Variable: Score_Differential
Absolute model (streak logarithmic variables) 2007-2008. Simultaneous entry of the 64 independent variables produced an equation that explained 21.4% of the variance in score differential (i.e., away team final score minus the home team final score) and was statistically significant (F64, 1040 = 4.427; p < 0.001). Displayed in Table 20 are the results from implementing a stepwise entry technique. The 13 variable equation was statistically significant (F13, 1091 = 17.171; p < 0.001) and predicted 17.0% of the variance in score differential. Away defensive rebounds allowed (β = -0.184) emerged as the equation‟s most influential independent variable. A 10% increase in the amount of away defensive rebounds allowed corresponds to a decrease of 2.726 in score differential. The negative regression coefficient signifies a shift of score differential in favor of the home team.
The second and third most impactful independent variables were away three pointers attempted allowed (β = 0.169) and home turnovers (β = 0.162) respectively.
In the hierarchal multiple regression analysis, as shown in Table 21, 14 absolute streak logarithmic performance measures entered the equation through a stepwise technique, which was followed by the simultaneous entry of the present investigation‟s
12 unique variables. The inclusion of the scheduling fatigue measures and team depth contribution factors explained an additional 1.3% of the variance in score differential, thus increasing the total variance explained to 18.2%. Of the variables entered in the second stage, only away one day prior (variable one; β = -3.279; p = 0.008) and home three days prior (β = 2.088; p = 0.033) reached statistical significance at the 0.05 level.
123
Variable Coefficient Std. Error β t
Constant -0.508 35.862 - -.014 Home assists -11.753** 2.979 -.118 -3.945 Away defensive rebounds allowed -28.601** 4.618 -.184 -6.193 Home turnovers 13.846** 2.482 .162 5.578 Away turnovers -12.210** 2.620 -.142 -4.660 Away turnovers allowed 11.994** 2.719 .134 4.411 Home defensive rebounds allowed 21.441** 4.585 .138 4.676 Home defensive rebounds -22.879** 4.815 -.143 -4.752 Away defensive rebounds allowed 15.184** 4.918 .093 3.088 Home steals -6.607** 1.750 -.109 -3.776 Home two point field goals attempted 12.932** 4.384 .084 2.950 Away three pointers attempted 2.220 -.076 -2.692 allowed -5.976** Away one day prior (variable one) -2.825** 1.088 -.072 -2.596 Away three pointers attempted 8.899** 1.568 .169 5.677
R 0.412 R² 0.170 Std. Error 12.506 * Significant at 0.05 level ** Significant at 0.01 level
Table 20. Stepwise multiple regression results for the absolute model (streak logarithmic variables) 2007-2008
124
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .190a .036 .035 13.403 .036 41.180 1 1103 .000
2 .247b .061 .059 13.234 .025 29.401 1 1102 .000
3 .280c .078 .076 13.117 .017 20.785 1 1101 .000
4 .303d .092 .088 13.028 .013 16.103 1 1100 .000
5 .318e .101 .097 12.968 .009 11.225 1 1099 .001
6 .330f .109 .104 12.918 .008 9.502 1 1098 .002
7 .344g .118 .113 12.853 .010 12.080 1 1097 .001
8 .353h .125 .118 12.811 .006 8.137 1 1096 .004
125 9 .368i .135 .128 12.742 .010 12.980 1 1095 .000
10 .377j .142 .134 12.695 .007 9.055 1 1094 .003
11 .386k .149 .140 12.650 .007 8.822 1 1093 .003
12 .385l .148 .140 12.650 -.001 1.020 1 1093 .313
13 .393m .154 .146 12.610 .006 7.955 1 1093 .005
14 .401n .161 .151 12.569 .006 8.133 1 1092 .004
15 .406o .165 .155 12.544 .004 5.352 1 1091 .021
16 .412p .169 .159 12.515 .005 6.124 1 1090 .013
17 .427q .182 .163 12.486 .013 1.426 12 1078 .148 Continued
Table 21. Hierarchal multiple regression results for the absolute model (streak logarithmic variables) 2007-2008
Table 21 continued
a. Predictors: (Constant), HASTA
b. Predictors: (Constant), HASTA, A3PMA
c. Predictors: (Constant), HASTA, A3PMA, ADREBAA
d. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA
e. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA
f. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, ATOA
g. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, ATOA, ATOAA
h. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, ATOA, ATOAA, HDREBAA
i. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA
j. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA
126 k. Predictors: (Constant), HASTA, A3PMA, ADREBAA, H2FGMAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA
l. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA
m. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA, H2FGAA
n. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA, H2FGAA, A3PAAA
o. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA, H2FGAA, A3PAAA,
A2FGAA
p. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA, H2FGAA, A3PAAA,
A2FGAA, A2FGMA
q. Predictors: (Constant), HASTA, A3PMA, ADREBAA, HTOA, ATOA, ATOAA, HDREBAA, HDREBA, ADREBA, HSTLA, H2FGAA, A3PAAA,
A2FGAA, A2FGMA, A2DP, H4DP, A4DP, HCONA, H1DPV1, H2DP, ACONA, A1DPV1, H3DP, A1DPV2, H1DPV2, A3DP
r. Dependent Variable: Score_Differential
Absolute model (streak logarithmic variables) 2008-2009. Entry of all 64 independent variables produced a statistically significant (F64, 1040 = 4.036; p < 0.001) equation, which explained 19.9% of the variance in score differential (i.e., away team final score subtracted by home team final score). Stepwise multiple regression results are presented in Table 22. With 16 independent variables, the equation explained 15.7% of the variance in score differential and was statistically significant (F16, 1088 = 12.684; p <
0.001). Away free throws attempted allowed (β = -0.209) was the equation‟s most important contributor. A 10% increase in the number of away free throws attempted allowed results in a -1.603 change in score differential. The negative regression coefficient represents an increase in the home team‟s favor. Intuitively this is sound, as the free throws attempted allowed statistic represents the average number of free throws attempted by the team‟s opponents. Following away free throws attempted allowed, home three pointers attempted (β = -0.163) and away personal fouls (β = 0.152) were the most significant contributors to the equation respectively.
The fist stage of the hierarchal multiple regression analysis entered 15 variables into the equation, via stepwise technique, and predicted 14.4% of the variance of score differential (as shown in Table 23). The addition of the present study‟s unique variables, in the second stage, increased the variance explained to a value of 17.2%. Away team depth contribution factor (β = 17.456; p = 0.001), away four days prior (β = -4.407; p =
0.019) and home three days prior (β = 2.282; p = 0.018) were the only unique variable to reach statistical significance at the 0.05 level.
127
Variable Coefficient Std. Error β t
Constant -71.078 37.881 - -1.876 Home assists allowed 14.820** 3.135 .142 4.728 Home defensive rebounds allowed 17.448** 4.272 .125 4.084 Away assists allowed -10.859** 3.029 -.107 -3.585 Home three pointers attempted -9.081** 1.806 -.163 -5.027 Away team depth contribution factor 19.206** 5.307 .103 3.619 Away free throws attempted allowed -16.826** 4.084 -.209 -4.120 Home three days prior 2.782** .826 .095 3.367 Home personal fouls allowed -16.749** 3.664 -.142 -4.571 Home free throws made allowed 8.341** 2.293 .114 3.638 Home two point field goals made -11.448** 4.460 -.087 -2.567 Away personal fouls 18.657** 6.271 .152 2.975 Away four days prior -3.711* 1.780 -.059 -2.084 Away defensive rebounds 9.692* 4.429 .065 2.188 Home turnovers 5.928* 2.566 .071 2.310 Home blocked shots -2.730* 1.308 -.059 -2.087 Away one day prior (variable one) -2.096* 1.057 -.057 -1.983
R 0.396 R² 0.157 Std. Error 12.111 * Significant at 0.05 level ** Significant at 0.01 level
Table 22. Stepwise multiple regression results for the absolute model (streak logarithmic variables) 2008-2009
128
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .174a .030 .029 12.903 .030 34.427 1 1103 .000
2 .223b .050 .048 12.780 .019 22.360 1 1102 .000
3 .259c .067 .065 12.666 .018 20.767 1 1101 .000
4 .285d .081 .078 12.578 .014 16.559 1 1100 .000
5 .304e .092 .088 12.506 .011 13.758 1 1099 .000
6 .314f .099 .094 12.468 .006 7.653 1 1098 .006
7 .326g .106 .101 12.421 .008 9.361 1 1097 .002
8 .338h .114 .108 12.371 .008 9.890 1 1096 .002
129 9 .345i .119 .112 12.343 .005 5.907 1 1095 .015
10 .355j .126 .118 12.300 .007 8.628 1 1094 .003
11 .360k .130 .121 12.277 .004 5.105 1 1093 .024
12 .365l .133 .124 12.259 .003 4.342 1 1092 .037
13 .371m .138 .127 12.234 .004 5.313 1 1091 .021
14 .375n .141 .130 12.216 .003 4.299 1 1090 .038
15 .380o .144 .133 12.198 .003 4.231 1 1089 .040
16 .415p .172 .151 12.067 .027 2.980 12 1077 .000 Continued
Table 23. Hierarchal multiple regression results for the absolute model (streak logarithmic variables) 2008-2009
Table 23 continued
a. Predictors: (Constant), HASTAA
b. Predictors: (Constant), HASTAA, HDREBAA
c. Predictors: (Constant), HASTAA, HDREBAA, AASTAA
d. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA
e. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA
f. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA
g. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA
h. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA
i. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA
j. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA
130 k. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, A3PAA
l. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, A3PAA, A2FGMA
m. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, A3PAA, A2FGMA, A2FGMAA
n. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, A3PAA, A2FGMA, A2FGMAA, HTOA
o. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, A3PAA, A2FGMA, A2FGMAA, HTOA, HBLKA
p. Predictors: (Constant), HASTAA, HDREBAA, AASTAA, H3PAA, AOREBAA, HPFAA, HFTAAA, H2FGMA, AFTAAA, APFA, A3PAA, A2FGMA, A2FGMAA, HTOA, HBLKA,
A1DPV2, H2DP, H1DPV1, HCONA, H4DP, ACONA, A1DPV1, A4DP, A2DP, H3DP, H1DPV2, A3DP
q. Dependent Variable: Score_Differential
Relative model 2007-2008. Basic descriptive statistics, including means, standard deviations, minimums and maximums, for the 2007-2008 relative model are presented in
Appendix C. As mentioned previously, relative performance measures are calculated as a series of ratios in which each team statistic is divided by its respective defensive counterpart (e.g., away assist ratio is found by dividing the away team‟s total assists by the away team‟s total assists allowed). Based on the aforementioned example, a ratio equal to a value of one would signify that over the games played, the away team has amassed an identical amount of assists as its numerous opponents. Alternatively, a ratio greater than one would represent an advantage in favor of the away team in terms of assists, whereas a ratio less than one would characterize a squad that has been outperformed by its opponents in the assist statistic. Therefore, each relative performance measure essentially puts into perspective the statistic in question by encapsulating a team‟s offensive and defensive prowess into a single figure.
Under the present conceptualization, in which team statistics throughout the season appear on both the away side and the home side of ledger, the mean ratios of all performance measures hover around a value of one, which intuitively is correct. A team and its corresponding statistics that are identified as an away team in one competition can be classified as the home team in the following game. Divergences from one are merely due to rounding as well as slight divergences in the number of home or away games each team has played over the 930 game observation set.
The simultaneous entry of all 32 independent variables resulted in a statistically significant equation (F32, 897 = 10.713; p < 0.001), which predicted 27.6% of the variance
131
in score differential (i.e., away team final score minus the home team final score). As shown in Table 24, the stepwise technique produced an equation with nine independent variables, which explained 25.9% of the variance in score differential, and was statically significant (F9, 920 = 36.136; p < 0.001). Home two point field goal ratio (β = -0.300) emerged as the most important relative contributor, as a one unit increase in home two point field goal ratio resulted in a 69.751 decrease in score differential. The negative coefficient of home two point field goal ratio coincides with movement in favor of the home team. While the large regression coefficient attached to home two point field goal ratio, at first glance seems absurd, it requires an additional explanation.
The two point field ratio variable is calculated as the team‟s field goal percentage
(i.e., two point field goals made divided by two point field goals attempted) divided by their opponents‟ two point field goal percentage. Therefore, a two point field goal ratio equal to one that undergoes a one unit increase, would represent a team with a two point field goal percentage double that of its opponents. At the NBA level, it is simply unrealistic to expect a team to be twice as successful as its opponents‟ in terms of two point field goal percentage. Consequently, for relative models the influence associated with regression coefficients will be discussed in terms of a 0.10 unit increase in the independent variable ratio. Based on the 2007-2008 relative stepwise equation, a 0.10 unit increase in home two point field goal ratio corresponds to -6.975 change in score differential.
132
Variable Coefficient Std. Error β t
Constant 32.298 17.293 - 1.868 Away two point field goal ratio 56.179** 8.867 .237 6.336 Home two point field goal ratio -69.751** 8.139 -.300 -8.570 Home steal ratio -9.300** 2.513 -.119 -3.701 Away steal ratio 6.914** 2.571 .088 2.689 Home offensive rebound ratio -13.988** 3.123 -.155 -4.480 Away three pointer ratio 18.931** 5.520 .119 3.430 Away offensive rebound ratio 10.057** 3.303 .107 3.045 Home free throw ratio -23.877* 9.893 -.079 -2.414 Away personal foul ratio -10.789* 4.983 -.070 -2.165
R 0.509 R² 0.259 Std. Error 11.807 * Significant at 0.05 level ** Significant at 0.01 level
Table 24. Stepwise multiple regression results for the relative model 2007-2008
133
The first stage of the hierarchal multiple regression analysis (see Table 25) entered nine relative performance variables into the equation and explained 25.9% of the variance in score differential. The second stage entered the 12 unique variables for a R2 change of 0.007. The resulting 21 variable statistically significant (F21, 908 = 15.640; p <
0.001) equation explained 26.6% of the variance in score differential. Away one day prior (variable one; β = -2.636; p = 0.043) was the only unique variables that reached the
0.05 level of statistical significance.
134
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .309a .096 .095 12.986 .096 98.153 1 928 .000
2 .429b .184 .182 12.343 .088 100.091 1 927 .000
3 .461c .212 .210 12.133 .028 33.366 1 926 .000
4 .476d .227 .224 12.025 .015 17.694 1 925 .000
5 .489e .239 .235 11.935 .012 15.077 1 924 .000
6 .494f .244 .240 11.902 .005 6.169 1 923 .013
7 .500g .250 .245 11.862 .006 7.203 1 922 .007
8 .505h .255 .249 11.830 .005 5.947 1 921 .015
135 9 .509i .259 .252 11.807 .004 4.688 1 920 .031
10 .515j .266 .249 11.830 .007 .697 12 908 .755 Continued
Table 25. Hierarchal multiple regression results for the relative model 2007-2008
Table 25 continued
a. Predictors: (Constant), A2FGR
b. Predictors: (Constant), A2FGR, H2FGR
c. Predictors: (Constant), A2FGR, H2FGR, HSTLR
d. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR
e. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR
f. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR
136 g. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR
h. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR, HFTR
i. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR, HFTR, APFR
j. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR, HFTR, APFR, H1DPV2, A1DPV2, H1DPV1, A2DP, H2DP, H4DP, A4DP, A1DPV1,
HCONR, H3DP, A3DP, ACONR
k. Dependent Variable: score_differential
Relative model 2008-2009. Descriptive statistics of the relative model for the
2008-2009 NBA campaign are provided within Appendix D. As was the case for the
2007-2008 relative model, league wide ratio performance measure means are essentially equal to a value of one for both away and home competitors. For example, the offensive rebound ratio for the away team going into the game in question was 1.007, whereas home offensive rebound ratio equaled 1.008. As stated previously, the underlying reason for the similarities in these values, as well as their proximity to one, is due to the fact that teams throughout the season appear on both sides of the ledger (i.e., as an away team and as a home team), contingent upon on the individual game in question. Divergences from one are merely the result of rounding and slight differences in the number of times a team appears under the away or home categorization over the observation set.
Simultaneous entry of all 32 independent variables yielded a statistically significant equation (F32, 897 = 8.922; p < 0.001) that predicted 24.1% of the variance in score differential (i.e., away team final score subtracted by home team final score).
Results of employing a stepwise entry technique are presented in Table 26. Nine independent variables entered the equation, which as a whole, was statistically significant
(F9, 920 = 29.827; p < 0.001). The equation explained 22.6% of the variance in score differential. Home defensive rebound ratio (β = -0.169) emerged as the most influential independent variable, as a 0.10 unit increase in home defensive rebound ratio resulted in a 3.167 decrease in score differential. The negative regression coefficient signifies that increasing the home team‟s defensive rebounds in relation to the home team‟s defensive
137
Variable Coefficient Std. Error β t
Constant -63.885** 14.264 - -4.479 Home defensive rebound ratio -31.674** 8.142 -.169 -3.890 Away three days prior 19.538** 4.632 .140 4.218 Home personal foul ratio 30.851** 6.550 .162 4.710 Away two point field goal ratio 33.922** 6.880 .164 4.931 Home assist ratio -14.029** 4.415 -.121 -3.178 Away offensive rebound ratio 9.625** 2.940 .097 3.273 Away four days prior -5.718** 1.877 -.089 -3.046 Home three days prior 2.452** .852 .084 2.877 Home turnover ratio 11.559** 4.153 .082 2.783
R 0.475 R² 0.226 Std. Error 11.412 * Significant at 0.05 level ** Significant at 0.01 level
Table 26. Stepwise multiple regression results for the relative model 2008-2009
138
rebounds allowed (i.e., increasing the home team‟s defensive rebound ratio), is advantageous for the home squad. Away two point field goal ratio (β = 0.164) and home personal foul ratio (β = 0.162) were the second and third most important contributors to the equation respectively.
The first stage of the hierarchal multiple regression analysis (as shown in Table
27) utilized a stepwise entry technique consisting solely of relative performance measures, which was followed by the simultaneous entry of the unique variables of this study. The second stage produced an equation with 19 independent variables that predicted an additional 2.2% of the variance in score differential, which raised the total explained to 23.3%. Of the 12 unique variables, only away four days prior (β = -5.678; p
= 0.005) and home three days prior (β = 2.122; p = 0.033) were statistically significant at the 0.05 level.
139
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .336a .113 .112 12.164 .113 117.998 1 928 .000
2 .393b .155 .153 11.881 .042 45.765 1 927 .000
3 .412c .169 .167 11.782 .015 16.566 1 926 .000
4 .429d .184 .181 11.684 .015 16.627 1 925 .000
5 .442e .195 .191 11.612 .011 12.522 1 924 .000
6 .452f .205 .199 11.548 .010 11.202 1 923 .001
7 .460g .211 .206 11.505 .007 8.017 1 922 .005
8 .483h .233 .217 11.418 .022 2.172 12 910 .011
140 a. Predictors: (Constant), HDREBR
b. Predictors: (Constant), HDREBR, A3PR
c. Predictors: (Constant), HDREBR, A3PR, HPFR
d. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR
e. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR
f. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR, AOREBR
g. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR, AOREBR, HTOR
h. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR, AOREBR, HTOR, A1DPV1, H2DP, H4DP, H1DPV1, A1DPV2, A4DP, A2DP,
ACONR, HCONR, H3DP, H1DPV2, A3DP
i. Dependent Variable: score_differential
Table 27. Hierarchal multiple regression results for the relative model 2008-2009
Relative model (streak variables) 2007-2008. Entry of all 32 independent variables produced a statistically significant equation (F32, 1072 = 7.059; p < 0.001) that explained 17.4% of the variance in score differential (i.e., away team final score minus the home team final score). Results from employing a stepwise entry technique are shown in Table 28. Seven independent variables entered the equation, which was statistically significant (F7, 1097 = 25.499; p < 0.001), and predicted 14.0% of the variance in score differential. Home two point field goal ratio (β = -0.252) emerged as the largest individual contributor. A 0.10 unit increase in home two point field goal ratio corresponded with a 3.202 decrease in score differential (i.e., a shift in score differential in favor of the home team). Following home two point field goal ratio, away two point field goal ratio (β = 0.207) and away steal ratio (β = 0.130) were the equation‟s most important independent variables respectively.
As displayed in Table 29, the first stage of the hierarchal multiple regression analysis entered seven relative streak performance measures into the equation through the stepwise approach, which explained 14.5% of the variance in score differential. The second stage of the statistical analysis subsequently entered the present study‟s 12 unique variables. The final equation, which consisted of 21 independent variables, explained
15.8% of the variance in score differential; thus the unique variables increased the variance explained by 1.2%. Away one day prior (variable one; β = -3.475, p= 0.005) was the lone unique variable to reach statistical significance at the 0.05 level.
141
Variable Coefficient Std. Error β t
Constant 9.569 8.038 - 1.190 Away two point field goal ratio 27.923** 3.796 .207 7.357 Home two point field goal ratio -32.018** 3.904 -.252 -8.201 Away steal ratio 5.779** 1.250 .130 4.622 Home turnover ratio 7.446** 2.033 .103 3.662 Home offensive rebound ratio -6.142** 1.500 -.127 -4.095 Home free throw ratio -15.737** 4.616 -.098 -3.410 Away one day prior (variable one) -2.515* 1.099 -.064 -2.289
R 0.374 R² 0.140 Std. Error 12.695 * Significant at 0.05 level ** Significant at 0.01 level
Table 28. Stepwise multiple regression results for the relative model (streak variables) 2007-2008
142
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .216a .047 .046 13.330 .047 53.817 1 1103 .000
2 .304b .092 .091 13.013 .046 55.384 1 1102 .000
3 .328c .108 .105 12.907 .015 19.114 1 1101 .000
4 .342d .117 .114 12.847 .009 11.332 1 1100 .001
5 .372e .138 .134 12.695 .022 27.487 1 1099 .000
6 .376f .142 .137 12.676 .003 4.272 1 1098 .039
7 .381g .145 .140 12.654 .004 4.877 1 1097 .027
8 .397h .158 .143 12.631 .012 1.332 12 1085 .194
143 a. Predictors: (Constant), A2FGR
b. Predictors: (Constant), A2FGR, HASTR
c. Predictors: (Constant), A2FGR, HASTR, ASTLR
d. Predictors: (Constant), A2FGR, HASTR, ASTLR, HDREBR
e. Predictors: (Constant), A2FGR, HASTR, ASTLR, HDREBR, HTOR
f. Predictors: (Constant), A2FGR, HASTR, ASTLR, HDREBR, HTOR, AASTR
g. Predictors: (Constant), A2FGR, HASTR, ASTLR, HDREBR, HTOR, AASTR, APFR
h. Predictors: (Constant), A2FGR, HASTR, ASTLR, HDREBR, HTOR, AASTR, APFR, A2DP, HCONR, H1DPV2, H1DPV1, ACONR, A4DP, A1DPV1,
H4DP, H2DP, A1DPV2, H3DP, A3DP
i. Dependent Variable: score_differential
Table 29. Hierarchal multiple regression results for the relative model (streak variables) 2007-2008
Relative model (streak variables) 2008-2009. The equation with all 32 independent variables was statistically significant (F32, 1072 = 5.750; p < 0.001) and predicted 14.7% of the variance in score differential (i.e., away team final score subtracted by the home team final score). Stepwise multiple regression (see Table 30), entered 10 variables into the statistically significant (F10, 1094 = 15.810; p < 0.001) equation, which explained 12.6% of the variance in score differential. Home defensive rebound ratio (β = -0.225) emerged as the most significant contributor, as a 0.10 unit increase in home defensive rebound ratio resulted in a 2.525 decrease in score differential. Coefficients corresponding to a decrease in score differential imply movement in favor of the home team. Following home rebound ratio, home turnover ratio (β = 0.143) and away two point field goal ratio (β = 0.143) tied as the equation‟s second most important contributors.
The first stage of the hierarchal multiple regression analysis utilized stepwise entry over a pool consisting solely of relative streak performance measures and produced an equation of six independent variables, which predicted 10.4% of the variance in score differential (see Table 31). The second stage simultaneously entered the 12 independent variables and increased the variance explained to 13.1%. Of the 12 unique independent variables, away team depth contribution factor ratio (β = 11.806; p = 0.010), away four days prior (β = -5.022; p = 0.009), away one day prior (variable 1; β = -2.747; p = 0.022), and home three days prior (β = 2.340; p = 0.016) reached statistical significance at the
0.05 level.
144
Variable Coefficient Std. Error β t
Constant -26.380** 7.124 - -3.703 Home defensive rebound ratio -25.255** 3.467 -.225 -7.284 Home turnover ratio 10.525** 2.231 .143 4.717 Away two point field goal ratio 18.758** 3.820 .143 4.910 Away offensive rebound ratio 6.150** 1.460 .122 4.213 Home three days prior 2.940** .835 .100 3.522 Away team depth contribution factor 4.537 .075 2.624 ratio 11.904** Away four days prior -3.989* 1.806 -.064 -2.208 Home blocked shot ratio -2.317** .847 -.080 -2.737 Away one day prior (variable one) -2.253** 1.068 -.061 -2.110 Away steal ratio 2.708* 1.340 .058 2.021
R 0.355 R² 0.126 Std. Error 12.298 * Significant at 0.05 level ** Significant at 0.01 level
Table 30. Stepwise multiple regression results for the relative model (streak variables) 2008-2009
145
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .202a .041 .040 12.833 .041 46.800 1 1103 .000
2 .250b .062 .061 12.694 .022 25.327 1 1102 .000
3 .280c .078 .076 12.591 .016 19.138 1 1101 .000
4 .294d .086 .083 12.541 .008 9.725 1 1100 .002
5 .304e .092 .088 12.505 .006 7.439 1 1099 .006
6 .314f .098 .093 12.470 .006 7.131 1 1098 .008
7 .323g .104 .098 12.435 .006 7.133 1 1097 .008
8 .322h .104 .099 12.432 .000 .445 1 1097 .505
146 9 .362i .131 .117 12.308 .027 2.852 12 1086 .001
Continued
Table 31. Hierarchal multiple regression results for the relative model (streak variables) 2008-2009
Table 31 continued
a. Predictors: (Constant), HDREBR
b. Predictors: (Constant), HDREBR, HTOR
c. Predictors: (Constant), HDREBR, HTOR, ADREBR
d. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR
e. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR, HBLKR
f. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR, HBLKR, AOREBR
g. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR, HBLKR, AOREBR, A2FGR
h. Predictors: (Constant), HDREBR, HTOR, ASTLR, HBLKR, AOREBR, A2FGR
147 i. Predictors: (Constant), HDREBR, HTOR, ASTLR, HBLKR, AOREBR, A2FGR, H2DP, A1DPV1, H1DPV1, H4DP, ACONR, HCONR, A1DPV2,
A4DP, A2DP, H3DP, H1DPV2, A3DP
j. Dependent Variable: score_differential
Relative model (logarithmic variables) 2007-2008. Entry of the 32 independent variables resulted in an equation that predicted 27.6% of the variance in score differential
(i.e., away team final score minus the home team final score) and was statistically significant (F32, 897 = 10.672; p < 0.001). Results of employing a stepwise entry technique are shown in Table 32. Nine independent variables entered the statistically significant (F9, 920 = 35.895; p < 0.001) equation, which explained 26.0% of the variance in score differential. As noted previously, one must exhibit caution when interpreting regression coefficients of logarithmic based models. Home two point field goal ratio (β =
-0.299) had the greatest influence on score differential. A 10% increase in home two point field goal ratio results in a 6.698 decrease in score differential. The negative regression coefficient signifies movement in favor of the home team. Following home two point field goal ratio, away two point field goal ratio (β = 0.230) and home offensive rebound ratio (β = -0.147) were the second and third most important independent variables respectively.
Hierarchal multiple regression results are shown in Table 33. Utilizing a stepwise technique, the first stage entered nine relative logarithmic measures of performance into the equation to predict 26.0% of the variance in score differential. Subsequently, the simultaneous entry of the current investigation‟s unique variables produced an R2 change of 0.007, thus the final equation explained 26.7% of the variance in score differential.
Away one day prior (variable one; β = -2.632; p = 0.043) was the lone unique variable to reach statistical significance at the 0.05 level.
148
Variable Coefficient Std. Error β t
Constant -3.438** .388 - -8.862 Away two point field goal ratio 55.282** 8.977 .230 6.159 Home two point field goal ratio -70.276** 8.238 -.299 -8.530 Home steal ratio -10.069** 2.497 -.129 -4.032 Away steal ratio 7.323** 2.592 .093 2.825 Home offensive rebound ratio -12.762** 2.989 -.147 -4.269 Away three days prior 19.494** 5.585 .121 3.490 Away offensive rebound ratio 9.354** 3.228 .102 2.898 Away personal foul ratio -12.124* 5.319 -.075 -2.279 Home free throw ratio -22.264* 9.811 -.074 -2.269
R 0.510 R² 0.260 Std. Error 11.798 * Significant at 0.05 level ** Significant at 0.01 level
Table 32. Stepwise multiple regression results for the relative model (logarithmic variables) 2007-2008
149
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .308a .095 .094 12.993 .095 96.972 1 928 .000
2 .429b .184 .182 12.341 .089 101.577 1 927 .000
3 .463c .215 .212 12.113 .031 36.260 1 926 .000
4 .479d .230 .226 12.005 .015 17.734 1 925 .000
5 .491e .241 .237 11.922 .011 13.989 1 924 .000
6 .496f .246 .241 11.886 .005 6.565 1 923 .011
7 .501g .251 .246 11.853 .005 6.232 1 922 .013
8 .506h .256 .249 11.825 .004 5.312 1 921 .021
150 9 .510i .260 .253 11.798 .004 5.150 1 920 .023
10 .516j .267 .250 11.823 .007 .683 12 908 .769 Continued
Table 33. Hierarchal multiple regression results for the relative model (logarithmic variables) 2007-2008
Table 33 continued
a. Predictors: (Constant), A2FGR
b. Predictors: (Constant), A2FGR, H2FGR
c. Predictors: (Constant), A2FGR, H2FGR, HSTLR
d. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR
e. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR
f. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR
g. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR
h. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR, APFR
i. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR, APFR, HFTR
j. Predictors: (Constant), A2FGR, H2FGR, HSTLR, ASTLR, HOREBR, A3PR, AOREBR, APFR, HFTR, H1DPV2, A1DPV2, H1DPV1, A2DP, H2DP,
151 H4DP, A4DP, A1DPV1, HCONR, H3DP, A3DP, ACONR
k. Dependent Variable: score_differential
Relative model (logarithmic variables) 2008-2009. Simultaneous entry of the 32 independent variables produced a statistically significant equation (F32, 897 = 8.994; p <
0.001), which explained 24.3% of the variance in score differential (i.e., away team final score subtracted by the home team final score). Displayed in Table 34 are the stepwise multiple regression results. The resulting nine variable equation was statistically significant (F9, 920 = 30.063; p < 0.001) and predicted 22.7% of the variance in score differential. Away two point field goal ratio (β = 0.165) emerged as the equation‟s most influential independent variable. A 10% increase in two point field goal ratio corresponds to a 3.295 increase in score differential (i.e., a shift in favor of the away team). The second and third most impactful independent variables were home personal foul ratio (β = 0.163) and home defensive rebound ratio (β = -0.162) respectively.
The first stage of the hierarchical multiple regression analysis (see Table 35) entered seven relative logarithmic performance measures into the equation, via stepwise entry, and predicted 21.3% of the variance in score differential. Subsequently, in the second stage, simultaneous entry of the present investigation‟s 12 unique variables produced an equation which explained an additional 2.2% of the variance in score differential, for a total of 23.5%. Of the 12 variables entered in second stage, only away four days prior (β = -5.639; p = 0.006) and home three days prior (β = 2.131; p = 0.032) reached statistical significance at the 0.05 level.
152
Variable Coefficient Std. Error β t
Constant -3.917** .444 - -8.826 Home defensive rebound ratio -30.777** 8.350 -.162 -3.686 Away three pointer ratio 19.102** 4.529 .139 4.218 Home personal foul ratio 30.910** 6.537 .163 4.729 Away two point field goal ratio 34.567** 6.968 .165 4.961 Home assist ratio -15.281** 4.526 -.131 -3.376 Away offensive rebound ratio 9.529** 3.005 .094 3.171 Away four days prior -5.669** 1.876 -.088 -3.021 Home three days prior 2.463** .851 .084 2.893 Home turnover ratio 11.884** 4.158 .084 2.858
R 0.477 R² 0.227 Std. Error 11.401 * Significant at 0.05 level ** Significant at 0.01 level
Table 34. Stepwise multiple regression results for the relative model (logarithmic variables) 2008-2009
153
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .338a .114 .113 12.155 .114 119.489 1 928 .000
2 .394b .155 .154 11.874 .041 45.470 1 927 .000
3 .412c .170 .167 11.779 .014 16.006 1 926 .000
4 .430d .185 .181 11.679 .015 16.965 1 925 .000
5 .444e .197 .192 11.599 .012 13.722 1 924 .000
6 .454f .206 .201 11.541 .009 10.431 1 923 .001
7 .461g .213 .207 11.494 .007 8.484 1 922 .004
8 .484h .235 .219 11.410 .022 2.140 12 910 .013
154 a. Predictors: (Constant), HDREBR
b. Predictors: (Constant), HDREBR, A3PR
c. Predictors: (Constant), HDREBR, A3PR, HPFR
d. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR
e. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR
f. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR, AOREBR
g. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR, AOREBR, HTOR
h. Predictors: (Constant), HDREBR, A3PR, HPFR, A2FGR, HASTR, AOREBR, HTOR, A1DPV1, H2DP, H4DP, H1DPV1, A1DPV2, A4DP, A2DP,
ACONR, H1DPV2, HCONR, H3DP, A3DP
i. Dependent Variable: score_differential
Table 35. Hierarchal multiple regression results for the relative model (logarithmic variables) 2008-2009
Relative model (streak logarithmic variables) 2007-2008. The simultaneous entry of all 32 independent variables produced an equation, which was statistically significant
(F32, 1072 = 7.056; p < 0.001), and explained 17.4% of the variance in score differential
(i.e., away team final score minus home team final score). Results of employing a stepwise entry technique are presented in Table 36. The 10 independent variables that entered the statistically significant (F10, 1094 = 19.790; p < 0.001) equation explained
15.3% of the variance in score differential. Home two point field goal ratio (β = -0.170) was the most important contributor to the equation. A 10% increase in home two point field goal ratio results in a 2.120 decrease in score differential. The negative regression coefficient represents an increase in the home team‟s favor. Following home two point field goal ratio, away two point field goal ratio (β = 0.156) and away steal ratio (β =
0.125) were the most significant contributors respectively.
With eight relative streak logarithmic performance variables, the first stage of the hierarchal multiple regression analysis (see Table 37) utilized a stepwise technique to produce an equation that predicted 14.5% of the variance of score differential. In the second stage, the addition of the present study‟s unique variables increased the variance explained by 1.2% for a total of 15.7%. Away one day prior (variable one; β = -3.376; p
= 0.007) was the sole unique variable to reach statistical significance at the 0.05 level.
155
Variable Coefficient Std. Error β t
Constant -3.166** .410 - -7.723 Home two point field goal ratio -22.237** 4.988 -.170 -4.458 Away two point field goal ratio 21.295** 4.842 .156 4.398 Away steal ratio 5.850** 1.311 .125 4.463 Home assist ratio -8.321** 2.465 -.122 -3.375 Home steal ratio -3.962** 1.309 -.085 -3.025 Home offensive rebound ratio -5.321** 1.515 -.110 -3.512 Home free throw ratio -14.078** 4.631 -.087 -3.040 Away one day prior (variable one) -2.766* 1.093 -.071 -2.531 Away assist ratio 5.362* 2.397 .079 2.237 Away personal foul ratio -6.331* 2.853 -.062 -2.219
R 0.391 R² 0.153 Std. Error 12.614 * Significant at 0.05 level ** Significant at 0.01 level
Table 36. Stepwise multiple regression results for the relative model (streak logarithmic variables) 2007-2008
156
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .217a .047 .046 13.326 .047 54.492 1 1103 .000
2 .301b .091 .089 13.024 .044 52.749 1 1102 .000
3 .324c .105 .102 12.927 .014 17.575 1 1101 .000
4 .344d .119 .115 12.833 .014 17.148 1 1100 .000
5 .358e .128 .124 12.772 .009 11.655 1 1099 .001
6 .368f .135 .130 12.724 .007 9.235 1 1098 .002
7 .377g .142 .136 12.681 .007 8.522 1 1097 .004
8 .381h .145 .139 12.664 .003 3.939 1 1096 .047
157 9 .396i .157 .141 12.643 .012 1.298 12 1084 .213
Continued
Table 37. Hierarchal multiple regression results for the relative model (streak logarithmic variables) 2007-2008
Table 37 continued
a. Predictors: (Constant), H2FGR
b. Predictors: (Constant), H2FGR, A2FGR
c. Predictors: (Constant), H2FGR, A2FGR, ASTLR
d. Predictors: (Constant), H2FGR, A2FGR, ASTLR, HASTR
e. Predictors: (Constant), H2FGR, A2FGR, ASTLR, HASTR, HSTLR
f. Predictors: (Constant), H2FGR, A2FGR, ASTLR, HASTR, HSTLR, HOREBR
g. Predictors: (Constant), H2FGR, A2FGR, ASTLR, HASTR, HSTLR, HOREBR, HFTR
h. Predictors: (Constant), H2FGR, A2FGR, ASTLR, HASTR, HSTLR, HOREBR, HFTR, HPFR
158 i. Predictors: (Constant), H2FGR, A2FGR, ASTLR, HASTR, HSTLR, HOREBR, HFTR, HPFR, A1DPV2, ACONR, H3DP, H1DPV1, HCONR, H4DP,
A2DP, A4DP, A1DPV1, H2DP, H1DPV2, A3DP
j. Dependent Variable: score_differential
Relative model (streak logarithmic variables) 2008-2009. Entry of all 32 independent variables resulted in a statistically significant equation (F32, 1072 = 5.721; p <
0.001), which predicted 14.6% of the variance in score differential (i.e., away team final score subtracted by home team final score). As shown in Table 38, stepwise entry produced a statistically significant equation (F10, 1094 = 15.603; p < 0.001), with ten independent variables, which explained 12.5% of the variance in score differential.
Home defensive rebound ratio (β = -0.218) emerged as the most important relative contributor to the equation, as a 10% increase resulted in a 2.362 decrease in score differential. The negative regression coefficient attached to home defensive rebound ratio is associated with movement in favor of the home team. Following home defensive rebound ratio, away two point field goal ratio (β = 0.145) and home turnover ratio (β =
0.135), were the most important respective independent variables.
As displayed in Table 39, the hierarchal multiple regression analysis initially entered eight relative streak logarithmic measures of performance into the equation through stepwise entry, which was followed by the simultaneous entry of measures of scheduling fatigue and team depth contribution variables. Entering the 12 unique variables increased the variance explained by 2.7% for a total 13.6%. Of these 12 unique variables, away team depth contribution factor ratio (β = 11.950; p = 0.009), away four days prior (β = -4.880; p = 0.011), away one day prior (variable one; β = -2.563; p =
0.033), and home three days prior (β = 2.351; p = 0.016) reached the 0.05 level of statistical significance.
159
Variable Coefficient Std. Error β t
Constant -3.690** .460 - -8.016 Home defensive rebound ratio -24.784** 3.519 -.218 -7.043 Home turnover ratio 10.135** 2.272 .135 4.461 Home three days prior 2.954** .836 .100 3.535 Away steals ratio 2.794* 1.410 .057 1.981 Home blocked shot ratio -2.631** .948 -.081 -2.775 Away offensive rebound ratio 6.563** 1.526 .125 4.301 Away two point field goal ratio 19.098** 3.840 .145 4.973 Away one day prior (variable one) -2.294* 1.068 -.062 -2.147 Away team depth contribution factor 4.534 .073 2.546 ratio 11.544* Away four days prior -3.833* 1.808 -.061 -2.120
R 0.353 R² 0.125 Std. Error 12.308 * Significant at 0.05 level ** Significant at 0.01 level
Table 38. Stepwise multiple regression results for the relative model (streak logarithmic variables) 2008-2009
160
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .200a .040 .039 12.837 .040 46.180 1 1103 .000
2 .246b .061 .059 12.705 .020 23.895 1 1102 .000
3 .277c .077 .074 12.600 .016 19.449 1 1101 .000
4 .291d .085 .081 12.552 .008 9.497 1 1100 .002
5 .302e .091 .087 12.515 .006 7.423 1 1099 .007
6 .311f .097 .092 12.480 .006 7.330 1 1098 .007
7 .321g .103 .097 12.443 .006 7.423 1 1097 .007
8 .321h .103 .098 12.439 .000 .329 1 1097 .566
161 9 .326i .106 .100 12.422 .003 4.027 1 1097 .045
10 .331j .110 .103 12.403 .004 4.466 1 1096 .035
11 .369k .136 .121 12.282 .027 2.808 12 1084 .001 Continued
Table 39. Hierarchal multiple regression results for the relative model (streak logarithmic variables) 2008-2009
Table 39 continued
a. Predictors: (Constant), HDREBR
b. Predictors: (Constant), HDREBR, HTOR
c. Predictors: (Constant), HDREBR, HTOR, ADREBR
d. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR
e. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR, HBLKR
f. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR, HBLKR, AOREBR
g. Predictors: (Constant), HDREBR, HTOR, ADREBR, ASTLR, HBLKR, AOREBR, A2FGR
h. Predictors: (Constant), HDREBR, HTOR, ASTLR, HBLKR, AOREBR, A2FGR
i. Predictors: (Constant), HDREBR, HTOR, ASTLR, HBLKR, AOREBR, A2FGR, HASTR
j. Predictors: (Constant), HDREBR, HTOR, ASTLR, HBLKR, AOREBR, A2FGR, HASTR, HPFR
162 k. Predictors: (Constant), HDREBR, HTOR, ASTLR, HBLKR, AOREBR, A2FGR, HASTR, HPFR, A2DP, H4DP, H1DPV1, ACONR, A4DP, H2DP,
HCONR, A1DPV1, A1DPV2, H3DP, H1DPV2, A3DP
l. Dependent Variable: score_differential
Possession model 2007-2008. Basic descriptive statistics, including means, standard deviations, minimums and maximums, for the 2007-2008 possession model are presented in Appendix E. Over the duration of the NBA season each team competes in an equal number of home and away contests. Thus under the present conceptualization, in which a team is classified as either the away or home squad depending on the game location, performance measure averages under the away / home categorization are essentially equal. For example, the mean away offensive rating equals 107.020, while the average home offensive rating was 107.071. Slight divergences in these values are the result of rounding as well as differences in the number of home or away games each team has played over the observation set.
Entry of all 18 independent variables yielded a statistically significant equation
(F18, 911 = 19.216; p < 0.001), which predicted 27.5% of the variance in score differential
(i.e., away team final score minus home team final score). Results of employing a stepwise entry technique are presented in Table 40. Four independent variables entered the equation, which was statistically significant (F4, 925 = 83.855; p < 0.001), and explained 26.6% of the variance in score differential. Home offensive rating (β = -0.299) emerged as the most influential independent variable, as a one unit increase in home offensive rating resulted in a 1.040 decrease in score differential. The negative regression coefficient represents movement in favor of the home team, which was to be expected given that an increase in efficiency in terms of the home team‟s offense would naturally result in a larger home team score differential. Away offensive rating (β =
163
Variable Coefficient Std. Error β t
Constant 46.048 28.621 1.609 Home offensive rating -1.040** .103 -.299 -10.061 Away offensive rating 0.794** .106 .224 7.477 Away defensive rating -0.901** .132 -.204 -6.806 Home defensive rating 0.684** .131 .155 5.223
R 0.516 R² 0.266 Std. Error 11.717 * Significant at 0.05 level ** Significant at 0.01 level
Table 40. Stepwise multiple regression results for the possession model 2007-2008
164
0.224) and away defensive rating (β = -0.204) were the second and third most important contributors to the equation respectively.
The first stage of the hierarchal multiple regression analysis (as shown in Table
41) utilized a stepwise technique to enter four independent variables, which was followed by the simultaneous entry of scheduling fatigue and team depth contribution factor measures. The inclusion of the current investigation‟s unique variables produced a 16 variable equation that predicted 27.4% of the variance in score differential, a 0.8% improvement over the first stage. Away one day prior (variable one; β = -2.849; p =
0.027) was sole unique variable to reach statistically significant at the 0.05 level.
165
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .348a .121 .120 12.801 .121 127.895 1 928 .000
2 .457b .209 .208 12.149 .088 103.354 1 927 .000
3 .494c .244 .242 11.882 .035 43.125 1 926 .000
4 .516d .266 .263 11.717 .022 27.285 1 925 .000
5 .523e .274 .261 11.730 .008 .831 12 913 .619
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, AOR
166 c. Predictors: (Constant), HOR, AOR, ADR
d. Predictors: (Constant), HOR, AOR, ADR, HDR
e. Predictors: (Constant), HOR, AOR, ADR, HDR, A1DPV1, H1DPV1, H2DP, H4DP, A4DP, HCONA, ACONA, A2DP, H1DPV2, A1DPV2, H3DP,
A3DP
f. Dependent Variable: score_differential
Table 41. Hierarchal multiple regression results for the possession model 2007-2008
Possession model 2008-2009. Descriptive statistics of the possession model for the 2008-2009 NBA campaign are provided within Appendix F. As was the case for the
2007-2008 possession model, league wide performance and pace measure means are essentially equivalent for the away and home competitors. For example, the average number of possessions for the away team going into the game in question was 91.761, whereas the home team‟s average possessions equaled 91.747. As stated previously, the similarities in these values stem from the fact that as teams progress through the season they appear on both sides of the ledger (i.e., as an away team and as a home team).
Entry of all 18 independent variables produced an equation that explained 22.5% of the variance in score differential (i.e., away team final score subtracted by home team final score) and was statistically significant (F18, 911 = 14.662; p < 0.001). Results from the stepwise multiple regression analysis are shown in Table 42. Six variables entered the equation, which was statistically significant (F6, 923 = 42.646; p < 0.001), and predicted 21.7% of the variance in score differential. Home defensive rating (β = 0.220) was the equation‟s largest individual contributor. A one unit increase in home defensive rating corresponded with a 0.835 increase in score differential (i.e., a shift in score differential in favor of the away team). It is imperative to remember that more efficient defensive teams possess lower defensive ratings. Therefore, a one unit increase in home defensive rating signifies that the home team has gotten worse on the defensive end.
Following home defensive rating, home offensive rating (β = -0.209) and away defensive rating (β = -0.191) were the most important independent variables respectively.
167
Variable Coefficient Std. Error β t
Constant 15.887 30.018 - .529 Home offensive rating -0.835** .125 -.209 -6.689 Away defensive rating -0.706** .116 -.191 -6.064 Home defensive rating 0.835** .119 .220 7.041 Away offensive rating 0.521** .124 .132 4.197 Home three days prior 2.817** .854 .096 3.299 Away four days prior -5.294** 1.877 -.082 -2.820
R 0.466 R² 0.217 Std. Error 11.458 * Significant at 0.05 level ** Significant at 0.01 level
Table 42. Stepwise multiple regression results for the possession model 2008-2009
168
As displayed in Table 43, the first stage of the hierarchal multiple regression analysis used a stepwise entry technique to produce an equation of four independent variables, which explained 20.2% of the variance in score differential. The second stage of the statistical analysis subsequently entered the present study‟s 12 unique variables and increased the variance explained to 22.4%. Of the unique variables, only away four days prior (β = -5.341; p = 0.009) and home three days prior (β = 2.344; p = 0.019) reached statistical significance at the 0.05 level.
169
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .299a .089 .088 12.325 .089 90.776 1 928 .000
2 .382b .146 .144 11.940 .057 61.927 1 927 .000
3 .434c .188 .186 11.648 .042 47.904 1 926 .000
4 .449d .202 .198 11.558 .013 15.578 1 925 .000
5 .474e .224 .211 11.467 .023 2.221 12 913 .009
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, ADR
170 c. Predictors: (Constant), HOR, ADR, HDR
d. Predictors: (Constant), HOR, ADR, HDR, AOR
e. Predictors: (Constant), HOR, ADR, HDR, AOR, H2DP, A1DPV2, H1DPV1, H4DP, HCONA, ACONA, A1DPV1, A4DP, A2DP, H3DP, H1DPV2,
A3DP
f. Dependent Variable: score_differential
Table 43. Hierarchal multiple regression results for the possession model 2008-2009
Possession model (streak variables) 2007-2008. Entry of all 18 independent variables produced a statistically significant equation (F18, 1086 = 42.646; p < 0.001) that predicted 16.8% of the variance in score differential (i.e., away team final score minus the home team final score). Results of the stepwise multiple regression analysis are presented in Table 44. The five variable equation was statistically significant (F5, 1099 =
41.916; p < 0.001) and explained 16.0% of the variance in score differential. Home offensive rating (β = -0.252), away offensive rating (β = 0.217) and home defensive rating (β = 0.166) were the most significant contributors to the equation respectively. A one unit increase in home offensive rating corresponded to a -0.547 change in score differential. As noted previously, coefficients corresponding to a decrease in score differential imply movement in favor of the home team.
The first stage of the hierarchal multiple regression analysis (see Table 45) entered four independent variables into the equation through a stepwise technique, which predicted 15.5% of the variance in score differential. In the second stage, the simultaneous entry of scheduling fatigue and team depth contribution variables increased the variance explained by 1.2%, for a total of 17.7%. Of the 12 unique independent variables, only away one day prior (variable 1; β = -3.529; p = 0.004), reached statistical significance at the 0.05 level.
171
Variable Coefficient Std. Error β t
Constant 6.876 12.984 - .530 Home offensive rating -0.547** .060 -.252 -9.094 Away offensive rating 0.451** .058 .217 7.819 Home defensive rating 0.383** .064 .166 5.999 Away defensive rating -0.378** .064 -.164 -5.900 Away one day prior (variable one) -2.756* 1.088 -.070 -2.534
R 0.400 R² 0.160 Std. Error 12.533 * Significant at 0.05 level ** Significant at 0.01 level
Table 44. Stepwise multiple regression results for the possession model (streak variables) 2007-2008
172
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .253a .064 .063 13.205 .064 75.711 1 1103 .000
2 .325b .105 .104 12.917 .041 50.774 1 1102 .000
3 .360c .130 .127 12.747 .024 30.648 1 1101 .000
4 .394d .155 .152 12.564 .026 33.306 1 1100 .000
5 .409e .167 .155 12.542 .012 1.320 12 1088 .201
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, AOR
173 c. Predictors: (Constant), HOR, AOR, HDR
d. Predictors: (Constant), HOR, AOR, HDR, ADR
e. Predictors: (Constant), HOR, AOR, HDR, ADR, A3DP, H4DP, H1DPV1, H2DP, HCONA, ACONA, A1DPV1, A4DP, A2DP, H1DPV2, H3DP,
A1DPV2
f. Dependent Variable: score_differential
Table 45. Hierarchal multiple regression results for the possession model (streak variables) 2007-2008
Possession model (streak variables) 2008-2009. Entry of all 18 independent variables resulted in an equation that predicted 13.2% of the variance in score differential
(i.e., away team final score subtracted by home team final score) and was statistically significant (F18, 1086 = 9.205; p < 0.001). In the stepwise multiple regression analysis (as shown in Table 46), seven independent variables entered the statistically significant equation (F7, 1097 = 21.507; p < 0.001) to explain 12.1% of the variance in score differential. Home offensive rating (β = -0.206) had the greatest influence on score differential. A one unit increase in home offensive rating resulted in a -0.459 change in score differential. The negative regression coefficient signifies movement in favor of the home team. Following home offensive rating, home defensive rating (β = 0.174) and away defensive rating (β = -0.144) were the most important independent variables respectively.
The first stage of the hierarchal multiple regression analysis (see Table 47), entered four independent variables via the stepwise technique, which predicted 9.9% of the variance in score differential. Subsequently, in the second stage, the simultaneous entry of the current investigation‟s unique variables produced an R2 change of 0.029.
Thus, the final equation explained 12.8% of the variance in score differential. Of the current study‟s unique independent variables, only away team depth contribution factor
(β = 2.346; p < 0.001), away four days prior (β = -5.089; p = 0.008), and home three days prior (β = 2.025; p = 0.038) reached statistical significance at the 0.05 level.
174
Variable Coefficient Std. Error β t
Constant 35.733* 14.862 - 2.404 Home offensive rating -0.459** .064 -.206 -7.181 Home defensive rating 0.390** .064 .174 6.095 Away defensive rating -0.316** .063 -.144 -5.004 Away team depth contribution factor 2.340** .655 .102 3.573 Home three days prior 2.403** .840 .082 2.862 Away four days prior -4.608** 1.770 -.074 -2.603 Away possessions -0.183* .093 -.057 -1.977
R 0.347 R² 0.121 Std. Error 12.320 * Significant at 0.05 level ** Significant at 0.01 level
Table 46. Stepwise multiple regression results for the possession model (streak variables) 2008-2009
175
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .200a .040 .039 12.838 .040 45.909 1 1103 .000
2 .263b .069 .067 12.648 .029 34.388 1 1102 .000
3 .308c .095 .092 12.478 .026 31.154 1 1101 .000
4 .315d .099 .096 12.454 .004 5.418 1 1100 .020
5 .357e .128 .115 12.321 .029 2.986 12 1088 .000
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, HDR
176 c. Predictors: (Constant), HOR, HDR, ADR
d. Predictors: (Constant), HOR, HDR, ADR, APOSA
e. Predictors: (Constant), HOR, HDR, ADR, APOSA, A2DP, H1DPV1, ACONA, H4DP, HCONA, A4DP, H2DP, A1DPV1, A1DPV2, H3DP, H1DPV2,
A3DP
f. Dependent Variable: score_differential
Table 47. Hierarchal multiple regression results for the possession model (streak variables) 2008-2009
Possession model (logarithmic variables) 2007-2008. Simultaneous entry of the
18 independent variables produced a statistically significant equation (F18, 911 = 19.174; p
< 0.001), which explained 27.5% of the variance in score differential (i.e., away team final score minus the home team final score). As displayed in Table 48, the stepwise technique entered four independent variables into the statistically significant (F4, 925 =
83.659; p < 0.001) equation, which predicted 22.7% of the variance in score differential.
Home offensive rating (β = -0.300) emerged as the equation‟s most influential independent variable. A 10% increase in home offensive rating corresponds to a 10.636 decrease in score differential. The negative regression coefficient signifies a shift in favor of the home team. The second and third most impactful independent variables were away offensive rating (β = 0.224) and away defensive rating (β = -0.204) respectively.
In the hierarchal multiple regression analysis (see Table 49), four possession based logarithmic performance measures entered the equation, which was followed by the unique variables of the present study. The simultaneous entry of scheduling fatigue measures and team depth contribution factors, explained an additional 0.8% of the variance in score differential, which raised the total to 27.4%. Of the 12 variables entered in second stage, only away one day prior (variable one; β = -2.835; p = 0.027) reached statistical significance at the 0.05 level.
177
Variable Coefficient Std. Error β t
Constant 233.673 132.573 1.763 Home offensive rating -111.594** 11.039 -.300 -10.109 Away offensive rating 85.102** 11.348 .224 7.499 Away defensive rating -95.416** 14.000 -.204 -6.815 Home defensive rating 71.141** 13.832 .153 5.143
R 0.515 R² 0.266 Std. Error 11.720 * Significant at 0.05 level ** Significant at 0.01 level
Table 48. Stepwise multiple regression results for the possession model (logarithmic variables) 2007-2008
178
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .348a .121 .120 12.802 .121 127.817 1 928 .000
2 .457b .209 .208 12.149 .088 103.400 1 927 .000
3 .495c .245 .242 11.880 .035 43.398 1 926 .000
4 .515d .266 .262 11.720 .021 26.453 1 925 .000
5 .523e .274 .261 11.733 .008 .830 12 913 .620
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, AOR
179 c. Predictors: (Constant), HOR, AOR, ADR
d. Predictors: (Constant), HOR, AOR, ADR, HDR
e. Predictors: (Constant), HOR, AOR, ADR, HDR, A1DPV1, H1DPV1, H2DP, H4DP, A4DP, HCONA, ACONA, A2DP, H1DPV2, A1DPV2, H3DP,
A3DP
f. Dependent Variable: score_differential
Table 49. Hierarchal multiple regression results for the possession model (logarithmic variables) 2007-2008
Possession model (logarithmic variables) 2008-2009. Entry of all 18 independent variables resulted in a statistically significant equation (F18, 911 = 14.609; p < 0.001), which predicted 22.4% of the variance in score differential (i.e., away team final score subtracted by home team final score). Results of employing the stepwise entry technique are presented in Table 50. The resulting six variable equation explained 21.7% of the variance in score differential and was statistically significant (F6, 923 = 42.542; p < 0.001).
Home defensive rating (β = 0.219) was as the most important contributor to the equation.
A 10% increase in home defensive rating results in an 8.538 change in score differential.
The positive regression coefficient represents an increase in the away team‟s favor.
Following home defensive rating, home offensive rating (β = -0.209) and away defensive rating (β = -0.191) were the most significant contributors to the equation respectively.
The hierarchal multiple regression analysis (as shown in Table 51) initially entered four possession based logarithmic performance measures through the stepwise technique to produce an equation that predicted 20.1% of the variance of score differential. In the second stage, the addition of the present study‟s unique variables increased the variance explained to a value of 22.4%. Away four days prior (β = -5.359; p = 0.008) and home three days prior (β = 2.333; p = 0.020) were the only unique variable to reach statistical significance at the 0.05 level.
180
Variable Coefficient Std. Error β t
Constant 92.333 139.983 - .660 Home offensive rating -90.137** 13.479 -.209 -6.687 Away defensive rating -76.055** 12.523 -.191 -6.073 Home defensive rating 89.585** 12.769 .219 7.016 Away offensive rating 56.014** 13.427 .131 4.172 Home three days prior 2.807** .854 .096 3.288 Away four days prior -5.307** 1.878 -.082 -2.826
R 0.465 R² 0.217 Std. Error 11.461 * Significant at 0.05 level ** Significant at 0.01 level
Table 50. Stepwise multiple regression results for the possession model (logarithmic variables) 2008-2009
181
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .298a .089 .088 12.326 .089 90.604 1 928 .000
2 .382b .146 .144 11.940 .057 61.961 1 927 .000
3 .433c .188 .185 11.650 .042 47.718 1 926 .000
4 .449d .201 .198 11.561 .013 15.427 1 925 .000
5 .473e .224 .210 11.471 .022 2.203 12 913 .010
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, ADR
1
82 c. Predictors: (Constant), HOR, ADR, HDR
d. Predictors: (Constant), HOR, ADR, HDR, AOR
e. Predictors: (Constant), HOR, ADR, HDR, AOR, H2DP, A1DPV2, H1DPV1, H4DP, HCONA, ACONA, A1DPV1, A4DP, A2DP, H3DP, H1DPV2,
A3DP
f. Dependent Variable: score_differential
Table 51. Hierarchal multiple regression results for the possession model (logarithmic variables) 2008-2009
Possession model (streak logarithmic variables) 2007-2008. Simultaneous entry of the 18 independent variables produced an equation, which explained 16.8% of the variance in score differential (i.e., away team final score minus home team final score), and was statistically significant (F18, 1086 = 12.153; p < 0.001). As shown in Table 52, the stepwise entry technique produced a statistically significant equation (F5, 1099 = 41.928; p
< 0.001) of five variables, which explained 16.0% of the variance in score differential.
Home offensive rating (β = -0.252) emerged as the most important relative contributor, as a 10% increase resulted in a 5.623 decrease in score differential (i.e., movement in favor of the home team). Following home offensive rating, away offensive rating (β = 0.216) and home defensive rating (β = 0.167), were the most important respective independent variables within the equation.
The first stage of the hierarchal multiple regression analysis (see Table 53) entered four possession-based streak logarithmic measures of performance via the stepwise technique to explain 15.5% of the variance in score differential. Entering the 12 unique variables simultaneously, in the second stage, explained an additional 1.2% of the variance, which increased the total to 16.7%. Away one day prior (variable one; β = -
3.565; p = 0.004) was the sole unique variable that reached the 0.05 level of statistical significance.
183
Variable Coefficient Std. Error β t
Constant 45.053 60.359 - .763 Home offensive rating -59.000** 6.486 -.252 -9.096 Away offensive rating 48.292** 6.202 .216 7.787 Home defensive rating 41.219** 6.855 .167 6.013 Away defensive rating -41.003** 6.891 -.165 -5.950 Away one day prior (variable one) -2.796** 1.088 -.071 -2.570
R 0.400 R² 0.160 Std. Error 12.533 * Significant at 0.05 level ** Significant at 0.01 level
Table 52. Stepwise multiple regression results for the possession model (streak logarithmic variables) 2007-2008
184
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .253a .064 .063 13.205 .064 75.729 1 1103 .000
2 .324b .105 .103 12.922 .041 49.912 1 1102 .000
3 .359c .129 .127 12.751 .024 30.814 1 1101 .000
4 .394d .155 .152 12.565 .026 33.833 1 1100 .000
5 .409e .167 .155 12.543 .012 1.318 12 1088 .202
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, AOR
185 c. Predictors: (Constant), HOR, AOR, HDR
d. Predictors: (Constant), HOR, AOR, HDR, ADR
e. Predictors: (Constant), HOR, AOR, HDR, ADR, A3DP, H4DP, H1DPV1, H2DP, HCONA, ACONA, A1DPV1, A4DP, A2DP, H1DPV2, H3DP,
A1DPV2
f. Dependent Variable: score_differential
Table 53. Hierarchal multiple regression results for the possession model (streak logarithmic variables) 2007-2008
Possession model (streak logarithmic variables) 2008-2009. Simultaneous entry of all 18 independent variables yielded a statistically significant equation (F18, 1086 =
9.243; p < 0.001), which predicted 13.3% of the variance in score differential (i.e., away team final score subtracted by the home team final score). Results of employing a stepwise entry technique are presented in Table 54. Seven independent variables entered the equation, which was statistically significant (F7, 1097 = 21.591; p < 0.001), and explained 12.1% of the variance in score differential. Home offensive rating (β = -0.207) was the most influential independent variable, as a 10% increase in home offensive rating resulted in a 4.814 decrease in score differential. The negative regression coefficient signifies movement in favor of the home team. Home defensive rating (β = 0.175) and away defensive rating (β = -0.144) were the second and third most important contributors to the equation respectively.
The first stage of the hierarchical multiple regression analysis (as shown in Table
55) entered four possession-based streak logarithmic pace / performance measures to explain 10.0% of the variance in score differential. In the second stage, simultaneous entry of the current investigation‟s 12 unique variables increased the total variance explained to 12.8%. Of the 12 unique variables, only away team depth contribution factor (β = 18.641; p = 0.001), away four days prior (β = -5.088; p = 0.008), and home three days prior (β = 2.030; p = 0.038), were statistically significant at the 0.05 level.
186
Variable Coefficient Std. Error β t
Constant 230.173** 63.994 - 3.597 Home offensive rating -50.505** 6.971 -.207 -7.245 Home defensive rating 42.641** 6.948 .175 6.137 Away defensive rating -34.148** 6.851 -.144 -4.984 Away team depth contribution factor 18.605** 5.328 .099 3.492 Home three days prior 2.397** .839 .082 2.856 Away four days prior -4.610** 1.769 -.074 -2.606 Away possessions -16.835* 8.542 -.057 -1.971
R 0.347 R² 0.121 Std. Error 12.320 * Significant at 0.05 level ** Significant at 0.01 level
Table 54. Stepwise multiple regression results for the possession model (streak logarithmic variables) 2008-2009
187
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .202a .041 .040 12.832 .041 46.942 1 1103 .000
2 .265b .070 .069 12.639 .030 35.048 1 1102 .000
3 .309c .096 .093 12.472 .025 30.738 1 1101 .000
4 .316d .100 .097 12.447 .004 5.306 1 1100 .021
5 .358e .128 .115 12.317 .028 2.942 12 1088 .000
a. Predictors: (Constant), HOR
b. Predictors: (Constant), HOR, HDR
188 c. Predictors: (Constant), HOR, HDR, ADR
d. Predictors: (Constant), HOR, HDR, ADR, APOSA
e. Predictors: (Constant), HOR, HDR, ADR, APOSA, A2DP, H1DPV1, ACONA, H4DP, HCONA, A4DP, H2DP, A1DPV1, A1DPV2, H3DP, H1DPV2,
A3DP
f. Dependent Variable: score_differential
Table 55. Hierarchal multiple regression results for the possession model (streak logarithmic variables) 2008-2009
Play model 2007-2008. Basic descriptive statistics, including means, standard deviations, minimums and maximums, for the 2007-2008 play model are presented in
Appendix G. With all NBA teams competing in an equal number of away and home competitions, team pace and performance measures averaged across the entire league, under the current conceptualization, are approximately equivalent. As the season progresses, each team‟s corresponding statistics are categorized as belonging to either the away team or home team depending on the game location. Consequently, a team‟s statistics appearing under the away heading on one night often will be categorized under the home classification the following night. Thus the averages of each individual statistical measure, regardless of its classification as an offensive or defensive statistic, regardless of location indicator, are essentially equal (e.g., AOPPP, ADPPP, HOPPP, and
HDPPP). Differences among these statistics are the result of rounding and slight divergences in the number of home or away games each team has played over the observation set.
Entry of all 20 independent variables produced a statistically significant equation
(F20, 909 = 17.472; p < 0.001), which explained 27.8% of the variance in score differential
(i.e., away team final score minus home team final score). Results from employing a stepwise entry technique are shown in Table 56. Consisting of six independent variables, the equation was statistically significant (F6, 923 = 54.888; p < 0.001) and predicted 26.3% of the variance in score differential. Home defensive plays (β = 0.418) emerged as the largest individual contributor. Each additional home defensive play corresponded with a
1.464 increase in score differential (i.e., a shift in score differential in favor of the away
189
Variable Coefficient Std. Error β t
Constant 67.792** 30.318 - 2.236 Home offensive points-per-play -122.133** 12.130 -.372 -10.069 Away offensive points-per-play 66.460** 10.426 .197 6.375 Away defensive points-per play -99.958** 15.122 -.205 -6.610 Home defensive points-per-play 87.914** 15.173 .181 5.794 Home defensive plays 1.464** .222 .418 6.590 Home offensive plays -1.536** .253 -.384 -6.076
R 0.513 R² 0.263 Std. Error 11.755 * Significant at 0.05 level ** Significant at 0.01 level
Table 56. Stepwise multiple regression results for the play model 2007-2008
190
team). Following home defensive plays, home offensive plays (β = -0.384) and home offensive points-per-play (β = -0.372) were the equation‟s most important independent variables.
As displayed in Table 57, hierarchal multiple regression analysis initially entered six play-based pace and performance variables in a stepwise manner, which was followed by the simultaneous entry of scheduling fatigue and team depth contribution measures.
The second entry stage of the statistical analysis increased the variance explained in score differential by 0.8%, raising the total explained to 27.0%. Of the 12 unique variables, only away one day prior (variable one; β = -2.569; p = 0.046) reached statistical significance at the 0.05 level.
191
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .297a .088 .087 13.039 .088 89.763 1 928 .000
2 .414b .171 .170 12.436 .083 93.102 1 927 .000
3 .454c .206 .203 12.182 .034 40.108 1 926 .000
4 .478d .228 .225 12.016 .022 26.730 1 925 .000
5 .483e .233 .229 11.981 .005 6.483 1 924 .011
6 .513f .263 .258 11.755 .029 36.914 1 923 .000
7 .520g .270 .256 11.771 .008 .782 12 911 .670
192 a. Predictors: (Constant), HOPPP
b. Predictors: (Constant), HOPPP, AOPPP
c. Predictors: (Constant), HOPPP, AOPPP, ADPPP
d. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP
e. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA
f. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA
g. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA, A2DP, H1DPV1, H1DPV2, ACONA, A4DP, A1DPV1, H2DP, H4DP,
HCONA, A1DPV2, H3DP, A3DP
h. Dependent Variable: score_differential
Table 57. Hierarchal multiple regression results for the play model 2007-2008
Play model 2008-2009. Descriptive statistics of the play model for the 2008-2009
NBA campaign are provided within Appendix H. Under the present conceptualization, as was the case for the 2007-2008 play model, league wide performance and pace measure means are essentially equivalent for both away and home competitors. For example, the average number of offensive plays for the away team going into the game in question was
104.341, whereas the home team‟s average offensive plays equaled 104.327. As stated previously, the similarities in these values is the result of teams appearing both under the away and home classification as the season progresses.
Simultaneous entry of all 20 independent variables resulted in a statistically significant equation (F20, 909 = 13.214; p < 0.001), which predicted 22.5% of the variance in score differential (i.e., away team final score subtracted by the home team final score).
In the stepwise multiple regression analysis (see Table 58), seven variables entered the equation to explain 20.8% of the variance in score differential. Home defensive points- per-play (β = 0.236) was as the equation‟s most significant contributor. The points-per- play statistic is measured on a relatively narrow scale and given that a discussion of one unit increases / decreases in points-per-play is not practical or realistic, the interpretation of regression coefficients will be conducted on the impact of a 0.10 unit increase / decrease. For the current equation, a 0.10 unit increase in home defensive points-per- play resulted in a 10.112 increase in score differential. Coefficients corresponding to an increase in score differential imply movement in favor of the away team. Following home defensive points-per-point, home offensive points-per-play (β = -0.177) and
193
Variable Coefficient Std. Error β t
Constant -25.948 29.786 - -.871 Home defensive points-per-play 101.119** 13.767 .236 7.345 Away defensive points-per play -72.547** 13.512 -.174 -5.369 Home offensive points-per-play -71.975** 13.107 -.177 -5.491 Away offensive points-per-play 46.712** 13.002 .118 3.593 Home three days prior 2.803** .859 .096 3.263 Away four days prior -5.523** 1.889 -.086 -2.923 Away team depth contribution factor 2.254* 1.062 .063 2.122
R 0.456 R² 0.208 Std. Error 11.531 * Significant at 0.05 level ** Significant at 0.01 level
Table 58. Stepwise multiple regression results for the play model 2008-2009
194
away defensive points-per-play (β = -0.174) were the second and third most important respective contributors.
The first stage of the hierarchal multiple regression analysis (see Table 59) used a stepwise entry technique, over a pool of play-based pace and performance measures, to produce an equation of four independent variables, which predicted 18.8% of the variance in score differential. The second stage simultaneously entered the present study‟s 12 unique independent variables to increase the total variance explained to
21.6%. Of the unique variables, away team depth contribution factor ratio (β = 2.267; p
= 0.034), away four days prior (β = -5.502; p = 0.007), and home three days prior (β =
2.265; p = 0.024), reached statistical significance at the 0.05 level.
195
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .309a .095 .095 12.282 .095 97.970 1 928 .000
2 .386b .149 .147 11.922 .053 57.913 1 927 .000
3 .419c .175 .173 11.741 .027 29.797 1 926 .000
4 .434d .188 .185 11.655 .013 14.774 1 925 .000
5 .464e .216 .202 11.531 .027 2.666 12 913 .002
a. Predictors: (Constant), HDPPP
b. Predictors: (Constant), HDPPP, ADPPP
196 c. Predictors: (Constant), HDPPP, ADPPP, HOPPP
d. Predictors: (Constant), HDPPP, ADPPP, HOPPP, AOPPP
e. Predictors: (Constant), HDPPP, ADPPP, HOPPP, AOPPP, H2DP, A1DPV2, H1DPV1, H4DP, HCONA, ACONA, A1DPV1, A4DP, A2DP, H3DP,
H1DPV2, A3DP
f. Dependent Variable: score_differential
Table 59. Hierarchal multiple regression results for the play model 2008-2009
Play model (streak variables) 2007-2008. Entry of the 20 independent variables resulted in an equation that predicted 17.2% of the variance in score differential (i.e., away team final score minus home team final score), and was statistically significant (F20,
1084 = 11.282; p < 0.001). Using a stepwise approach (see Table 60), seven independent variables entered the statistically significant equation (F7, 1097 = 29.565; p < 0.001), which explained 15.9% of the variance in score differential. Home offensive points-per-play (β
= -0.291) had the greatest influence on score differential. A 0.10 unit increase in home offensive points-per-play resulted in a -6.512 change in score differential (i.e., movement in favor of the home team). Following home offensive points-per-play, home defensive plays (β = 0.248) and away offensive points-per-play (β = 0.210) were the most important independent variables respectively.
The first stage of the hierarchal multiple regression analysis (as shown in Table
61) entered six play-based measures of pace / performance, via the stepwise technique, to predict 15.5% of the variance in score differential. Subsequently, the simultaneous entry of scheduling fatigue and team depth contribution factors, in the second stage, produced an R2 change of 0.012, which increased the variance explained to 16.7%. Of these 12 unique independent variables, only away one day prior (variable 1; β = -3.237; p = 0.009) reached statistical significance at the 0.05 level.
197
Variable Coefficient Std. Error β t
Constant 0.568 15.055 - .038 Home offensive points-per-play -65.122** 6.723 -.291 -9.686 Away offensive points-per-play 44.834** 5.922 .210 7.570 Away defensive points-per-play -34.513** 6.802 -.141 -5.074 Home defensive points-per-play 41.895** 7.236 .174 5.790 Home defensive plays 0.666** .125 .248 5.324 Home offensive plays -0.583** .138 -.200 -4.231 Away one day prior (variable one) -2.530* 1.089 -.065 -2.323
R 0.398 R² 0.159 Std. Error 12.555 * Significant at 0.05 level ** Significant at 0.01 level
Table 60. Stepwise multiple regression results for the play model (streak variables) 2007- 2008
198
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .248a .061 .060 13.226 .061 72.015 1 1103 .000
2 .320b .102 .100 12.941 .041 50.072 1 1102 .000
3 .345c .119 .117 12.824 .017 21.255 1 1101 .000
4 .365d .133 .130 12.727 .014 17.938 1 1100 .000
5 .376e .141 .138 12.672 .008 10.513 1 1099 .001
6 .393f .155 .150 12.580 .013 17.058 1 1098 .000
7 .408g .167 .153 12.560 .012 1.300 12 1086 .212
199 a. Predictors: (Constant), HOPPP
b. Predictors: (Constant), HOPPP, AOPPP
c. Predictors: (Constant), HOPPP, AOPPP, ADPPP
d. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP
e. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA
f. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA
g. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA, H3DP, A4DP, A2DP, ACONA, H1DPV1, H4DP, HCONA, A1DPV1,
H2DP, A1DPV2, H1DPV2, A3DP
h. Dependent Variable: score_differential
Table 61. Hierarchal multiple regression results for the play model (streak variables) 2007-2008
Play model (streak variables) 2008-2009. Simultaneous entry of all 20 variables produced a statistically significant equation (F20, 1084 = 8.252; p < 0.001) that explained
13.2% of the variance in score differential (i.e., away team final score subtracted by the home team final score). Displayed in Table 62 are the stepwise multiple regression analysis results. Nine independent variables entered the equation, which was statistically significant (F9, 1095 = 16.479; p < 0.001), and predicted 11.9% of the variance in score differential. Home offensive points-per-play (β = -0.211) was the equation‟s most influential independent variable. A 0.10 unit increase in the amount of home offensive points-per-play corresponds to a 4.997 decrease in score differential. The negative regression coefficient signifies a shift in score differential in favor of the home team. The second and third most impactful independent variables were home defensive plays (β =
0.205) and home defensive points-per-play (β = 0.185) respectively.
Hierarchal multiple regression analysis (as shown in Table 63) initially used a stepwise entry technique to create an equation of six play steak pace and performance measures, which predicted 9.8% of the variance in score differential. Subsequently in the second stage of the analysis, the simultaneous entry of the present investigation‟s 12 unique variables increased the total variance explained to 12.6%. Of the scheduling fatigue and team depth contribution measures, away team depth contribution factor (β =
2.371; p < 0.001), away four days prior (β = -5.004; p = 0.009) and home three days prior
(β = 2.016; p = 0.040) reached statistical significance at the 0.05 level.
200
Variable Coefficient Std. Error β t
Constant 26.398 17.031 - 1.550 Home offensive points-per-play -49.974** 7.235 -.211 -6.907 Home defensive points-per-play 44.965** 7.296 .185 6.163 Away defensive points-per-play -29.788** 6.835 -.125 -4.358 Away defensive plays -0.234** .074 -.090 -3.143 Away team depth contribution factor 2.355** .657 .102 3.582 Home three days prior 2.414** .845 .082 2.858 Away four days prior -4.550** 1.774 -.073 -2.565 Home defensive plays 0.536** .139 .205 3.853 Home offensive plays -0.457** .147 -.169 -3.114
R 0.345 R² 0.119 Std. Error 12.341 * Significant at 0.05 level ** Significant at 0.01 level
Table 62. Stepwise multiple regression results for the play model (streak variables) 2008- 2009
201
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .181a .033 .032 12.886 .033 37.403 1 1103 .000
2 .238b .057 .055 12.732 .024 27.752 1 1102 .000
3 .268c .072 .069 12.634 .015 18.251 1 1101 .000
4 .292d .085 .082 12.550 .013 15.709 1 1100 .000
5 .299e .089 .085 12.527 .004 5.030 1 1099 .025
6 .313f .098 .093 12.474 .009 10.507 1 1098 .001
7 .355g .126 .112 12.345 .028 2.922 12 1086 .001
202 a. Predictors: (Constant), HOPPP
b. Predictors: (Constant), HOPPP, HDPPP
c. Predictors: (Constant), HOPPP, HDPPP, ADPPP
d. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA
e. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA, HDPLA
f. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA, HDPLA, HOPLA
g. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA, HDPLA, HOPLA, A2DP, H1DPV1, H1DPV2, ACONA, HCONA, A4DP, A1DPV2, H2DP,
H4DP, A1DPV1, H3DP, A3DP
h. Dependent Variable: score_differential
Table 63. Hierarchal multiple regression results for the play model (streak variables) 2008-2009
Play model (logarithmic variables) 2007-2008. Simultaneous entry of all 20 independent variables resulted in a statistically significant equation (F20, 909 = 17.388; p <
0.001), which predicted 27.7% of the variance in score differential (i.e., away team final score minus home team final score). Results of employing the stepwise entry technique are presented in Table 64. The six variable equation explained 26.2% of the variance in score differential and was statistically significant (F6, 923 = 54.644; p < 0.001). Home defensive plays (β = 0.413) was the most important contributor to the equation. A 10% increase in home defensive plays results in a 14.702 change in score differential. It is important to recognize that a 10% increase in the amount of defensive plays signifies a substantial increase. The positive regression coefficient represents movement in the away team‟s favor. Following home defensive plays, home offensive plays (β = -0.381) and home offensive points-per-play (β = -0.371) were the most significant contributors to the equation respectively.
Utilizing a stepwise entry technique, the first stage of the hierarchal multiple regression analysis (see Table 65), produced an equation of six independent variables, which predicted 26.2% of the variance of score differential. In the second stage, the addition of the present study‟s 12 unique independent variables increased the variance explained to a value of 27.0%. Away one day prior (variable one; β = -2.555; p = 0.048) was the lone unique variable to reach statistical significance at the 0.05 level.
203
Variable Coefficient Std. Error β t
Constant 27.057 57.368 - .472 Home offensive points-per-play -114.784** 11.365 -.371 -10.100 Away offensive points-per-play 62.628** 9.823 .197 6.376 Away defensive points-per-play -93.590** 14.094 -.205 -6.640 Home defensive points-per-play 81.102** 14.148 .179 5.732 Home defensive plays 154.260** 23.464 .413 6.574 Home offensive plays -161.677** 26.639 -.381 -6.069
R 0.512 R² 0.262 Std. Error 11.761 * Significant at 0.05 level ** Significant at 0.01 level
Table 64. Stepwise multiple regression results for the play model (logarithmic variables) 2007-2008
204
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .297a .088 .087 13.041 .088 89.524 1 928 .000
2 .413b .171 .169 12.440 .083 92.791 1 927 .000
3 .454c .206 .203 12.183 .035 40.500 1 926 .000
4 .477d .227 .224 12.022 .022 25.901 1 925 .000
5 .482e .233 .229 11.987 .005 6.406 1 924 .012
6 .512f .262 .257 11.761 .029 36.834 1 923 .000
7 .519g .270 .255 11.779 .007 .776 12 911 .676
205 a. Predictors: (Constant), HOPPP
b. Predictors: (Constant), HOPPP, AOPPP
c. Predictors: (Constant), HOPPP, AOPPP, ADPPP
d. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP
e. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA
f. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA
g. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA, A2DP, H1DPV1, H1DPV2, ACONA, A4DP, A1DPV1, H2DP, H4DP,
HCONA, A1DPV2, H3DP, A3DP
h. Dependent Variable: score_differential
Table 65. Hierarchal multiple regression results for the play model (logarithmic variables) 2007-2008
Play model (logarithmic variables) 2008-2009. Entry of all 20 independent variables resulted in a statistically significant equation (F20, 909 = 13.178; p < 0.001), which predicted 22.5% of the variance in score differential (i.e., away team final score subtracted by home team final score). As shown in Table 66, stepwise entry produced an equation of seven variables, which explained 20.7% of the variance in score differential, and was statistically significant (F7, 922 = 34.425; p < 0.001). Home defensive points-per- play (β = 0.236) emerged as the most important relative contributor, as a 10% increase resulted in a 9.100 increase in score differential. The positive regression coefficient attached to home defensive points-per-play is associated with movement of score differential in favor of the away team. Following home defensive points-per-play, home offensive points-per-play (β = -0.177) and away defensive points-per-play (β = -0.175), were the most important respective independent variables within the equation.
As displayed in Table 67, the first stage of the hierarchal multiple regression analysis entered four play based logarithmic measures of performance into the equation, which explained 18.8% of the variance in score differential. Entering the scheduling fatigue measures and team depth contribution factors in the second stage resulted in a R2 change of 0.027, raising the total variance explained to 21.5%. Of the 12 unique variables, away team depth contribution factor (β = 18.113; p = 0.010), away four days prior (β = -5.506; p = 0.007), and home three days prior (β = 2.263; p = 0.024) reached the 0.05 level of statistical significance.
206
Variable Coefficient Std. Error β t
Constant -42.026* 18.664 - -2.252 Home defensive points-per-play 95.478** 13.007 .236 7.340 Away defensive points-per-play -68.697** 12.742 -.175 -5.392 Home offensive points-per-play -67.925** 12.370 -.177 -5.491 Away offensive points-per-play 43.941** 12.271 .118 3.581 Home three days prior 2.798** .860 .096 3.256 Away four days prior -5.534** 1.890 -.086 -2.928 Away team depth contribution factor 18.000* 8.733 .061 2.061
R 0.455 R² 0.207 Std. Error 11.536 * Significant at 0.05 level ** Significant at 0.01 level
Table 66. Stepwise multiple regression results for the play model (logarithmic variables) 2008-2009
207
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .308a .095 .094 12.285 .095 97.485 1 928 .000
2 .385b .148 .147 11.923 .053 58.123 1 927 .000
3 .418c .175 .172 11.743 .026 29.697 1 926 .000
4 .433d .188 .184 11.658 .013 14.580 1 925 .000
5 .464e .215 .201 11.535 .027 2.649 12 913 .002
a. Predictors: (Constant), HDPPP
b. Predictors: (Constant), HDPPP, ADPPP
208 c. Predictors: (Constant), HDPPP, ADPPP, HOPPP
d. Predictors: (Constant), HDPPP, ADPPP, HOPPP, AOPPP
e. Predictors: (Constant), HDPPP, ADPPP, HOPPP, AOPPP, H2DP, A1DPV2, H1DPV1, H4DP, HCONA, ACONA, A1DPV1, A4DP, A2DP, H3DP,
H1DPV2, A3DP
f. Dependent Variable: score_differential
Table 67. Hierarchal multiple regression results for the play model (logarithmic variables) 2008-2009
Play model (streak logarithmic variables) 2007-2008. Simultaneous entry of all
20 independent variables yielded a statistically significant equation (F20, 1084 = 11.261; p <
0.001), which predicted 17.2% of the variance in score differential (i.e., away team final score minus the home team final score). Stepwise multiple regression analysis (see Table
68), entered seven variables into the equation, which explained 15.9% of the variance in score differential, and was statistically significant (F7, 1097 = 29.535; p < 0.001). Home offensive points-per-play (β = -0.290) was the most influential independent variable, as a
10% increase in home offensive points-per-play resulted in a 5.864 decrease in score differential (i.e. a shift in favor of the home team). Home defensive plays (β = 0.248) and away offensive points-per-play (β = 0.209) were the second and third most important contributors to the equation respectively.
The first stage of the hierarchal multiple regression analysis (as shown in Table
69) employed a stepwise entry technique, over a pool of play-based streak logarithmic pace and performance measures, and entered six independent variables into the equation to explain 15.4% of the variance in score differential. Simultaneous entry of the current investigation‟s 12 unique variables, in the second stage, increased the variance explained by 1.2%, raising the total to 16.6%. Of the 12 unique variables, only away one day prior
(variable one; β = -3.293; p = 0.008) was statistically significant at the 0.05 level.
209
Variable Coefficient Std. Error β t
Constant -49.158 41.553 - -1.183 Home offensive points-per-play -61.524** 6.370 -.290 -9.659 Away offensive points-per-play 42.298** 5.624 .209 7.521 Away defensive points-per-play -33.075** 6.456 -.143 -5.123 Home defensive points-per-play 39.390** 6.845 .173 5.755 Home defensive plays 70.447** 13.149 .248 5.357 Home offensive plays -60.672** 14.428 -.198 -4.205 Away one day prior (variable one) -2.587* 1.090 -.066 -2.374
R 0.398 R² 0.159 Std. Error 12.556 * Significant at 0.05 level ** Significant at 0.01 level
Table 68. Stepwise multiple regression results for the play model (streak logarithmic variables) 2007-2008
210
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .247a .061 .060 13.229 .061 71.587 1 1103 .000
2 .318b .101 .100 12.948 .040 49.299 1 1102 .000
3 .344c .119 .116 12.827 .017 21.850 1 1101 .000
4 .364d .133 .130 12.731 .014 17.801 1 1100 .000
5 .376e .141 .137 12.673 .009 11.054 1 1099 .001
6 .393f .154 .150 12.583 .013 16.791 1 1098 .000
7 .408g .166 .152 12.563 .012 1.291 12 1086 .218
211 a. Predictors: (Constant), HOPPP
b. Predictors: (Constant), HOPPP, AOPPP
c. Predictors: (Constant), HOPPP, AOPPP, ADPPP
d. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP
e. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA
f. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA
g. Predictors: (Constant), HOPPP, AOPPP, ADPPP, HDPPP, HDPLA, HOPLA, H3DP, A4DP, A2DP, ACONA, H1DPV1, H4DP, HCONA, A1DPV1,
H2DP, A1DPV2, H1DPV2, A3DP
h. Dependent Variable: score_differential
Table 69. Hierarchal multiple regression results for the play model (streak logarithmic variables) 2007-2008
Play model (streak logarithmic variables) 2008-2009. Entry of all 20 independent variables produced a statistically significant equation (F20, 1084 = 8.321; p <
0.001), which explained 13.3% of the variance in score differential (i.e., away team final score subtracted by home team final score). Results of the stepwise multiple regression analysis are shown in Table 70. The nine variable equation was statistically significant
(F9, 1095 = 16.597; p < 0.001) and predicted 12.0% of the variance in score differential.
Home offensive points-per-play (β = -0.214) was the equation‟s largest individual contributor. A 10 % increase in home offensive points-per-play corresponded with a
4.621 decrease in score differential (i.e., a shift in favor of the home team). Home defensive plays (β = 0.207) and home defensive points-per-play (β = 0.186) were the second and third most important variables respectively.
In the hierarchal multiple regression analysis (see Table 71), six play-based model streak logarithmic pace and performance measures entered the equation in a stepwise manner, which was followed by the simultaneous entry of scheduling fatigue and team depth contribution factors. In the second stage, these 12 unique variables increased the variance explained by 2.8%, raising the total to 12.7%. Of the unique independent variables, away team depth contribution factor (β = 18.821; p < 0.001), away four days prior (β = -4.999; p = 0.009), and home three days prior (β = 2.011; p = 0.041), reached statistical significance at the 0.05 level.
212
Variable Coefficient Std. Error β t
Constant 30.431 56.751 - .536 Home offensive points-per-play -48.484** 6.929 -.214 -6.997 Home defensive points-per-play 43.298** 6.967 .186 6.215 Away defensive points-per-play -28.347** 6.537 -.124 -4.337 Away defensive plays -24.502** 7.826 -.090 -3.131 Away team depth contribution factor 18.705** 5.348 .100 3.498 Home three days prior 2.402** .844 .082 2.846 Away four days prior -4.547** 1.773 -.073 -2.565 Home defensive plays 56.761** 14.593 .207 3.889 Home offensive plays -48.407** 15.349 -.171 -3.154
R 0.346 R² 0.120 Std. Error 12.336 * Significant at 0.05 level ** Significant at 0.01 level
Table 70. Stepwise multiple regression results for the play model (streak logarithmic variables) 2008-2009
213
Change Statistics
Adjusted R Std. Error of the R Square
Model R R Square Square Estimate Change F Change df1 df2 Sig. F Change
1 .184a .034 .033 12.878 .034 38.724 1 1103 .000
2 .241b .058 .056 12.723 .024 28.159 1 1102 .000
3 .270c .073 .071 12.626 .015 17.930 1 1101 .000
4 .293d .086 .083 12.544 .013 15.543 1 1100 .000
5 .300e .090 .086 12.520 .004 5.057 1 1099 .025
6 .315f .099 .094 12.464 .009 10.895 1 1098 .001
7 .356g .127 .112 12.339 .028 2.870 12 1086 .001
214 a. Predictors: (Constant), HOPPP
b. Predictors: (Constant), HOPPP, HDPPP
c. Predictors: (Constant), HOPPP, HDPPP, ADPPP
d. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA
e. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA, HDPLA
f. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA, HDPLA, HOPLA
g. Predictors: (Constant), HOPPP, HDPPP, ADPPP, ADPLA, HDPLA, HOPLA, A2DP, H1DPV1, H1DPV2, ACONA, HCONA, A4DP, A1DPV2, H2DP,
H4DP, A1DPV1, H3DP, A3DP
h. Dependent Variable: score_differential
Table 71. Hierarchal multiple regression results for the play model (streak logarithmic variables) 2008-2009
Logistic Regression
Goodness-of-fit indices of the null model for the 2007-2008 NBA season are presented in Appendix I. The null model consists solely of a constant and is constructed to maximize the hit ratio by predicting that the dependent variable always belongs to the larger group. Over the 930 observations for the 2007-2008 NBA season, the away team was victorious 364 times (i.e., 39.1%), while the home team won 566 times (i.e., 60.9%).
Therefore, by predicting a home team victory in all cases, the classification accuracy baseline is established for 2007-2008 non-streak models at 60.9%. To further facilitate model comparisons, a secondary measure of overall model fit was established. More specifically, the null model contained a -2 Log Likelihood (-2LL) valued at 1245.027.
As previously noted, lower -2LL values are associated with better model goodness-of-fit.
Measures of goodness-of-fit for the 2008-2009 NBA season null model are provided within Appendix J. Over the final 930 games of the 2008-2009 NBA season, the away team totaled 352 victories (i.e., 37.8%), with the home squad winning 579 contests (i.e., 62.2%). Hence, the construction of a model, which predicted that the home team would be victorious over the entire observation set, would result in a success rate of
62.2%. In regards to an alternative measure of overall fit, the null model produced a -
2LL valued at 1233.780.
Displayed in Appendix K are the measures of goodness-of-fit for 2007-2008 streak models. During the final 1105 games of the 2007-2008 NBA season, the away team won a total of 437 times (i.e., 39.5%) and the home team emerged victorious 668 times (i.e., 60.5%). To further facilitate model comparisons for streak models, a
215
secondary measure of overall model fit was established. The -2LL for the null model over the last 1105 games of the 2007-2008 campaign was valued at 1483.207.
Goodness-of-fit indices of the null streak model for the 2008-2009 NBA season are shown in Appendix L. Based on the classification matrix, the hit ratio for the null model equaled 61.1%. In other words, for the final 1105 games of the 2008-2009 NBA season, the away team victories totaled 430 (i.e., 38.9%), while the home team compiled
675 wins (i.e., 61.1%). As a secondary goodness-of-fit benchmark, the -2LL statistic for the 2008-2009 null streak model was established at 1477.080.
Absolute model 2007-2008. The simultaneous entry of all 64 independent variables produced an equation that accurately predicted 73.4% of the final 930 NBA games for the 2007-2008 NBA season. The equation correctly categorized 59.6% of away team victories (i.e., 217 games) and 82.3% of home team wins (i.e., 466 games).
As a whole, the equation was statistically significant (Chi-Squared = 250.699; df = 64; p
< 0.001) and produced a -2LL valued at 994.327.
Results of the stepwise logistic regression analysis are presented in Table 72. The
16 variable equation possessed a -2LL value of 1033.616 and was statistically significant
(Chi-Squared = 211.411; df = 16; p < 0.001). The equation accurately predicted 70.9% of the observed games, with 53.6% of the away team victories and 82.0% of the home team victories correctly classified. Away steals (B = -0.406) was the most important contributor to the equation. As noted previously, interpretation of the relationship between an independent variable and the dependent variables for logistic regression
216
Variable B Std. Error Wald Exp(B)
Constant -5.548 3.727 2.216 .004 Away three pointers made -0.304* .063 23.187 .738 Away defensive rebounds -0.140* .062 4.992 .870 Away steals -0.406* .085 22.672 .666 Away personal fouls 0.190* .055 11.956 1.209 Away two point field goals made .091 .051 3.168 1.095 allowed Away three pointers made allowed 0.302* .108 7.866 1.353 Away defensive rebounds allowed 0.178* .065 7.520 1.194 Home two point field goals 0.110* .045 5.944 1.116 attempted Home three pointers attempted 0.169* .047 13.147 1.184 Home free throws made 0.166* .047 12.431 1.180 Home assists .077 .047 2.693 1.080 Home steals 0.201* .100 4.043 1.223 Home turnovers -0.272* .075 13.292 .762 Home defensive rebounds allowed -0.260* .071 13.224 .771 Home assists allowed -0.157* .072 4.716 .855 Home one day prior (variable two) .380 .224 2.891 1.463
Hit Ratio 70.9% -2 Log Likelihood 1033.616 * Significant at 0.05 level
Table 72. Stepwise logistic regression results for the absolute model 2007-2008
217
analysis is more easily understood through the calculation of percentage change in odds, which utilizes exponential coefficients. Holding all other independent variables constant, a one unit increase in the number of away steals decreases the odds of a home team victory by 33.4%. Following away steals, home one day prior (variable two; B = 0.380), and away three pointers made (B = -0.304), were the most significant contributors respectively.
Hierarchal logistic regression was utilized to assess the influence of scheduling fatigue variables and team depth contribution measures (as shown in Table 73). With a total of 16 independent variables, the first stage used a stepwise technique to produce an equation with a -2LL of 1036.566, which accurately predicted 70.1% of the 930 games.
The addition of the present study‟s unique variables, in the second stage, resulted in a model containing a -2LL valued at 1017.888 and at hit ratio to 72.5%. Of the 12 unique variables, away one day prior (variable one; B = 0.614; p = 0.017), away one day prior
(variable two; B = 0.570; p = 0.013), home three days prior (B = -0.406; p = 0.044), and home two days prior (B = 0.382; p = 0.048) were deemed statistically significant at the
0.05 level.
218
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 189 175 51.9 Home Team 103 463 81.8 Overall Percentage 70.1
-2 Log Likelihood 1036.566 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 205 159 56.3 Home Team 97 469 82.9 Overall Percentage 72.5
-2 Log Likelihood 1017.888 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of absolute performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 73. Hierarchal logistic regression goodness-of-fit measures for the absolute model 2007-2008
219
Absolute model 2008-2009. Simultaneous entry of all 64 independent variables resulted in a statistically significant equation (Chi-Squared = 250.798; df = 64; p < 0.001) with a -2LL value of 982.982 and a hit ratio of 72.9%. The equation correctly classified
55.1% of away team wins and 83.7% of home team victories over the 930 observations.
As shown in Table 74, the stepwise logistic regression produced a 17 variable equation with a -2LL of 1015.713 and a hit ratio of 72.6%. The statistically significant equation (Chi-Squared = 218.067; df = 17; p < 0.001) accurately predicted 196 of the away team wins (55.7%) and 479 of the home team victories (82.9%). Away four days prior (B = 0.750) was the most impactful variable within the equation. It is imperative to note that the interpretation of dichotomous independent variables within logistic regression analysis differs from continuous variables, for dichotomous variables represent the presence or absence of a characteristic and do not vary over a range of values (Hair et al., 2006). Percentage change in odds for dichotomous variables is calculated in an identical fashion as percentage change in odds for metric independent variables (i.e., subtracting 1.0 from the exponentiated coefficient and multiplying the result by 100), but differ in their interpretation. For the current equation, when the away team is entering their fourth competition in five days, the odds of a home team victory are 111.7% greater.
The higher likelihood of a home victory associated with the away team playing four games in a series of five days serves as a testament to the detrimental impact of fatigue.
As the interpretation of independent variable impact is centered on odds, percentage changes larger than 100 are more than acceptable (Hair et al.).
220
Variable B Std. Error Wald Exp(B)
Constant 1.748 3.794 .212 5.744 Away free throws attempted -0.090* .036 6.332 .914 Away defensive rebounds -0.222* .065 11.613 .801 Away two point field goals made 0.113* .049 5.385 1.119 allowed Away free throws attempted 0.099* .034 8.681 1.104 allowed Away assists allowed .097 .058 2.851 1.102 Away blocked shots allowed .189 .122 2.391 1.208 Away four days prior .750 .428 3.073 2.117 Home two point field goals 0.152* .046 11.085 1.164 attempted Home three pointers made 0.486* .115 17.754 1.627 Home steals 0.386* .127 9.180 1.471 Home personal fouls -0.310* .072 18.775 .733 Home two point field goals made -.103 .064 2.576 .902 allowed Home defensive rebounds allowed -0.181* .075 5.796 .835 Home assists allowed -0.280* .059 22.823 .756 Home personal fouls allowed 0.209* .086 5.967 1.232 Home three days prior -0.690* .189 13.300 .501 Home two days prior .334 .189 3.145 1.397
Hit Ratio 72.6% -2 Log Likelihood 1015.713 * Significant at 0.05 level
Table 74. Stepwise logistic regression results for the absolute model 2008-2009
221
For example, entering a contest, if the probability of a home team victory was
60%, the probability of an away team victory would be 40%. In converting probability to odds, which is accomplished by taking the probability that the event occurs and dividing it by one minus the probability that the event occurs, we find that the odds of a home victory are 1.5 (i.e., 0.6 / [1 – 0.6]). When the away team is entering their fourth game in five days, the odds of a home team victory are 111.7% greater, which in the present example would mean that the odds of a home team victory would be increased to 3.1755.
We can easily reconvert odds into probability by taking the odds of the event occurring and dividing it by one plus the odds of the event occurring. Therefore, in the situation when the home team typically has a 60% chance of emerging victorious and the away team is playing their fourth game in five days, the probability of a home team victory increases to roughly 76% (i.e., 3.1755 / [1 + 3.1755]). Following away four days prior, home three days prior (B = -0.690) and home three pointers made (B = 0.486), were the most impactful independent variables within the equation.
As displayed in Table 75, hierarchal logistic regression was utilized to ascertain the contribution of scheduling fatigue and team depth contribution variables. The 14 absolute measures of performance that entered the equation, via a stepwise technique in first stage, contained a -2LL of 1031.270 and a hit ratio of 70.8%. In the second stage, entering the 12 unique variables resulted in an equation with a -2LL of 1012.175 and hit ratio of 73.3%. Home three days prior (B = -0.669; p = 0.001) was the only unique variable that reached the 0.05 level of statistical significance.
222
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 188 164 53.4 Home Team 108 470 81.3 Overall Percentage 70.8
-2 Log Likelihood 1031.27 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 196 156 55.7 Home Team 92 486 84.1 Overall Percentage 73.3
-2 Log Likelihood 1012.175 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of absolute performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 75. Hierarchal logistic regression goodness-of-fit measures for the absolute model 2008-2009
223
Absolute model (streak variables) 2007-2008. Simultaneous entry of all 64 independent variables yielded an equation with a -2LL valued at 1282.659, which correctly predicted 69.9% of the 1105 game outcomes. The statistically significant equation (Chi-Squared = 200.547; df = 64; p < 0.001) accurately predicted 51.7% of away team victories and 81.7% of home team wins.
Results of employing a forward stepwise entry technique based on the likelihood ratio are presented in Table 76. The 19 variable statistically significant (Chi-Squared =
170.091; df = 19; p < 0.001) equation correctly predicted 68.3% of game outcomes and contained a -2LL of 1313.116. Away one day prior (variable one; B = 0.387) was the most influential independent variable. When the away team is playing the second leg of back-to-back games and the previous game was at home, the odds of a home team victory are 47.2% greater. Away one day prior (variable two; B = 0.254) and home turnovers (B
= -0.175) were the second and third most impactful independent variables respectively.
The first stage of the hierarchal logistic regression analysis (see Table 77) entered
18 absolute streak performance measures in a stepwise manner, which produced an equation that accurately classified 69.2% of the 1105 observations and contained a -2LL of 1315.226. Simultaneous entry of the current investigation‟s 12 unique variables in the second stage created an equation that predicted 70.1% of game outcomes and had a -2LL of 1301.720. Of the 12 unique variables, only away one day prior (variable one; B=
0.475; p = 0.033) and home three days prior (B = -0.362; p = 0.041) were statistically significant at the 0.05 level.
224
Variable B Std. Error Wald Exp(B)
Constant -4.571* 1.981 5.323 .010 Away two point field goals made -0.147* .032 20.660 .864 Away two point field goals 0.080* .022 13.237 1.083 attempted Away three pointers made -0.124* .045 7.412 .884 Away steals -0.128* .045 8.134 .880 Away two point field goals made 0.102* .028 12.948 1.108 allowed Away three pointers attempted 0.082* .024 11.744 1.085 allowed Away free throws made allowed 0.049* .022 4.995 1.050 Away steals allowed .080 .050 2.501 1.083 Away one day prior (variable one) .387 .198 3.799 1.472 Away one day prior (variable two) .254 .176 2.083 1.289 Home three pointers attempted .030 .016 3.431 1.031 Home defensive rebounds 0.084* .031 7.596 1.088 Home assists .047 .025 3.514 1.048 Home steals 0.124* .045 7.466 1.132 Home turnovers -0.175* .039 20.511 .839 Home personal fouls .048 .034 2.018 1.049 Home defensive rebounds allowed -0.121* .030 16.344 .886 Home assists allowed -.045 .027 2.832 .956 Home personal fouls allowed .048 .030 2.552 1.049
Hit Ratio 68.3% -2 Log Likelihood 1313.116 * Significant at 0.05 level
Table 76. Stepwise logistic regression results for the absolute model (streak variables) 2007-2008
225
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 205 232 46.9 Home Team 108 560 83.8 Overall Percentage 69.2
-2 Log Likelihood 1315.226 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 218 219 49.9 Home Team 111 557 83.4 Overall Percentage 70.1
-2 Log Likelihood 1301.72 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of absolute streak performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 77. Hierarchal logistic regression goodness-of-fit measures for the absolute streak model 2007-2008
226
Absolute model (streak variables) 2008-2009. Goodness-of-fit indices for the equation with all 64 independent variables included a -2LL valued at 1285.443 and a hit ratio of 68.6%. The equation was statistically significant (Chi-Squared = 191.637; df =
64; p < 0.001) and correctly categorized 47.9% of the away team wins, along with 81.8% of home team victories over the 1105 games.
Results of the stepwise logistic regression analysis are shown in Table 78. The 20 variable equation was statistically significant (Chi-Squared = 166.538; df = 20; p <
0.001), possessed a -2LL of 1310.542, and predicted 68.1% of the game outcomes.
Home one day prior (variable one; B = -1.133) emerged as the largest individual contributor. A home team, which is entering its second consecutive home game (i.e., home one day prior [variable one] = 1), has a 67.8% lower odds of achieving victory.
Following home one day prior (variable one), away four days prior (B = 0.563) and home three days prior (B = -0.502) were the equation‟s most important independent variables respectively.
The first stage of the hierarchal logistic regression analysis (as shown in Table 79) used stepwise entry, over pool of absolute steak performance measures, to produce a 15 variable equation with a -2LL of 1331.447 and a hit ratio of 67.4%. In the second stage, the present study‟s 12 unique variables were entered into the equation. The resulting 27 variable equation contained a -2LL valued at 1306.788 and accurately predicted 68.5% of game outcomes. Of these unique variables, away team depth contribution factor (B = -
0.245; p = 0.043) and home three days prior (B = -0.536; p = 0.002) reached statistical significance at the 0.05 level.
227
Variable B Std. Error Wald Exp(B)
Constant -.656 2.103 .097 .519 Away three pointers attempted -0.033* .017 3.903 .968 Away steals -0.205* .077 7.123 .815 Away free throws attempted 0.044* .018 5.983 1.045 allowed Away offensive rebounds allowed 0.087* .039 4.993 1.091 Away defensive rebounds allowed .040 .026 2.370 1.041 Away assists allowed 0.082* .026 9.794 1.086 Away turnovers allowed .103 .055 3.507 1.108 Away team depth contribution -0.268* .121 4.939 .765 factor Away four days prior .563 .358 2.478 1.756 Away one day prior (variable one) .305 .199 2.342 1.356 Home two point field goals made 0.064* .027 5.559 1.066 Home three pointers attempted 0.099* .019 27.024 1.104 Home blocked shots .089 .054 2.770 1.093 Home turnovers -0.077* .036 4.531 .926 Home free throws made allowed -0.081* .022 13.175 .923 Home defensive rebounds allowed -0.096* .026 13.265 .908 Home assists allowed -0.105* .027 14.649 .900 Home personal fouls allowed 0.128* .032 15.682 1.137 Home three days prior -0.502* .150 11.249 .605 Home one day prior (variable one) -1.133 .679 2.783 .322
Hit Ratio 68.1% -2 Log Likelihood 1310.542 * Significant at 0.05 level
Table 78. Stepwise logistic regression results for the absolute model (streak variables) 2008-2009
228
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 180 250 41.9 Home Team 110 565 83.7 Overall Percentage 67.4
-2 Log Likelihood 1331.477 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 194 236 45.1 Home Team 112 563 83.4 Overall Percentage 68.5
-2 Log Likelihood 1306.788 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of absolute streak performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 79. Hierarchal logistic regression goodness-of-fit measures for the absolute streak model 2008-2009
229
Relative model 2007-2008. The simultaneous entry of all 32 independent variables produced an equation that accurately predicted 71.8% of the final 930 NBA games for the 2007-2008 NBA season. The equation correctly categorized 54.9% of away team victories (i.e., 200 games) and 82.7% of home team wins (i.e., 468 games).
As a whole, the equation was statistically significant (Chi-Squared = 207.977; df = 32; p
< 0.001) and produced a -2LL valued at 1037.050.
Results of forward stepwise entry based on the likelihood ratio are presented in
Table 80. The 12 variables equation possessed a -2LL value of 1053.381 and was statistically significant (Chi-Squared = 191.646; df = 12; p < 0.001). The equation accurately predicted 70.2% of the observed games, with 52.5% of the away team victories and 81.6% of the home team victories correctly classified. Home two point field goal ratio (B = 8.499), away two point field goal ratio (B = -8.275), and home free throw ratio (B = 3.500) were the most important respective contributors. Positive logistic coefficients coincide with an increased probability of a home team win.
In the first stage of the hierarchal logistic regression analysis (as shown in Table
81), 10 relative performance variables entered the equation through a stepwise technique to produce a -2LL of 1059.051 and a hit ratio 69.8% of the 930 games. Simultaneously entering the present study‟s unique variables, in the second stage, created an equation containing a -2LL valued at 1042.963 and at hit ratio to 71.5%. Of the 12 unique variables only away one day prior (variable two; B = 0.448; p = 0.045) reached statistically significant at the 0.05 level.
230
Variable B Std. Error Wald Exp(B)
Constant -1.278 3.796 .113 .279 Away two point field goal ratio -8.275* 1.715 23.270 .000 Away three pointer ratio -2.656* 1.042 6.493 .070 Away offensive rebound ratio -1.818* .634 8.218 .162 Away steal ratio -1.167* .492 5.629 .311 Away personal foul ratio 1.929* 1.025 3.540 6.882 Home two point field goal ratio 8.499* 2.032 17.492 4910.214 Home free throw ratio 3.500* 1.908 3.366 33.125 Home offensive rebound ratio 2.378* .683 12.121 10.786 Home assist ratio 1.524* .781 3.812 4.592 Home steal ratio 1.214* .508 5.704 3.367 Home team depth contribution -3.392* 2.143 2.506 .034 factor ratio Home one day prior (variable two) 0.385* .221 3.043 1.469
Hit Ratio 70.2% -2 Log Likelihood 1053.381 * Significant at 0.05 level
Table 80. Stepwise logistic regression results for the relative model 2007-2008
231
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 186 178 51.1 Home Team 103 463 81.8 Overall Percentage 69.8
-2 Log Likelihood 1059.051 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 198 166 54.4 Home Team 99 467 82.5 Overall Percentage 71.5
-2 Log Likelihood 1042.963 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of relative performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 81. Hierarchal logistic regression goodness-of-fit measures for the relative model 2007-2008
232
Relative model 2008-2009. Simultaneous entry of all 32 independent variables resulted in a statistically significant equation (Chi-Squared = 212.507; df = 32; p < 0.001) with a -2LL value of 1021.272 and a hit ratio of 71.4%. Over the 930 observations, the equation correctly classified 52.6% of away team wins and 82.9% of home team victories.
As shown in Table 82, the stepwise entry technique produced an equation of 12 independent variables, with a -2LL of 1027.539 and a hit ratio of 71.1%. The statistically significant equation (Chi-Squared = 206.240; df = 12; p < 0.001) accurately predicted
182 of the away team wins (51.7%) and 479 of the home team victories (82.9%). Away two point field goal ratio (B = -6.351) emerged as the most important contributor to the equation. The negative logistic coefficient indicates a decrease in the probability of a home team win. Following away two point field goal ratio, home defensive rebound ratio
(B = 5.069) and home personal foul ratio (B = -3.726) were the most important respective independent variables.
As displayed in Table 83, a two stage hierarchal logistic regression analysis was utilized to ascertain the contribution of scheduling fatigue and team depth contribution variables. Entering relative measures of performance in a stepwise manner produced an equation of eight variables, which contained a -2LL of 1044.969 and a hit ratio of 71.3%.
Entering the 12 unique variables in the second stage resulted in an equation with a -2LL of 1025.362 and hit ratio of 71.0%. Home three days prior (B = -0.654; p = 0.001) was the sole unique variable that reached the 0.05 level of statistical significance.
233
Variable B Std. Error Wald Exp(B)
Constant 5.336 3.823 1.949 207.775 Away two point field goal ratio -6.351* 1.535 17.120 .002 Away three pointer ratio -2.999* .993 9.111 .050 Away offensive rebound ratio -1.693* .616 7.560 .184 Away personal foul ratio 1.902 1.313 2.099 6.698 Away four days prior .813 .427 3.617 2.254 Home two point field goal ratio 3.273 2.370 1.907 26.380 Home defensive rebound ratio 5.069* 2.055 6.083 158.983 Home assist ratio 2.245* 1.031 4.744 9.437 Home turnover ratio -2.455* .918 7.155 .086 Home personal foul ratio -3.726* 1.367 7.434 .024 Home three days prior -0.656* .186 12.475 .519 Home two days prior .272 .186 2.147 1.313
Hit Ratio 71.1% -2 Log Likelihood 1027.539 * Significant at 0.05 level
Table 82. Stepwise logistic regression results for the relative model 2008-2009
234
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 188 164 53.4 Home Team 103 475 82.2 Overall Percentage 71.3
-2 Log Likelihood 1044.969 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 184 168 52.3 Home Team 102 476 82.4 Overall Percentage 71.0
-2 Log Likelihood 1025.362 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of relative performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 83. Hierarchal logistic regression goodness-of-fit measures for the relative model 2008-2009
235
Relative model (streak variables) 2007-2008. Simultaneous entry of all 32 independent variables yielded an equation with a -2LL valued at 1317.65 and a hit ratio of 68.1% over the 1105 game outcomes. The statistically significant equation (Chi-
Squared = 165.556; df = 32; p < 0.001) accurately predicted 47.6% of away team victories and 81.4% of home team wins.
Results of employing a forward stepwise entry technique based on the likelihood ratio are presented in Table 84. Seven independent variables entered the statistically significant equation (Chi-Squared = 138.668; df = 7; p < 0.001). As a whole, the equation correctly predicted 66.2% of game outcomes and contained a -2LL of 1344.539.
Away two point field goal ratio (B = -4.237) emerged as the most influential independent variable. The negative logistic coefficient signifies a decrease in the probability of a home team victory. Home defensive rebound ratio (B = 3.415) and home turnover ratio
(B = -1.833) were the second and third most impactful independent variables respectively.
The first stage of the hierarchal regression analysis (see Table 85) entered five relative streak performance measures in a stepwise manner into the equation. The resulting equation accurately classified 65.4% of the observations and possessed a -2LL equal to 1350.221. Simultaneous entry of the current investigation‟s 12 unique variables in the second stage produced an equation which predicted 66.7% of game outcomes and had a -2LL of 1336.482. Of the 12 unique variables, only away one day prior (variable one; B= 0.513; p = 0.019) was statistically significant at the 0.05 level.
236
Variable B Std. Error Wald Exp(B)
Constant 2.676* .956 7.827 14.525 Away two point field goal ratio -4.237* .669 40.064 .014 Away steal ratio -1.027* .215 22.845 .358 Away one day prior (variable one) .368 .192 3.668 1.444 Home offensive rebound ratio .348 .242 2.068 1.417 Home defensive rebound ratio 3.415* .778 19.248 30.429 Home assist ratio 1.133* .419 7.316 3.105 Home turnover ratio -1.833* .413 19.664 .160
Hit Ratio 66.2% -2 Log Likelihood 1344.539 * Significant at 0.05 level
Table 84. Stepwise logistic regression results for the relative model (streak variables) 2007-2008
237
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 173 264 39.6 Home Team 118 550 82.3 Overall Percentage 65.4
-2 Log Likelihood 1350.221 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 188 249 43.0 Home Team 119 549 82.2 Overall Percentage 66.7
-2 Log Likelihood 1336.482 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of relative streak performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 85. Hierarchal logistic regression goodness-of-fit measures for the relative model (streak variables) 2007-2008
238
Relative model (streak variables) 2008-2009. Goodness-of-fit indices for the equation with all 32 independent variables included a -2LL valued at 1337.439 and a hit ratio of 66.1%. The equation was statistically significant (Chi-Squared = 139.641; df =
32; p < 0.001) and correctly categorized 41.6% of the away team wins along with 81.6% of home team victories over the 1105 games.
Results of the stepwise logistic regression analysis are shown in Table 86. The 12 variable equation was statistically significant (Chi-Squared = 124.745; df = 12; p <
0.001), possessed a -2LL of 1352.335, and predicted 66.0% of the game outcomes.
Away two point field goal ratio (B = -3.068) was as the largest individual contributor.
The negative logistic coefficient indicates a decrease in the probability of a home team win. Following away two point field goal ratio, home defensive rebound ratio (B =
2.281) and home personal foul ratio (B = -1.445) emerged as the equation‟s most impactful independent variables respectively.
The first stage of the hierarchal logistic regression analysis (as shown in Table 87) used stepwise entry, over a pool of relative steak performance measures, to produce an equation of nine independent variables with a -2LL of 1371.666 and a hit ratio of 65.8%.
Subsequently, in the second stage, the present study‟s 12 unique variables were entered into the equation. The resulting 21 variable equation contained a -2LL valued at
1345.149 and accurately predicted 66.6% of game outcomes. Of these unique variables, away four days prior (B = 0.765; p = 0.037) and home three days prior (B = -0.547; p =
0.001) reached statistical significance at the 0.05 level.
239
Variable B Std. Error Wald Exp(B)
Constant 3.871* 1.334 8.424 47.988 Away two point field goal ratio -3.068* .684 20.137 .047 Away three pointer ratio -.482 .306 2.481 .617 Away offensive rebound ratio -0.927* .260 12.701 .396 Away steal ratio -0.554* .235 5.577 .574 Away four days prior 0.680* .345 3.889 1.975 Home two point field goal ratio 1.399 1.000 1.956 4.050 Home defensive rebound ratio 2.281* .916 6.199 9.791 Home assist ratio .917 .485 3.578 2.503 Home turnover ratio -1.297* .435 8.908 .273 Home personal foul ratio -1.445* .553 6.824 .236 Home three days prior -0.516* .145 12.586 .597 Home one day prior (variable one) -1.097 .672 2.668 .334
Hit Ratio 66.0% -2 Log Likelihood 1352.335 * Significant at 0.05 level
Table 86. Stepwise logistic regression results for the relative model (streak variables) 2008-2009
240
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 164 266 38.1 Home Team 112 563 83.4 Overall Percentage 65.8
-2 Log Likelihood 1371.666 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 182 248 42.3 Home Team 121 554 82.1 Overall Percentage 66.6
-2 Log Likelihood 1345.149 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of relative streak performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 87. Hierarchal logistic regression goodness-of-fit measures for the relative model (streak variables) 2008-2009
241
Possession model 2007-2008. The simultaneous entry of all 18 independent variables produced an equation that accurately predicted 70.6% of the final 930 NBA games for the 2007-2008 NBA season. The equation correctly categorized 53.3% of away team victories (i.e., 194 games) and 81.8% of home team wins (i.e., 463 games).
As a whole, the equation was statistically significant (Chi-Squared = 199.803; df = 18; p
< 0.001) and produced a -2LL valued at 1045.224.
Results of the stepwise logistic regression analysis are presented in Table 88.
Five independent variables entered the statistically significant equation (Chi-Squared =
187.288; df = 5; p < 0.001), which possessed a -2LL value of 1057.739. The equation accurately predicted 70.5% of the observed games, with 52.5% of the away team victories and 82.2% of the home team victories correctly classified. Home one day prior
(variable two; B = 0.376) was the most impactful variable in the equation. Holding all other variables constants, when the home team is entering back-to-back games and the previous game was on the road, the odds of a home team victory are 45.6% greater.
Following home one day prior (variable two), away defensive rating (B = 0.156) and home offensive rating (B = 0.152) were the most significant respective contributors.
The first stage of the hierarchal logistic regression analysis (as shown in Table 89) entered four variables into the equation in a stepwise manner. The resulting equation had a -2LL of 1060.721 and accurately predicted 69.8% of the 930 games. In the second stage, the addition of the present study‟s unique variables created an equation containing
242
Variable B Std. Error Wald Exp(B)
Constant -10.164 5.654 3.231 .000 Away offensive rating -0.100* .021 23.719 .904 Away defensive rating 0.156* .027 34.118 1.169 Home offensive rating 0.152* .021 54.232 1.164 Home defensive rating -0.108* .027 15.661 .898 Home one day prior (variable two) .376 .220 2.920 1.456
Hit Ratio 70.5% -2 Log Likelihood 1057.739 * Significant at 0.05 level
Table 88. Stepwise logistic regression results for the possession model 2007-2008
243
Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 188 176 51.6 Home Team 105 461 81.4 Overall Percentage 69.8
-2 Log Likelihood 1060.721 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 196 168 53.8 Home Team 102 464 82.0 Overall Percentage 71.0
-2 Log Likelihood 1045.628 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of possession model based pace and performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 89. Hierarchal logistic regression goodness-of-fit measures for the possession model 2007-2008
244
a -2LL valued at 1045.628 and at hit ratio to 71.0%. Of the 12 unique variables, away one day prior (variable one; B = 0.525; p = 0.038) and away one day prior (variable two;
B = 0.484; p = 0.031) were deemed statistically significant at the 0.05 level.
245
Possession model 2008-2009. Simultaneous entry of all 18 independent variables resulted in a statistically significant equation (Chi-Squared = 209.381; df = 18; p < 0.001) with a -2LL value of 1024.398 and a hit ratio of 72.3%. Over the 930 observations, the equation correctly classified 53.7% of away team wins and 83.6% of home team victories.
As shown in Table 90, the stepwise entry technique produced a six variable equation, with a -2LL of 1031.888 and a hit ratio of 71.4%. The statistically significant equation (Chi-Squared = 201.892; df = 6; p < 0.001) accurately predicted 185 of the away team wins (52.6%) and 479 of the home team victories (82.9%). Away four days prior
(B = 0.727) was the most impactful variable in the equation. When the away team is entering their fourth competition in five days, the odds of a home team victory are
107.0% greater. Following away four days prior, home three days prior (B = -0.589) and home defensive rating (B = -0.161), were the most influential independent variables respectively.
As displayed in Table 91, a two stage hierarchal logistic regression analysis was utilized to ascertain the contribution of scheduling fatigue and team depth contribution variables. In the first stage, entering possession measures of pace and performance in a stepwise manner produced an equation with four variables, which contained a -2LL of
1046.275 and a hit ratio of 71.4%. Entering the 12 unique variables in the second stage resulted in an equation with a -2LL of 1025.904 and hit ratio of 72.2%. Home three days prior (B = -0.687; p = 0.001) was the sole unique variable that reached the 0.05 level of statistical significance.
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Variable B Std. Error Wald Exp(B)
Constant -1.393 6.075 .053 .248 Away offensive rating -0.105* .025 17.161 .900 Away defensive rating 0.132* .024 29.856 1.141 Away four days prior .727 .427 2.896 2.070 Home offensive rating 0.155* .026 36.199 1.167 Home defensive rating -0.161* .026 39.424 .851 Home three days prior -0.589* .170 12.043 .555
Hit Ratio 71.4% -2 Log Likelihood 1031.888 * Significant at 0.05 level
Table 90. Stepwise logistic regression results for the possession model 2008-2009
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 182 170 51.7 Home Team 96 482 83.4 Overall Percentage 71.4
-2 Log Likelihood 1046.275 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 185 167 52.6 Home Team 92 486 84.1 Overall Percentage 72.2
-2 Log Likelihood 1025.904 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of possession model based pace and performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 91. Hierarchal logistic regression goodness-of-fit measures for the possession model 2008-2009
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Possession model (streak variables) 2007-2008. Simultaneous entry of all 18 independent variables yielded an equation with a -2LL valued at 1338.205, which correctly predicted 66.3% of the 1105 game outcomes. The statistically significant equation (Chi-Squared = 145.002; df = 18; p < 0.001) accurately predicted 43.0% of away team victories and 81.6% of home team wins.
Results of employing a forward stepwise entry technique based on the likelihood ratio are presented in Table 92. Five independent variables entered the statistically significant equation (Chi-Squared = 133.525; df = 5; p < 0.001), which correctly predicted 66.3% of game outcomes and contained a -2LL of 1349.681. Away one day prior (variable one; B = 0.365) emerged as the most influential independent variable.
When the away team is playing the second leg of back-to-back games and the previous game was at home, the odds of a home team victory are 44.1% greater. Home offensive rating (B = 0.078) was the second most impactful independent variable.
In the first stage of the hierarchal logistic regression analysis (see Table 93) four possession streak performance measures entered the equation in a stepwise manner. The equation accurately classified 66.5% of the observations and contained a -2LL of
1353.390. Simultaneous entry of the current investigation‟s 12 unique variables, in the second stage, resulted in an equation that predicted 66.9% of game outcomes and had a -
2LL of 1340.583. Of the 12 unique variables, only away one day prior (variable one; B=
0.528; p = 0.016) was statistically significant at the 0.05 level.
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Variable B Std. Error Wald Exp(B)
Constant -2.483 2.265 1.202 .083 Away offensive rating -0.061* .010 35.093 .941 Away defensive rating 0.061* .011 28.578 1.063 Away one day prior (variable one) .365 .192 3.628 1.441 Home offensive rating 0.078* .011 51.139 1.081 Home defensive rating -0.051* .011 20.322 .950
Hit Ratio 66.3% -2 Log Likelihood 1349.681 * Significant at 0.05 level
Table 92. Stepwise logistic regression results for the possession model (streak variables) 2007-2008
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 182 255 41.6 Home Team 115 553 82.8 Overall Percentage 66.5
-2 Log Likelihood 1353.390 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 189 248 43.2 Home Team 118 550 82.3 Overall Percentage 66.9
-2 Log Likelihood 1340.583 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of possession model based streak pace and performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 93. Hierarchal logistic regression goodness-of-fit measures for the possession model (streak variables) 2007-2008
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Possession model (streak variables) 2008-2009. Goodness-of-fit indices for the equation with all 18 independent variables included a -2LL valued at 1342.591 and a hit ratio of 64.8%. The equation was statistically significant (Chi-Squared = 134.489; df =
18; p < 0.001) and correctly categorized 39.8% of the away team wins along with 80.7% of home team victories over the 1105 games.
Results of the stepwise logistic regression analysis are shown in Table 94. The nine variable equation was statistically significant (Chi-Squared = 129.269; df = 9; p <
0.001), possessed a -2LL of 1347.811, and predicted 64.8% of the game outcomes correctly. Home one day prior (variable one; B = -1.112) was the most impactful of the independent variables. A home team, which is entering its second consecutive home game, has 67.1% lower odds of achieving victory. Following home one day prior
(variable one), away four days prior (B = 0.732) and home three days prior (B = -0.460) were the equation‟s most influential independent variables respectively.
The first stage of the hierarchal logistic regression analysis (as shown in Table 95) utilized stepwise entry, over a pool of possession model based steak performance and pace measures, to produce an equation of five independent variables with a -2LL of
1369.654 and a hit ratio of 65.0%. Subsequently, in the second stage, the present study‟s
12 unique variables were entered simultaneously. The resulting 17 variable equation contained a -2LL valued at 1343.836 and accurately predicted 64.0% of game outcomes.
Of these unique variables, away team depth contribution factor (B = -0.255; p = 0.029), away four days prior (B = 0.793; p = 0.032) and home three days prior (B = -0.496; p =
0.004) reached statistical significance at the 0.05 level.
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Variable B Std. Error Wald Exp(B)
Constant -3.964 2.819 1.977 .019 Away possessions 0.038* .017 5.331 1.039 Away offensive rating -0.032* .012 7.953 .968 Away defensive rating 0.049* .011 18.510 1.050 Away team depth contribution -0.251* .117 4.641 .778 factor Away four days prior 0.732* .349 4.397 2.080 Home offensive rating 0.077* .012 43.215 1.080 Home defensive rating -0.065* .012 30.268 .937 Home three days prior -0.460* .146 9.992 .631 Home one day prior (variable one) -1.112 .659 2.847 .329
Hit Ratio 64.8% -2 Log Likelihood 1347.811 * Significant at 0.05 level
Table 94. Stepwise logistic regression results for the possession model (streak variables) 2008-2009
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 158 272 36.7 Home Team 115 560 83.0 Overall Percentage 65.0
-2 Log Likelihood 1369.654 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 165 265 38.4 Home Team 133 542 80.3 Overall Percentage 64.0
-2 Log Likelihood 1343.836 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of possession model based streak pace and performance measures. Block 2 simultaneously entered scheduling fatigue measures and team depth contribution factors.
Table 95. Hierarchal logistic regression goodness-of-fit measures for the possession model (streak variables) 2008-2009
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Play model 2007-2008. Entry of all 20 independent variables produced an equation that predicted 71.2% of the final 930 NBA games for the 2007-2008 NBA season. The equation correctly categorized 54.7% of away team victories (i.e., 199 games) and 81.8% of home team wins (i.e., 463 games). The equation was statistically significant (Chi-Squared = 202.450; df = 20; p < 0.001) and produced a -2LL valued at
1042.577.
Results of forward stepwise entry based on the likelihood ratio are presented in
Table 96. The seven variable equation possessed a -2LL value of 1061.382 and was statistically significant (Chi-Squared = 183.644; df = 7; p < 0.001). The equation accurately predicted 72.4% of the observed games, with 55.2% of the away team victories and 83.4% of the home team victories correctly classified. Away defensive points-per-play (B = 18.278), home offensive points-per-play (B = 17.112), and home defensive points-per-play (B = -14.404) were the most influential variables. Positive logistic coefficients coincide with an increased probability of a home team win.
Hierarchal logistic regression analysis was used to assess the influence of scheduling fatigue variables and team depth contribution measures (see Table 81). In the first stage, six independent variables entered the equation through a stepwise technique.
The resulting equation had a -2LL of 1064.428 and accurately predicted 71.5% of game outcomes. The addition of the present study‟s unique variables in the second stage produced an equation with a -2LL valued at 1049.875 and at hit ratio to 72.3%. Of the 12 unique variables only away one day prior (variable two; B = 0.482; p = 0.032) reached statistically significant at the 0.05 level.
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Variable B Std. Error Wald Exp(B)
Constant -13.802 6.001 5.289 .000 Away offensive points-per-play -7.422 1.988 13.932 .001 Away defensive points-per-play 18.278 3.057 35.746 86704251.203 Home offensive plays .217 .049 19.945 1.242 Home defensive plays -.202 .043 21.708 .817 Home offensive points-per-play 17.112 2.421 49.948 27010546.040 Home defensive points-per-play -14.404 3.082 21.847 .000 Home one day prior (variable two) .380 .220 2.980 1.462
Hit Ratio 72.4% -2 Log Likelihood 1061.382 * Significant at 0.05 level
Table 96. Stepwise logistic regression results for the play model 2007-2008
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 194 170 53.3 Home Team 95 471 83.2 Overall Percentage 71.5
-2 Log Likelihood 1064.428 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 200 164 54.9 Home Team 94 472 83.4 Overall Percentage 72.3
-2 Log Likelihood 1049.875 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of play model based pace and performance measures. Block 2 simultaneous entered scheduling fatigue measures and team depth contribution factors.
Table 97. Hierarchal logistic regression goodness-of-fit measures for the play model 2007-2008
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Play model 2008-2009. Simultaneous entry of all 20 independent variables resulted in a statistically significant equation (Chi-Squared = 213.012; df = 20; p < 0.001) with a -2LL value of 1020.768 and a hit ratio of 72.5%. The equation correctly classified
54.5% of away team wins and 83.4% of home team victories over the 930 observations.
As shown in Table 98, the stepwise entry technique produced an equation of 11 independent variables, which had a -2LL of 1026.326 and a hit ratio of 70.8%. The statistically significant equation (Chi-Squared = 207.453; df = 11; p < 0.001) accurately predicted 185 of the away team wins (52.6%) and 473 of the home team victories
(81.8%). Home defensive points-per-play (B = -21.059) emerged as the most influential contributor. The negative logistic coefficient indicates a decrease in the probability of a home team win. Following home defensive points-per-play, home offensive points-per- play (B = 16.077) and away defensive points-per-play (B = 13.744) were the most impactful respective independent variables within the equation.
As displayed in Table 99, the first stage of the hierarchal logistic regression analysis entered solely play measures of pace and performance in a stepwise manner to produce an equation of six variables. The resulting equation contained a -2LL of
1048.365 and a hit ratio of 70.0%. Entering the 12 unique variables in the second stage resulted in an equation with a -2LL of 1025.775 and hit ratio of 71.2%. Home three days prior (B = -0.659; p = 0.001) was the sole unique variable that reached the 0.05 level of statistical significance.
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Variable B Std. Error Wald Exp(B)
Constant .757 6.859 .012 2.132 Away offensive plays -0.131* .055 5.748 .877 Away defensive plays 0.146* .051 8.159 1.158 Away offensive points-per-play -13.427* 2.916 21.202 .000 Away defensive points-per-play 13.744* 2.829 23.596 930840.897 Away four days prior .834 .441 3.580 2.304 Home offensive plays 0.136* .054 6.228 1.145 Home defensive plays -0.108* .049 4.827 .897 Home offensive points-per-play 16.077* 2.917 30.386 9597785.451 Home defensive points-per-play -21.059* 3.063 47.275 .000 Home three days prior -0.555* .172 10.348 .574 Home one day prior (variable one) -.921 .726 1.609 .398
Hit Ratio 70.8% -2 Log Likelihood 1026.326 * Significant at 0.05 level
Table 98. Stepwise logistic regression results for the play model 2008-2009
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 171 181 48.6 Home Team 98 480 83.0 Overall Percentage 70.0
-2 Log Likelihood 1048.365 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 184 168 52.3 Home Team 100 478 82.7 Overall Percentage 71.2
-2 Log Likelihood 1025.775 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of play model based pace and performance measures. Block 2 simultaneous entered scheduling fatigue measures and team depth contribution factors.
Table 99. Hierarchal logistic regression goodness-of-fit measures for the play model 2008-2009
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Play model (streak variables) 2007-2008. Simultaneous entry of all 20 independent variables yielded an equation with a -2LL valued at 1335.738, which correctly predicted 66.5% of the 1105 game outcomes. The statistically significant equation (Chi-Squared = 147.469; df = 20; p < 0.001) accurately predicted 42.8% of away team victories and 82.0% of home team wins.
Results of employing a forward stepwise entry technique based on the likelihood ratio are presented in Table 100. The nine variable equation was statistically significant
(Chi-Squared = 138.738; df = 9; p < 0.001), correctly predicted 66.4% of game outcomes and contained a -2LL of 1344.469. Home offensive points-per-play (B = 8.994) was the most influential independent variable. The positive logistic coefficient coincides with an increase in the probability of a home team victory. Away offensive points-per-play (B =
-7.144) and away defensive points-per-play (B = 6.487) were the second and third most impactful independent variables respectively.
In the first stage of the hierarchal logistic regression analysis (see Table 101) eight play streak pace and performance measures entered the equation in a stepwise manner. The constructed equation accurately classified 65.6% of the 1105 observations and had a -2LL equal to 1348.156. In the second stage, simultaneous entry of the current investigation‟s 12 unique variables resulted in the aforementioned model, which consisted of all 20 independent variables. Of the 12 unique variables, only home one day prior (variable one; B= 0.521; p = 0.017) was statistically significant at the 0.05 level.
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Variable B Std. Error Wald Exp(B)
Constant -3.592 3.023 1.412 .028 Away offensive plays -.040 .024 2.660 .961 Away defensive plays 0.065* .023 8.211 1.067 Away offensive points-per-play -7.144* 1.160 37.922 .001 Away defensive points-per-play 6.487* 1.282 25.602 656.660 Away one day prior (variable one) .365 .192 3.608 1.440 Home offensive plays 0.088* .024 13.183 1.092 Home defensive plays -0.097* .022 18.593 .908 Home offensive points-per-play 8.994* 1.230 53.489 8051.275 Home defensive points-per-play -5.987* 1.298 21.283 .003
Hit Ratio 66.4% -2 Log Likelihood 1344.469 * Significant at 0.05 level
Table 100. Stepwise logistic regression results for the play model (streak variables) 2007- 2008
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 178 259 40.7 Home Team 121 547 81.9 Overall Percentage 65.6
-2 Log Likelihood 1348.156 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 187 250 42.8 Home Team 120 548 82.0 Overall Percentage 66.5
-2 Log Likelihood 1335.738 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of play model based streak pace and performance measures. Block 2 simultaneous entered scheduling fatigue measures and team depth contribution factors.
Table 101. Hierarchal logistic regression goodness-of-fit measures for the play model (streak variables) 2007-2008
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Play model (streak variables) 2008-2009. Goodness-of-fit indices for the equation of all 20 independent variables included a -2LL valued at 1341.935 and a hit ratio of 65.1%. The equation was statistically significant (Chi-Squared = 135.145; df =
20; p < 0.001) and correctly categorized 38.8% of the away team wins along with 81.8% of home team victories over the 1105 games.
Results of the stepwise logistic regression analysis are shown in Table 102. The
12 variable equation was statistically significant (Chi-Squared = 131.152; df = 12; p <
0.001), possessed a -2LL of 1345.928, and was able to predict 65.4% of the game outcomes for season. Home offensive points-per-play (B = 8.226) was the most impactful independent variable. The positive logistic coefficient indicates an increase in the probability of a home team win. Following home offensive points-per-play, home defensive points-per-play (B = -7.448) and away defensive points-per-play (B = 5.242) were as the equation‟s most influential independent variables respectively.
The first stage of the hierarchal logistic regression analysis (as shown in Table
103) used stepwise entry, over a pool of play model based steak pace and performance measures, to produce an eight variable equation with a -2LL of 1366.387 and a hit ratio of 65.2%. Subsequently, in the second stage, the present study‟s 12 unique variables were entered into the model. With all 20 variables, the model mirrors the aforementioned simultaneous entry analysis. Of these unique variables, away team depth contribution factor (B = -0.248, p = 0.035), away four days prior (B = 0.770; p = 0.037) and home three days prior (B = -0.483; p = 0.005) reached statistical significance at the 0.05 level.
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Variable B Std. Error Wald Exp(B)
Constant -2.019 3.170 .406 .133 Away offensive plays -0.051* .026 3.887 .951 Away defensive plays 0.087* .025 12.100 1.091 Away offensive points-per-play -3.984* 1.306 9.300 .019 Away defensive points-per-play 5.242* 1.268 17.082 189.093 Away team depth contribution -0.244* .117 4.350 .784 factor Away four days prior 0.714* .349 4.197 2.043 Home offensive plays 0.051* .026 3.845 1.052 Home defensive plays -0.062* .025 6.204 .940 Home offensive points-per-play 8.226* 1.316 39.101 3737.087 Home defensive points-per-play -7.448* 1.338 30.971 .001 Home three days prior -0.450* .146 9.457 .638 Home one day prior (variable one) -1.077 .661 2.654 .340
Hit Ratio 65.4% -2 Log Likelihood 1345.928 * Significant at 0.05 level
Table 102. Stepwise logistic regression results for the play model (streak variables) 2008- 2009
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Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 1 Game Winner Away Team 154 276 35.8 Home Team 109 566 83.9 Overall Percentage 65.2
-2 Log Likelihood 1366.387 Predicted Game Winner Percentage Observed Away Team Home Team Correct Block 2 Game Winner Away Team 167 263 38.8 Home Team 123 552 81.8 Overall Percentage 65.1
-2 Log Likelihood 1341.935 Note: Block 1 of the hierarchal procedure consisted of forward stepwise entry based on the Likelihood Ratio for a pool of play model based streak pace and performance measures. Block 2 simultaneous entered scheduling fatigue measures and team depth contribution factors.
Table 103. Hierarchal logistic regression goodness-of-fit measures for the play model (streak variables) 2008-2009
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Chapter 5: Discussion
The present study examined the relationship between a set of variables (i.e., previously compiled performance statistics, indicators related to scheduling fatigue, and measures pertaining to team depth contributions) and the outcome of an “un-played”
National Basketball Association contest. Basketball research, in general, has used independent variables of the traditional box score statistic variety, which were operationalized in either absolute or relative terms. Expanding upon this line of research, the current study provided insight into other factors that influence team success, by exploring the influence of measures of scheduling fatigue and team depth contributions.
Additionally, alternative methods of operationalizing performance were explored as possession and play models were constructed via composite pace and efficiency measures. Finally, unlike Loeffelholz et al.‟s (2009) neural network based research, the individual contributions of independent variables in relation to future outcomes (i.e., score differential or game winner) was examined. Given the deficiencies within previous studies, the current investigation offers a new perspective on evaluating performance within the context of basketball. A summary of the numerous models and the rationale behind their construction will be discussed in the following paragraphs.
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Consisting of 64 independent variables, absolute models include the most extensive array of performance measures for the two competing teams. Performance variables within these absolute models are traditional box score statistics in an unmodified form. Free from any manipulation, absolute models present the greatest potential to explain and predict NBA outcomes. The prospect of producing a model capable of explaining large proportions of the variance in score differential and/or accurately predicting game outcomes at a high success rate, ultimately, was the reason behind the construction of the absolute models. Offensive and defensive statistics for the away and the home team are included in two point field goals made, two point field goals attempted, three point field goals made, three point field goals attempted, free throws made, free throws attempted, offensive rebounds, defensive rebounds, assists, steals, blocked shots, turnovers, and personal fouls.
With 32 independent variables, relative models contain performance measures operationally defined in ratios. Representing performance measures in this manner is advantageous, as both the team‟s offensive and defensive abilities are encapsulated in a single variable, which places strengths and weaknesses into perspective through a comparative approach. For both the away and home team, ratios are computed in the following categories: two point field goal percentage, three point field goal percentage, free throw percentage, offensive rebounds, defensive rebounds, assists, steals, blocked shots, turnovers, and personal fouls.
Composed of 18 independent variables, possession models were based on pace and efficiency. Possession models include the number of possessions for the competing
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teams as an indication of pace, and team offensive and defensive ratings as composite measures of efficiency. The number of possessions for a given team can be calculated with knowledge of field goals attempted, offensive rebounds, defensive rebounds, turnovers and free throws; while offensive and defensive ratings can be found through the aforementioned statistics, plus the number of points a team has scored / allowed. Given that previous production function research in basketball has failed to explore the value of these composite measures, the present study examines their relationship with game outcomes. The adoption of an alternative perspective, such as that provided by the possession model, can be critical to the further acceptance of composite measures as an evaluation tool at the team level.
Similar to the possession models, play models were devised with a focus on pace and efficiency. Ultimately, the two model categorizations differ in the manner offensive rebounds are classified. The action of gathering an offensive rebound to maintain control of the ball constitutes one possession, but two plays. Play models contain the number of offensive plays and defensive plays for the two competing teams as measures of pace, and offensive points-per-play and defensive points-per-play as indicators of efficiency.
As a whole, the play models offer a different conceptualization of basketball production through pace and efficiency measures.
While the various models differ in the manner performance measures are operationalized, they share several common threads. First, as previously noted, each model includes a host of innovative quantifiable measures that focus on scheduling fatigue and team depth contributions. Unique variables include 10 dichotomous
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scheduling fatigue variables, which chronicle the events of the two competing teams‟ five previous days, as well as a team depth contribution factor for both squads. Therefore, with the exception of the relative models, which quantify team depth contribution factors as ratios, each model includes a total of 12 identical variables.
Second, general model categorizations (i.e., absolute, relative, possession and play) can be further segmented in those models which are constructed from performance statistics amassed over all previously “played” games, and streak models, which include performance measures only from each team‟s previous five contests. Streak models were conceptualized with the goal of capturing positive and negative fluctuations in team play, as well as the ramifications stemming from injuries. Calculating performance measures as five game moving averages can be beneficial, given that levels of play often varies over the duration of the NBA season.
Third, the numerous models can be further classified into those modeled with the linear functional form and those of the Cobb-Douglas variety. The linear estimation of the Cobb-Douglas production function is accomplished by taking the natural logarithm of the variables, hence logarithmic based models in the present study coincide with the
Cobb-Douglas functional form. While linear production functions contain constant returns to scale and an infinite elasticity of substitution, the Cobb-Douglas function has the ability to represent varying returns to scale and supports a unitary elasticity of substitution. As the present conceptualization is exploratory and lacks an established functional form, both the linear and Cobb-Douglas formulations were used.
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Fourth, game outcomes for the four general model classifications were operationalized in two distinct fashions (i.e., as score differential or as a binary variable that indicates game winner). As a result, two different statistical analyses were employed, in which multiple regression analysis was used to examine the “un-played” competition‟s score differential and logistic regression analysis was used to explore whether the away team or the home emerged victorious in the “un-played” game. While conceptually similar, the decision to investigate two dependent variables nearly doubled the number of statistical analyses and thus deserves further explanation.
From one perspective, the bottom line in any basketball game is a win or a loss, which consequently supports the use of logistic regression analysis. Logistic regression analysis is advantageous in the present context because the logistic regression equation can be used to calculate the probability of winning an upcoming contest based on the two team‟s previously compiled performance measures as well as scheduling fatigue and team depth contribution measures. Alternatively, from a second perspective, score differential can be used as a quantifiable measure to compare the abilities of the two competing squads. A truly dominant team, not only will regularly beat their opponents, they will do so convincingly, which would be demonstrated through score differential.
Furthermore, the multiple regression analyses of the present study uses the ordinary least squares method for estimating regression parameters, which has an accepted procedure for calculating standardized coefficients; whereas the calculation of standardized logistic regression coefficients can be accomplished in numerous ways and does not have a consensus (Menard, 2004). Standardized coefficients facilitate the process of
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determining the relative influence of different predictors, especially when the independent variables lack a natural metric (Menard). Hence, determining the influence of the individual independent variables is more easily accomplished and accepted for the multiple regression analyses. As each statistical analysis procedure is beneficial in its own right, both were employed within the present investigation. For an “un-played” contest, individuals interested in determining the probability of a team victory should look at the logistic regression results, while multiple regression results should be used if score differential is of interest.
Fifth, the process of entering each model‟s respective independent variables into the regression equation was executed in three distinct ways (i.e., simultaneous, stepwise, and hierarchal). Simultaneous entry of all independent variables was used to determine each model‟s overall predictive capabilities. Unfortunately, simultaneous entry disregards the statistical significance of the independent variables and leads to the least parsimonious equations. As an alternative technique, stepwise entry objectively selects variables based on explanatory contribution along with statistical significance.
Constructed in this manner, stepwise equations reduce the number of independent variables and therefore are more parsimonious. Finally, as a third technique, hierarchal entry was used to ascertain the influence of scheduling fatigue and team depth contribution factors. In the first stage of the hierarchal analysis, stepwise entry over a pool of performance measures was executed, which was followed by the simultaneous entry of the present study‟s unique variables. As evidenced, each of the entry procedures was performed with the intention of accomplishing a different task.
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Sixth and finally, the numerous statistical analyses of the present research investigation were conducted over the 2007-2008 and 2008-2009 NBA seasons, which were identified as two populations. Use of a census is advantageous for all elements are included and issues pertaining to sampling are removed. Given that the research was exploratory in nature, the 2008-2009 NBA season served as a confirmatory population of the research findings that emerged from the 2007-2008 NBA campaign. Simply stated, dual populations were used to test generalizability and the stability in the production functions from one season to the next.
In discussing the results, the following sections will provide a series of detailed explanations on the overall predictive capabilities of the various models. Comparisons between models are made to identify superior models. Furthermore, the influence and consistency of the performance related variables, as well as the present study‟s scheduling fatigue and team depth contribution measures, are examined. First however, it is important to discuss issues pertaining to the interpretation of independent variables within the absolute and relative models, as well as the regression equation selection process.
Absolute Models
The interpretation of the importance of the independent variables within the absolute models is problematic due to the inherent high correlations among the independent variables (e.g., shot attempted statistics highly correlate with shot made statistics). For example, for the 2007-2008 absolute models, the correlation between away two point field goals made and away two point field goals attempted was 0.729,
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away three pointers made and away three pointers attempted was 0.971, and away free throws attempted and away free throws made was 0.931. Intuitively this is logical because, the number of made field goals / free throws ultimately depends upon the number of field goals / free throws attempted. In a similar manner, high correlations for shooting statistics reside among away team allowed statistics, home team statistics, and the home team allowed statistics. Please note that correlation tables for independent variables are not included due to the large number of variables within the models (e.g., the correlation matrix for absolute models would consist of 4096 entries).
In addition, high correlations exist for each team‟s turnover statistic and its number of steals allowed. This finding is hardly surprising, for each instance when a team is credited with a steal, its opponent is correspondingly charged with a turnover. In other words, when a player from the away team fortuitously intercepts a home team‟s pass, the away team is credited with a steal at the same exact moment as the home team is credited a turnover. In fact every team steal is accompanied by an opponent turnover.
Coinciding with the above example, for the 2007-2008 absolute models, the correlation between away steals and away turnovers allowed was equal to 0.876. High correlation values are additionally present between away turnovers and away steals allowed, home steals and home turnovers allowed, and home turnovers and home steals allowed.
Finally, the absolute models suffer from collinearity issues in regards to a team‟s personal foul statistic, and the measures of free throws attempted allowed and free throws made allowed. Once again, the high level of correlation between these independent variables was expected, given that the majority of free throws are awarded as the result of
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an opponent‟s personal foul. For example, when a player from the away team fouls a player from the home team in the act of attempting a two point field goal, the statistician records a personal foul under the away team, two free throws attempted for the home team, and between zero and two made free throws for the home team. Following the above example, for the 2007-2008 absolute models, the correlation between away personal fouls and away free throws attempted allowed was 0.927, and the correlation between away personal fouls and away free throws made allowed was 0.933. Similar high levels of correlations for a team and its opponent exists for away free throws made / away free throws attempted and away personal fouls allowed, home personal fouls and home free throws made allowed / home free throws attempted allowed, and home free throws made / home free throws attempted and home personal fouls allowed.
Multicollinearity is a serious concern when conducting regression analyses. A model plagued by multicollinearity contains independent variables whose predictive power is shared by the other independent variables, thus the unique variance attributable to each independent variable diminishes as shared variance percentages increase (Hair et al., 2006). When shared variance is present, determining the relative influence of the individual independent variables becomes increasingly difficult. High degrees of multicollinearity can result in the incorrect estimation of regression coefficients as well as an incorrect assignment of coefficient signs (Hair et al.). It is imperative to note, that the presence of multicollinearity additionally affects sequential search methods (e.g., stepwise entry techniques). In the situation when several independent variables are highly correlated and one of the variables has been selected for the equation, it becomes
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very unlikely that the other correlated variable(s) will enter due to a lack of unique contribution (Hair et al.). Therefore, with multicollinearity present for the absolute models, individuals should exercise caution before declaring that the independent variables not selected during sequential entry are inconsequential.
Relative Models
While multicollinearity is not an issue for relative models, the interpretation of individual relative performance measures deserves an additional explanation, for they are operationalized as ratios. The increase or decrease in ratio based performance statistics can be achieved through several avenues. For example an increase in home two point field goal ratio can be accomplished by (a) increasing the home team‟s two point field goal percentage, (b) decreasing the home team‟s opponents‟ two point field goal percentage, or (c) a mixture of the first and second. In a hypothetical situation, in which the home team possesses a two point field goal percentage of 45.0% and allows it‟s opponents‟ to shoot 47.0% from two point range, a 0.10 unit increase in the ratio can be achieved by (a) increasing the home team‟s two point field goal percentage to 49.7%, while holding constant the home team‟s opponents‟ two point field goal percentage, (b) decreasing the home team‟s opponents‟ two point field goal percentage to a value of
42.1%, while holding the home team‟s two point field goal percentage constant, or (c) a combination of increasing the home team‟s two point field goal percentage and decreasing its opponents‟ two point field goal percentage. Therefore, in order to utilize the relative production functions, an individual must be aware that while the coefficient attached to a relative performance measure reveals its effect on the “un-played” game‟s
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dependent variable (i.e., score differential or game winner), altering the ratio to achieve a desired result can be done in different ways.
Regression Equation Selection
Ultimately, regression equation selection criterion is contingent upon the goodness-of-fit measures (i.e., the variance explained or the hit ratio) and the parsimony of the equation. In general increasing the number of parameters inevitably improves goodness-of-fit indicators; however improvements can stand at the expense of adding variables with miniscule contributions (Forster, 2000). Based on this notion, the fact that absolute model equations, which contained the greatest number of independent variables, explained the largest proportions of variance in score differential and contained the most successful hit ratios, was to be expected. While hardly an exact science, the best models were identified by taking into account predictive capabilities as well as the number of included independent variables.
Research Questions
1) With what accuracy can the absolute model, the relative model, the possession
model, and the play model explain and predict NBA score differentials for the
2007-2008 and 2008-2009 seasons?
Table 104 provides the summary of the various multiple regression analyses carried out with the 2007-2008 NBA season data. To recall, score differential was the dependent variable in these models. Of all the models, the possession model with stepwise entry and the possession model with stepwise entry (logarithmic variables) were
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Variance Model Entry # I.V. Explained
Absolute Simultaneous 64 32.1% Absolute Stepwise 14 28.5% Absolute Hierarchal 26 29.4% Absolute (Streak) Simultaneous 64 21.1% Absolute (Streak) Stepwise 14 16.9% Absolute (Streak) Hierarchal 26 18.2% Absolute (Logarithmic) Simultaneous 64 31.9% Absolute (Logarithmic) Stepwise 13 27.3% Absolute (Logarithmic) Hierarchal 24 28.1% Absolute (Streak Logarithmic) Simultaneous 64 21.4% Absolute (Streak Logarithmic) Stepwise 13 17.0% Absolute (Streak Logarithmic) Hierarchal 26 18.2% Relative Simultaneous 32 27.6% Relative Stepwise 9 25.9% Relative Hierarchal 21 26.6% Relative (Streak) Simultaneous 32 17.4% Relative (Streak) Stepwise 7 14.0% Relative (Streak) Hierarchal 21 15.8% Relative (Logarithmic) Simultaneous 32 27.6% Relative (Logarithmic) Stepwise 9 26.0% Relative (Logarithmic) Hierarchal 21 26.7% Relative (Streak Logarithmic) Simultaneous 32 17.4% Relative (Streak Logarithmic) Stepwise 10 15.3% Relative (Streak Logarithmic) Hierarchal 20 15.7% Possession Simultaneous 18 27.5% Possession Stepwise 4 26.6% Possession Hierarchal 16 27.4% Possession (Streak) Simultaneous 18 16.8%
Continued
Table 104. Summary of multiple regression analyses 2007-2008
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Table 104 continued
Possession (Streak) Stepwise 5 16.0% Possession (Streak) Hierarchal 16 16.7% Possession (Logarithmic) Simultaneous 18 27.5% Possession (Logarithmic) Stepwise 4 26.6% Possession (Logarithmic) Hierarchal 16 27.4% Possession (Streak Logarithmic) Simultaneous 18 16.8% Possession (Streak Logarithmic) Stepwise 5 16.0% Possession (Streak Logarithmic) Hierarchal 16 16.7% Play Simultaneous 20 27.8% Play Stepwise 6 26.3% Play Hierarchal 18 27.0% Play (Streak) Simultaneous 20 17.2% Play (Streak) Stepwise 7 15.9% Play (Streak) Hierarchal 18 16.7% Play (Logarithmic) Simultaneous 20 27.7% Play (Logarithmic) Stepwise 6 26.2% Play (Logarithmic) Hierarchal 18 27.0% Play (Streak Logarithmic) Simultaneous 20 17.2% Play (Streak Logarithmic) Stepwise 7 15.9% Play (Streak Logarithmic) Hierarchal 18 16.6%
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the best models in terms of variance explained and parsimony. Given that these models contained an identical number of variables and explained the same amount of variance, the non-logarithmic model was selected for further explanation due to the innate difficulties associated with interpreting logarithmic variables.
The production function for the 2007-2008 possession model with stepwise entry is shown in equation 16.
(16) Score differential = 46.048 – 1.040 * Home offensive rating + 0.794 *
Away offensive rating – 0.901 * Away defensive rating + 0.684 * Home
defensive rating
Included within the above production function are all four measures of efficiency. Over the 930 observations, 26.6% of the variance in score differential can be explained by the away team‟s defensive rating, the away team‟s offensive rating, the home team‟s defensive rating, and the home team‟s offensive rating. It is critical to remember that offensive ratings and defensive rating are conceptual opposites. While an offensive rating of large magnitude indicates a highly efficient offensive team, a defensive rating of large magnitude indicates a highly inefficient defensive team. With this in mind, the signs attached to the four measures of efficiency are as expected. The positive signs associated with away offensive rating and home defensive rating correspond with movement of score differential that is in favor of the away team, while the negative signs attached to home offensive rating and away defensive rating signify a potential shift of score differential in favor of the home team.
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Focusing on the standardized coefficients allows for an examination of the relative importance of the independent variables. Interestingly, the most impactful variables in determining score differential were the offensive efficiency measures. Both home offensive rating (β = - 0.299) and away offensive rating (β = 0.224), were of greater importance than home defensive rating (β = 0.155) and away defensive rating (β = -
0.204). Based on these findings, offensive efficiency measures are more influential determinants of score differential than defensive efficiency measures. Hence, the possession model with stepwise entry for 2007-2008, serves to support the notion that teams interested in increasing the margin they defeat their opponents by should focus on increasing offensive efficiency measures as opposed to defensive efficiency measures.
However, it must be cautioned, that while producing large score differentials for the occasional game may be impressive, over the course of the season this may not translate into an increase in winning percentage. Further, it must be understood that a team‟s offensive rating is a composite measure which includes points scored as well as the team‟s field goals attempted, field goals made, free throws attempted, offensive rebounds, turnovers, and their opponents‟ defensive rebounds. Therefore, an increase / decrease in offensive rating can be accomplished in several ways.
Multiple regression analysis results for the 2008-2009 models are presented in
Table 105. Upon taking into account both variance explained, as a goodness-of-fit measure, and the number of independent variables, as an indication of parsimony, the possession model with stepwise entry and the possession model with stepwise entry
(logarithmic variables) were identified as the best models. As these models explained the
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Variance Model Entry # I.V. Explained
Absolute Simultaneous 64 27.0% Absolute Stepwise 10 22.2% Absolute Hierarchal 21 23.2% Absolute (Streak) Simultaneous 64 19.6% Absolute (Streak) Stepwise 14 15.0% Absolute (Streak) Hierarchal 24 16.0% Absolute (Logarithmic) Simultaneous 64 27.1% Absolute (Logarithmic) Stepwise 13 23.1% Absolute (Logarithmic) Hierarchal 23 23.9% Absolute (Streak Logarithmic) Simultaneous 64 19.9% Absolute (Streak Logarithmic) Stepwise 16 15.7% Absolute (Streak Logarithmic) Hierarchal 27 17.2% Relative Simultaneous 32 24.1% Relative Stepwise 9 22.6% Relative Hierarchal 19 23.3% Relative (Streak) Simultaneous 32 14.7% Relative (Streak) Stepwise 10 12.6% Relative (Streak) Hierarchal 18 13.1% Relative (Logarithmic) Simultaneous 32 24.3% Relative (Logarithmic) Stepwise 9 22.7% Relative (Logarithmic) Hierarchal 19 23.5% Relative (Streak Logarithmic) Simultaneous 32 14.6% Relative (Streak Logarithmic) Stepwise 10 12.5% Relative (Streak Logarithmic) Hierarchal 20 13.6% Possession Simultaneous 18 22.5% Possession Stepwise 6 21.7% Possession Hierarchal 16 22.4% Possession (Streak) Simultaneous 18 13.2%
Continued
Table 105. Summary of multiple regression analyses 2008-2009
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Table 105 continued
Possession (Streak) Stepwise 7 12.1% Possession (Streak) Hierarchal 16 12.8% Possession (Logarithmic) Simultaneous 18 22.4% Possession (Logarithmic) Stepwise 6 21.7% Possession (Logarithmic) Hierarchal 16 22.4% Possession (Streak Logarithmic) Simultaneous 18 13.3% Possession (Streak Logarithmic) Stepwise 7 12.1% Possession (Streak Logarithmic) Hierarchal 16 12.8% Play Simultaneous 20 22.5% Play Stepwise 7 20.8% Play Hierarchal 16 21.6% Play (Streak) Simultaneous 20 13.2% Play (Streak) Stepwise 9 11.9% Play (Streak) Hierarchal 18 12.6% Play (Logarithmic) Simultaneous 20 22.5% Play (Logarithmic) Stepwise 7 20.7% Play (Logarithmic) Hierarchal 16 21.5% Play (Streak Logarithmic) Simultaneous 20 13.3% Play (Streak Logarithmic) Stepwise 9 12.0% Play (Streak Logarithmic) Hierarchal 18 12.7%
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same amounts of variance and included the same number of variables, the non- logarithmic possession stepwise model was selected for additional discussion due to the difficulties associated with interpreting logarithmic variables.
The production function for the 2008-2009 possession model with stepwise entry is displayed in equation 17.
(17) Score differential = 15.887 – 0.835 * Home offensive rating – 0.706 *
Away defensive rating + 0.835 * Home defensive rating + 0.521 * Away
offensive rating + 2.817 * Home three days prior – 5.294 * Away four days prior
The regression equation explained 21.7% of the variance in score differential and included all four composite measures of efficiency, along with two of the scheduling fatigue measures. As was the case with the 2007-2008 possession models, multicollinearity was not an issue for the 2008-2009 possession model varieties.
In terms of relative importance, home defensive rating (β = 0.220) followed by home offensive rating (β = -0.209) were the equation‟s most influential independent variables. The fact that home team efficiency measures impacted score differential to a greater extent than away team efficiency measures serves as a testament to the influence of home court advantage. Instead of determining the value of a home court constant, the present conceptualization modifies each performance measure individually by attaching an unstandardized coefficient, which is based on the location of the game. The divergence between home team efficiency measures and away team efficiency measures is empirical evidence of the benefits of playing at home.
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The 2008-2009 possession stepwise regression equation includes two scheduling fatigue variables (i.e., home three days prior and away four days prior). Holding all other variables constant, the scenario in which the home team is entering their third game in four consecutive days will result in a 2.817 change in score differential (i.e., shift in favor of the away team). In a similar manner, in the situation where the away team is playing their fourth contest in a five day span, score differential decreases by 5.294 (i.e., movement in the direction of the home team). Thus the cumulative effect of fatigue stemming from the away team entering their fourth game in five days is worth more than five points. Over the 2008-2009 observation set, the emergence of these scheduling fatigue measures as statistically significant and valuable contributors in terms of variance explained is noteworthy, for it permits for the quantification of the detrimental impact of fatigue.
Given that the best models for the 2007-2008 and 2008-2009 NBA seasons, when taking into account score differential variance explained as well as parsimony, were of the possession variety, lends credence to offensive and defensive ratings as composite efficiency measures. As macro-level measures of performance, a team‟s offensive and defensive ratings can serve as a valuable tool for evaluation. Absolute and relative models include a diverse array of performance measures, which consequently increases the difficulty in determining importance. In the analysis of team statistics, one squad may be successful as the result of an outstanding three point field goal percentage, while another team may flourish due to their proficiency at offensive rebounding. Conversely, offensive and defensive ratings provide an indication of success with merely two
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variables. With access to offensive and defensive ratings, an individual can easily compare efficiency levels of competing squads. As these composite measures gain further acceptance, members of the journalism and broadcasting communities can easily inform the public as to which teams are more efficient on the offensive and defensive ends of the floor. Moreover, the success of the possession model serves to support its use for research within the academic community. As noted previously, basketball production function research has solely relied on absolute and relative conceptualizations.
2) At what rate of success can the absolute model, the relative model, the possession
model, and the play model classify individual game winners for the 2007-2008
and 2008-2009 NBA seasons?
A summary of results for the numerous logistic regression analyses over the 2007-
2008 NBA season are provided in Table 106. Accessing overall fit for logistic regression equations can be accomplished through a variety of approaches, including statistical measures, pseudo R2 measures and classification accuracy (Hair et al., 2006). It must be noted that while these different approaches tend to yield similar conclusions, logistic regression analyses utilize maximum likelihood for estimation, and measures fit with the
-2 log likelihood measure (Hair et al.). Although the inclusion of additional variables will generally reduce the value of the logistic equation‟s -2LL, it does not necessarily increase its classification accuracy, which is of primary concern for the present investigation. Therefore, in some cases, an equation containing fewer independent
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variables will produce a more successful hit ratio than a similar equation with a greater number of independent variables.
In taking into account parsimony along with classification success, the play model with stepwise entry, which included seven independent variables and correctly predicted
72.4% of game outcomes, was identified as the best model for the 2007-2008 NBA season. Estimating the probability of a home team victory for the 2007-2008 NBA season can be found through the following equation, 1 / (1 + e –z), where z for the play model with stepwise entry, is shown in equation 18.
(18) z = -13.802 – 7.422 * Away offensive points-per-play + 18.278 * Away
defensive points-per-play + 0.217 * Home offensive plays – 0.202 * Home
defensive plays + 17.112 * Home offensive points-per-play – 14.404 * Home
defensive points-per-play + 0.380 * Home one day prior (variable two)
All four play model efficiency point-per-play measures are included within the equation, as well as the both home pace measures and one scheduling fatigue variable. By entering the numerous independent variables into the equation, the probability of a home team victory can be ascertained. If the calculated probability exceeds a value 0.50, the equation predicts that the home team will emerge victorious. When the probability of an away team win is desired, the probability of the home team is simply subtracted from one. In terms of the sheer magnitude of the coefficients, away defensive points-per-play had the greatest influence on predicting which team is victorious in an “un-played” contest. However, it must be noted that these coefficients are not standardized; thus any
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Model Entry # I.V. Hit Ratio
Absolute Simultaneous 64 73.4% Absolute Stepwise 16 70.9% Absolute Hierarchal 28 72.5% Absolute (Streak) Simultaneous 64 69.9% Absolute (Streak) Stepwise 19 68.3% Absolute (Streak) Hierarchal 30 70.1% Relative Simultaneous 32 71.8% Relative Stepwise 12 70.2% Relative Hierarchal 22 71.5% Relative (Streak) Simultaneous 32 68.1% Relative (Streak) Stepwise 7 66.2% Relative (Streak) Hierarchal 17 66.7% Possession Simultaneous 18 70.6% Possession Stepwise 5 70.5% Possession Hierarchal 16 71.0% Possession (Streak) Simultaneous 18 66.3% Possession (Streak) Stepwise 5 66.3% Possession (Streak) Hierarchal 16 66.9% Play Simultaneous 20 71.2% Play Stepwise 7 72.4% Play Hierarchal 18 72.3% Play (Streak) Simultaneous 20 66.5% Play (Streak) Stepwise 9 66.4% Play (Streak) Hierarchal 20 66.5%
Table 106. Summary of logistic regression analyses 2007-2008
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comparisons between coefficients should be done with caution for efficiency variables, pace variables, and scheduling fatigue variables operate on different scales.
Interestingly, the inclusion of both home pace measures (i.e., home offensive plays and home defensive plays) supports the idea that a game‟s outcome is more contingent upon the home team‟s speed of play than the away team‟s speed of play.
Taking these findings a step further lends credence to the notion that the act of controlling the speed of a game is often easier for the home team than the away team.
Members of the media and the coaching communities frequently mention that dictating pace is more easily accomplished on a team‟s home floor, which coincides with the empirical findings of the play stepwise entry model.
Table 107 provides the summary of the various logistic regression analyses carried out with the 2008-2009 NBA season data. Upon taking into account the number of independent variables within the respective models, as well as classification accuracy, the possession model with stepwise entry was identified as the best model for the 2008-
2009 NBA season. With six independent variables, the possession model with stepwise entry accurately predicted 71.4% of game outcomes.
The probability of a home team victory for the 2008-2009 NBA season can be estimated through the following equation, 1 / (1 + e –z), where z for the possession model with stepwise entry, is displayed in equation 19.
(19) z = -1.393 – 0.105 * Away offensive rating + 0.132 * Away defensive
rating + 0.727 * Away four days prior + 0.155 * Home offensive rating – 0.161 *
Home defensive rating – 0.589 * Home three days prior
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Model Entry # I.V. Hit Ratio
Absolute Simultaneous 64 72.9% Absolute Stepwise 17 72.6% Absolute Hierarchal 26 73.3% Absolute (Streak) Simultaneous 64 68.6% Absolute (Streak) Stepwise 20 68.1% Absolute (Streak) Hierarchal 27 68.5% Relative Simultaneous 32 71.4% Relative Stepwise 12 71.1% Relative Hierarchal 20 71.3% Relative (Streak) Simultaneous 32 66.1% Relative (Streak) Stepwise 12 66.0% Relative (Streak) Hierarchal 21 66.6% Possession Simultaneous 18 72.3% Possession Stepwise 6 71.4% Possession Hierarchal 16 72.2% Possession (Streak) Simultaneous 18 64.8% Possession (Streak) Stepwise 9 64.8% Possession (Streak) Hierarchal 17 64.0% Play Simultaneous 20 72.5% Play Stepwise 11 70.8% Play Hierarchal 18 71.2% Play (Streak) Simultaneous 20 65.1% Play (Streak) Stepwise 12 65.4% Play (Streak) Hierarchal 20 65.1%
Table 107. Summary of logistic regression analyses 2008-2009
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When the calculated probability of a home team victory exceeds 0.50, the equation predicts a win for the home squad. Conversely, calculated probabilities valued at less than 0.50 correspond with the prediction of an away team victory. Included within the equation are all four efficiency measures, as well as two scheduling fatigue variables.
Examining the logistic regression coefficients of the four efficiency measures reveals, that on the basis of magnitude, home efficiency measures (i.e., home offensive rating and home defensive rating) are of greater importance than away efficiency measures (i.e., away offensive rating and away defensive rating), when predicting a game‟s outcome.
The difference in coefficients between home and away efficiency measures empirically supports the influence of game location. In addition, the inclusion of the two scheduling fatigue measures (i.e., away four days prior and home three days prior) is empirical evidence of the detrimental impact of fatigue. In comparing the coefficients of the scheduling fatigue variables, the fact that the away four days prior variable had a greater influence on game outcome than the home three days prior variable is logical. Naturally, a team competing in their fourth game in five days should experience greater fatigue than a team playing their third game in four days.
In terms of predicting game outcomes the best models for 2007-2008 and 2008-
2009 seasons were the play stepwise entry model and the possession stepwise entry model respectively. While these models differ in the manner their pace and efficiency measures were operationalized, it must be noted that both are similar in their inclusion of composite measures. The success of these composite pace and efficiency measures in predicting NBA game outcomes is yet another step forward in their overall acceptance.
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In using these equations, an individual could simply enter the away and home team statistics and determine each team‟s probability of victory. Based on the knowledge that a team‟s probability of achieving victory was miniscule, a head coach could alter game strategies with the intention of increasing the likelihood of winning. Through the adoption of less conventional tactics (e.g., in the NBA, the employment of a zone defense), an inferior team could present problems for a superior foe. While this notion of using a riskier strategy to topple a better opponent is hardly revolutionary, the utilization of the aforementioned equations can be a guide to determine the efficiency benchmarks necessary to complete the upset.
3) Do streak models predict the dependent variable (i.e., score differential or game
winner) better than their non-streak counterparts?
As shown in Table 104 and Table 105, the amount of variance in score differential explained by each streak model equation is substantially less than its respective non-streak counterpart. Streak models differ from non-streak models both in the manner performance measures are operationalized (i.e., performance statistics for streak models are computed over each team‟s previous five games, whereas performance statistics for non-streak models are calculated over all of each team‟s previous games), and the number of observations responsible for creating the equation (i.e., streak models were constructed from 1105 games, whereas non-streak models were constructed from
930 games). Differences in explained variance between streak and non-streak model
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equations varied based on the model conceptualization, along with the variable entry method. In terms of the amount of variance explained in score differential, the largest disparities between streak and non-streak model equations exceeded 10.0% in some cases. For example, over the 2007-2008 NBA season, the variance explained for the relative model with simultaneous entry, stepwise entry and hierarchal entry equaled
27.6%, 25.9%, and 26.6% respectively; while the variance explained for the relative streak model with simultaneous entry, stepwise entry and hierarchal entry was 17.4%,
14.0%, and 15.8% respectively.
Examining the logistic regression results displayed in Table 106 and Table 107 tells a similar story in regards to the superior predictive capabilities of non-streak models.
However, the difference in hit ratios for classifying game outcomes is less pronounced than the divergence in variance explained for score differential. As noted previously, logistic regression analyses contain several goodness-of-fit measures, where classification accuracy does not necessarily improve with the addition of variables.
Typically, over the two year period, the difference in prediction accuracy between a model and its respective streak counterpart is several percentage points, and at the extremes exceeded 5%. For example, over the 2008-2009 NBA season, the hit ratio for the play model with simultaneous entry, stepwise entry and hierarchal entry equaled
72.5%, 70.8%, and 71.2% respectively; while the hit ratio for the play streak model with simultaneous entry, stepwise entry and hierarchal entry was 65.1%, 65.4%, and 65.1% respectively.
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Based on these findings, it is clear that performance measures accumulated over the previous five games are not as effective at predicting score differentials or game outcomes as performance measures amassed over all previous played games.
Researchers in the future are advised to avoid constructing streak models in favor of models that encapsulate performance measures from all previous contests. It is hypothesized, that while the utilization of streak models can be effective under certain circumstances, in the majority of situations the streak statistics become over-dependent upon the last five opponents. For example, if over a span of five games a team faces the top five teams in the league, their streak performance measures will fail to represent the true abilities of the team in question. A similar problem arises when a team plays five bottom rung squads over the previous five games. While a five game set allows for the representation of recent surge or lulls in play, it is simultaneously too small of a sample and thus does not permit statistics to converge to their actual performance levels. When a performance measure converges slowly, it inevitably fails to be representative of the team‟s true ability in the area in question. Evaluating performance with streak measures can be in deceiving, as streak statistics may not necessarily be completely indicative of team proficiencies.
4) How are NBA game score differentials associated with the competing teams‟
average number of contributing players and fatigue stemming from playing and/or
traveling in the preceding days?
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A two stage hierarchal regression analysis was used to assess the influence of scheduling fatigue measures and team depth contribution variables. A comparison was made between the amount of variance explained in score differential between the first stage, which entered performance statistics in a stepwise manner, and the second stage, which simultaneously entered the present study‟s 12 unique variables. While the impact of these unique variables can also be explored in regards to the classification of game outcomes (i.e., through logistic regression analysis), as noted previously, the hit ratio is not as straightforward as the variance explained measures of multiple linear regression.
The hit ratio is one of several goodness-of-fit measures for logistic regression and while the addition of relevant variables will reduce a model‟s -2LL value, it will not necessarily increase its classification accuracy, which inevitably muddles the present interpretation.
For example, over the 2008-2009 NBA season, entering the scheduling fatigue and team depth variables increased the hit ratio for the absolute streak model from 67.4% to 68.5%, whereas entering the variables for the play streak model decreased the hit ratio from
65.2% to 65.1%.
A summary of the second stage of hierarchal entry for the numerous multiple regression models are presented in Table 108. The additional variance explained by the present study‟s unique variables fluctuated based on the season as well as the model.
Additional variance explained ranged from as low as 0.7% in the relative model for 2007-
2008, to as high as 2.9% in the possession streak model for 2008-2009. Based on these results scheduling fatigue and team depth contribution had a greater influence on score differential for the 2008-2009 NBA season. The difference in the magnitude of
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2007-2008 2008-2009 Additional Statistically Additional Statistically variance significant variance significant Model explained variables explained variables A4DP, Absolute 0.9% A1DPV1 2.3% H3DP ACONA, A1DPV1, A4DP, Absolute (streak) 1.4% H3DP 2.8% H3DP ACONA, A4DP, Absolute (logarithmic) 1.4% A1DPV1 2.2% H3DP ACONA, A1DPV1, A4DP, Absolute (streak logarithmic) 1.3% H3DP 2.7% H3DP A4DP, Relative 0.7% A1DPV1 2.2% H3DP ACONA, A4DP, A1DPV1, Relative (streak) 1.2% A1DPV1 2.7% H3DP A4DP, Relative (logarithmic) 0.7% A1DPV1 2.2% H3DP ACONA, A4DP, A1DPV1, Relative (streak logarithmic) 1.2% A1DPV1 2.7% H3DP A4DP, Possession 0.8% A1DPV1 2.3% H3DP ACONA, A4DP, Possession (streak) 1.2% A1DPV1 2.9% H3DP
Continued
Table 108. Summary of hierarchal multiple regression analyses
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Table 108 continued
A4DP, Possession (logarithmic) 0.8% A1DPV1 2.2% H3DP ACONA, A4DP, Possession (streak logarithmic) 1.2% A1DPV1 2.8% H3DP ACONA, A4DP, Play 0.8% A1DPV1 2.7% H3DP ACONA, A4DP, Play (streak 1.2% A1DPV1 2.8% H3DP ACONA, A4DP, Play (logarithmic) 0.7% A1DPV1 2.7% H3DP ACONA, A4DP, Play (streak logarithmic) 1.2% A1DPV1 2.8% H3DP
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additional variance explained as a result the inclusion of the unique variables, illustrates the difference between the two populations and is a blow to the generalizability of the models.
While the majority of the independent variables added in the second stage of the hierarchal multiple regression analyses failed to reach statistical significance, several of the scheduling fatigue variables consistently reached significant status. More specifically, away one day prior (variable one) was statistically significant at the 0.05 level for all 2007-2008 models, and away four days prior and home three days prior reached significance for all 2008-2009 models. Interestingly, statistical significance of the scheduling fatigue variables varied substantially between the two populations. The lack of consistency in regards to the unique variables that reach statistical significance is additional evidence of the lack of generalizability between the two seasons.
As a whole, the increase in variance explained as a result of entering the current study‟s unique variables empirically supports the influence of scheduling fatigue and depth contribution factors within the National Basketball Association. While the magnitudes vary by regression equation, coefficients attached to the scheduling fatigue measures reveal the detrimental effect of fatigue. Researchers in the future are encouraged to examine the impact of these scheduling fatigue variables for other NBA seasons as well as other professional leagues (e.g., Euroleague Basketball or the
Women‟s National Basketball Association). Based on these findings, individuals in NBA front office positions would be wise to reexamine travel procedures, in an effort to counteract the detrimental impact of playing game in close succession. From the
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coaching perspective, altering player rotations in an effort to proactively battle the cumulative effect of scheduling fatigue could be beneficial over the duration of the regular season.
5) In terms of relative importance and variable inclusion, are regression equations
consistent between the 2007-2008 and 2008-2009 NBA season?
Stability of regression equations between the two populations is explored by comparing the relative contribution comparisons of the independent variables. Based on the absence of an accepted standardization procedure for logistic regression analysis, the consistency between the two NBA seasons will be explored solely for stepwise multiple regression based models.
As anticipated, the presence of multicollinearity within the absolute models makes relative importance of performance measures from one year to the next incredibly unstable. For example, the three most important contributors for the absolute model with logarithmic variables over the 2007-2008 NBA season was, home defensive rebounds allowed (β = 0.237), away steals (β = 0.192), and away assists allowed (β = -0.190); while the three most influential predictors of the same model over the 2008-2009 NBA season was home personal fouls allowed (β = -0.246), home personal fouls (β = 0.242), and home assists allowed (β = 0.194). None of the top three most impactful variables from either season appears within the top three of the other season. In fact, of the six variables, only away steals, is selected for entry within the regression equation of both
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seasons. A comparison of the most influential predictors within the other absolute regression equations further supports the lack of consistency that plagues the absolute models. Therefore, absolute models under the present conceptualization, serve little purpose.
With fewer concerns of multicollinearity, the relative models are slightly more consistent over the two NBA seasons. The most influential predictors for the relative model over the 2007-2008 season were home two point field goal ratio (β = -0.300), away two point field goal ratio (β = 0.237), and home offensive rebound ratio (β = -
0.155), while the most impactful independent variables for the same model over the
2008-2009 season were home defensive rebound ratio (β = -0.169), away two point field goal ratio (β = 0.164), and home personal foul ratio (β = 0.162). While the magnitudes of the standardized coefficients differ between the two regression equations, both identify away two point field goal ratio as one of the critical predictors of score differential.
Furthermore, away two point field goal ratio appears as one of the three most impactful independent variables for all relative models. Unfortunately, the independent variables selected for entry into the various relative model regression equations varies substantially. Taken as a whole, the relative models are not overly useful as currently conceptualized due to the absence of consistency in regards to the included regression equation variables.
Constructed with four measures of efficiency, possession models are the most stable of the four model categories. Regardless of whether performance measures are streak or non-streak, logarithmic or non-logarithmic, each of the stepwise entry
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possession model regression equations include away defensive rating, away offensive rating, home defensive rating, and home offensive rating, as independent variables. In terms of relative importance, the three most impactful contributors for the possession model over 2007-2008 were home offensive rating (β = -0.299), away offensive rating (β
= 0.224), and away defensive rating (β = -0.204); while the most important predictors for the same model over the 2008-2009 season were home defensive rating (β = 0.220), home offensive rating (β = -0.209), and away defensive rating (β = -0.191). While the relative contribution order of importance differs between the two regression equations, both include home offensive rating and away defensive rating among the most influential contributors. In fact, home offensive rating was one of the most important independent variables for all stepwise entry possession models. The inclusion of composite measures of efficiency, simplifies the production function, which consequently increases stability.
In adopting the perspective of a prognosticator, the consistency of the possession model and its corresponding offensive and defensive efficiency measures, is appealing.
Stability, in regards to the prediction of variance in score differential, has potentially lucrative implications for individuals involved in the activity of wagering on sport. More specifically, these composite measures of efficiency can serve as a valuable tool capable of facilitating an increase in the accuracy of predicting National Basketball Association contests.
As an alternative pace and efficiency based conceptualization, play models exhibit a level of consistency between the relative models and the possession models. As a measure of offensive efficiency, home offensive points-per-play is among the most
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influential independent variables for all play models. For example, the three most impactful predictors for the play model with stepwise entry over the 2007-2008 NBA season were home defensive plays (β = 0.418), home offensive plays (β = -0.384), and home offensive points-per-play (β = -0.372); while the three most influential independent variables for the same model over the 2008-2009 season were home defensive points-per- play (β = 0.236), home offensive points-per-play (β = -0.177), and away defensive points- per-play (β = -0.174). In fact, the four point-per-play efficiency measures were selected for entry for all play model regression equations, except for the 2008-2009 play model
(streak variables) and the 2008-2009 play model (streak logarithmic variables), both of which failed to include away offensive points-per-play. As a whole, the inconsistent inclusion of pace measures, which appear in several of the regression equations and not in others, raises concerns regarding stability.
Limitations
Upon conducting a research investigation it is pivotal that the researcher acknowledge the presence of limitations. As the current investigation attempts to predict game outcomes through the construction of four distinct models (i.e., absolute, relative, possession and play), it must be noted that the included independent variables are hardly all-inclusive. As a whole, team success is composed of an innumerable number of determinants, many of which are qualitative (e.g., player attitudes, leadership capabilities and team chemistry; Sánchez et al., 2007). Of the quantitative variety, the present study has disregarded measures of coaching productivity (e.g., allocation of minutes in relation to individual performance measures; Clement & McCormick, 1989) as well as the
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possibility that statistics from a previous game between the two teams are prophetic of a future encounter.
A second limitation pertains specifically to the streak models and their potentially problematic relationship with injuries. For example, if a star player is injured for a five game period, the streak variables will portray the injury‟s effect on the team in upcoming games. However, upon returning, the streak variables will be indicative of the team‟s performance without the star player, even though his return is imminent. Similar problems arise with suspensions and minor injuries, both of which inescapably influence a team‟s streak statistics. Although streak performance measures accurately account for some situations, they fail to account for others, and therefore are not without their drawbacks.
A third limitation involves the operationalization of scheduling fatigue variables.
While the scheduling fatigue variables capture the potentially detrimental effect of games occurring in close succession, they fail to account for the possibility of additional fatigue due to variations in travel distance between contests. The crossing of multiple time zones can desynchronize circadian rhythms, which consequentially negatively affects performance (Reilly, 2009). Unfortunately, without detailed team travel log information, the present study is forced to take a rather simplistic approach. Furthermore, one day prior scheduling fatigue variables do not account for differences in the level of competition in the first game of back-to-back contests. For example, it would be expected that the negative effect of fatigue from a triple overtime thriller would be significantly greater than a game that starters played limited minutes, which ended in a
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blowout. Under the present conceptualization, fatigue from previous games is assumed to have equal effect, regardless of the previous game‟s happenings.
Conclusion
The purpose of the current study was to explore the relationship between numerous input variables (i.e., previously accumulated performance statistics, scheduling fatigue measures, and team depth contribution factors) and the outcome (i.e., score differential or game winner) of an “un-played” National Basketball Association game. A multitude of models were constructed under four general classifications (i.e., absolute models, relative models, possession models, and play models). Models can be further segmented in those that include streak performance measures and those that operationalize performance statistics over all previously played games, and those that use logarithmic performance variables and those that include non-logarithmic performance measures.
Over the 2007-2008 and 2008-2009 NBA seasons, the possession model with stepwise entry was identified as the best model in terms of explaining variance in score differential. Over the 2007-2008 season, the play model with stepwise entry was deemed the best model in terms of classifying game outcomes, whereas the possession model with stepwise entry was chosen as the best model in regards to predicting game winner for the 2008-2009 campaign. As a whole, streak models were not as successful as non- streak models at explaining variance in score differential or predicting game winners.
Additionally, while the inclusion of scheduling fatigue variables and team depth contribution factors increased the variance explained in score differential, the amount
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fluctuated by both model and by season, as did the variables that reached statistical significance.
In summary, the results of the present investigation illustrate a lack of generalizability, as the derived regression equations differ from one season to the next.
Thus, these results demonstrate that utility of the regression equations is minimal.
However, the overall success in terms of goodness-of-fit measures for the pace and efficiency based models is noteworthy. More specifically, over the 2007-2008 and 2008-
2009 NBA seasons, the possession models not only were among the best models, they exhibited the highest levels of stability. Composite measures of pace and efficiency can facilitate team evaluations on the macro-level, which has implications for the members in management as well as the media and academic communities.
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Appendix A: Descriptive statistics for absolute model 2007-2008 NBA season
Mean SD Minimum Maximum A2FGMA 30.424 1.743 26.348 36.154 A2FGAA 63.301 3.822 52.480 74.185 A3PMA 6.398 1.522 3.523 9.761 A3PAA 17.869 3.755 10.633 27.783 AFTMA 18.983 2.121 14.333 23.952 AFTAA 25.179 2.736 18.143 31.409 AOREBA 11.256 1.271 8.353 14.265 ADREBA 30.801 1.473 28.000 35.318 AASTA 21.531 2.038 16.741 27.933 ASTLA 7.375 0.975 5.469 10.227 ABLKA 4.798 0.851 2.191 7.455 ATOA 13.993 1.295 10.875 16.636 APFA 21.322 1.522 18.143 25.350 A2FGMAA 30.408 1.788 25.500 35.783 A2FGAAA 63.177 3.364 56.476 73.974 A3PMAA 6.435 0.796 4.840 8.333 A3PAAA 17.976 1.759 13.795 22.810 AFTMAA 18.947 2.201 14.615 25.192 AFTAAA 25.142 2.841 18.933 32.400 AOREBAA 11.241 1.001 8.800 14.431 ADREBAA 30.810 1.724 27.046 35.048 AASTAA 21.569 1.680 16.885 25.551 ASTLAA 7.371 0.887 5.297 9.429 ABLKAA 4.820 0.673 3.400 6.297 ATOAA 13.999 1.245 11.644 18.364 APFAA 21.312 1.611 17.400 25.273 ACONA 8.220 0.343 7.194 9.096 A4DP 0.048 0.215 0.000 1.000 A3DP 0.378 0.485 0.000 1.000
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A2DP 0.833 0.373 0.000 1.000 A1DPV1 0.132 0.339 0.000 1.000 A1DPV2 0.198 0.399 0.000 1.000 H2FGMA 30.420 1.773 26.000 36.429 H2FGAA 63.197 3.879 52.292 74.088 H3PMA 6.428 1.553 3.521 9.833 H3PAA 17.947 3.814 10.655 27.818 HFTMA 18.940 2.088 14.600 23.947 HFTAA 25.145 2.703 18.636 31.619 HOREBA 11.228 1.300 8.327 14.333 HDREBA 30.784 1.479 27.790 35.238 HASTA 21.535 2.084 16.684 28.091 HSTLA 7.364 0.976 5.415 10.286 HBLKA 4.803 0.844 2.220 7.395 HTOA 13.986 1.275 10.889 16.600 HPFA 21.309 1.524 18.282 25.286 H2FGMAA 30.427 1.821 25.708 35.773 H2FGAAA 63.218 3.453 57.091 74.020 H3PMAA 6.426 0.795 4.917 8.550 H3PAAA 17.930 1.758 13.909 23.200 HFTMAA 18.976 2.175 14.604 25.148 HFTAAA 25.180 2.822 19.000 32.524 HOREBAA 11.238 1.026 8.885 14.544 HDREBAA 30.795 1.709 26.857 35.050 HASTAA 21.542 1.684 17.000 25.688 HSTLAA 7.369 0.880 5.324 9.500 HBLKAA 4.803 0.682 3.429 6.364 HTOAA 13.975 1.249 11.593 18.429 HPFAA 21.310 1.602 17.444 25.211 HCONA 8.214 0.336 7.200 9.078 H4DP 0.013 0.113 0.000 1.000 H3DP 0.270 0.444 0.000 1.000 H2DP 0.740 0.439 0.000 1.000
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H1DPV1 0.008 0.086 0.000 1.000 H1DPV2 0.139 0.346 0.000 1.000 Away Final Score 98.443 12.755 54.000 151.000 Home Final Score 102.006 13.045 65.000 168.000
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Appendix B: Descriptive statistics for absolute model 2008-2009 NBA season
Mean SD Minimum Maximum A2FGMA 30.220 1.937 25.364 34.434 A2FGAA 62.625 4.180 51.594 72.125 A3PMA 6.558 1.535 3.581 11.050 A3PAA 18.064 3.688 9.739 29.850 AFTMA 19.097 1.708 14.676 24.000 AFTAA 24.799 2.318 19.432 31.489 AOREBA 11.078 1.120 8.216 14.273 ADREBA 30.323 1.430 26.682 34.250 AASTA 20.893 1.488 16.095 25.046 ASTLA 7.328 0.800 4.826 10.000 ABLKA 4.859 0.874 1.800 7.048 ATOA 13.732 1.264 11.150 16.227 APFA 21.179 1.491 18.256 25.696 A2FGMAA 30.221 2.044 26.077 37.250 A2FGAAA 62.665 3.574 55.115 71.600 A3PMAA 6.550 0.824 4.550 8.667 A3PAAA 18.038 1.780 14.226 21.904 AFTMAA 19.068 2.023 14.590 24.800 AFTAAA 24.774 2.585 18.949 32.174 AOREBAA 11.088 1.071 8.714 14.950 ADREBAA 30.310 1.820 25.905 35.952 AASTAA 20.850 1.733 17.480 25.150 ASTLAA 7.325 0.800 5.580 8.952 ABLKAA 4.866 0.751 3.217 6.818 ATOAA 13.734 1.115 11.173 16.276 APFAA 21.195 1.408 18.238 25.000 ACONA 8.298 0.362 7.188 9.368 A4DP 0.042 0.201 0.000 1.000
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A3DP 0.342 0.475 0.000 1.000 A2DP 0.819 0.385 0.000 1.000 A1DPV1 0.140 0.347 0.000 1.000 A1DPV2 0.172 0.378 0.000 1.000 H2FGMA 30.214 1.907 25.389 34.407 H2FGAA 62.656 4.148 51.557 72.318 H3PMA 6.571 1.512 3.524 10.680 H3PAA 18.060 3.632 9.238 29.840 HFTMA 19.121 1.721 14.629 24.455 HFTAA 24.812 2.310 19.389 31.727 HOREBA 11.080 1.121 8.250 14.381 HDREBA 30.275 1.421 26.957 34.500 HASTA 20.838 1.468 16.100 24.973 HSTLA 7.315 0.787 4.846 9.762 HBLKA 4.855 0.848 2.036 6.917 HTOA 13.685 1.249 11.124 16.191 HPFA 21.193 1.492 18.286 25.708 H2FGMAA 30.240 2.007 26.222 36.704 H2FGAAA 62.655 3.522 55.200 70.760 H3PMAA 6.567 0.825 4.781 8.700 H3PAAA 18.048 1.773 14.216 22.222 HFTMAA 19.131 2.065 14.692 24.875 HFTAAA 24.823 2.628 19.098 32.208 HOREBAA 11.085 1.093 8.862 15.048 HDREBAA 30.313 1.795 25.950 35.593 HASTAA 20.918 1.694 17.500 25.095 HSTLAA 7.298 0.807 5.522 9.143 HBLKAA 4.843 0.742 3.125 6.647 HTOAA 13.707 1.108 11.188 16.125 HPFAA 21.182 1.383 18.282 25.091 HCONA 8.305 0.361 7.179 9.300 H4DP 0.016 0.126 0.000 1.000 H3DP 0.266 0.442 0.000 1.000
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H2DP 0.728 0.445 0.000 1.000 H1DPV1 0.011 0.103 0.000 1.000 H1DPV2 0.142 0.349 0.000 1.000 Away Final Score 98.739 12.606 66.000 154.000 Home Final Score 102.243 12.706 63.000 144.000
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Appendix C: Descriptive statistics for relative model 2007-2008 NBA season
Mean SD Minimum Maximum A2FGR 1.001 .058 .878 1.165 A3PR .999 .086 .809 1.271 AFTR 1.002 .045 .880 1.122 AOREBR 1.010 .145 .579 1.431 ADREBR 1.002 .057 .854 1.193 AASTR 1.006 .139 .756 1.398 ASTLR 1.014 .174 .588 1.519 ABLKR 1.021 .255 .376 1.901 ATOR 1.004 .104 .776 1.279 APFR 1.005 .089 .840 1.347 ACONR 1.000 .040 .869 1.085 A4DP .048 .215 .000 1.000 A3DP .378 .485 .000 1.000 A2DP .833 .373 .000 1.000 A1DPV1 .132 .339 .000 1.000 A1DPV2 .198 .399 .000 1.000 H2FGR 1.002 .059 .874 1.161 H3PR .998 .086 .820 1.265 HFTR 1.000 .045 .891 1.119 HOREBR 1.008 .151 .580 1.451 HDREBR 1.002 .057 .867 1.220 HASTR 1.008 .143 .756 1.401 HSTLR 1.013 .175 .594 1.538 HBLKR 1.027 .262 .394 1.933 HTOR 1.006 .104 .776 1.272 HPFR 1.004 .087 .828 1.360 HCONR 1.000 .039 .866 1.084 H4DP .013 .113 .000 1.000
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H3DP .270 .444 .000 1.000 H2DP .740 .439 .000 1.000 H1DPV1 .008 .086 .000 1.000 H1DPV2 .139 .346 .000 1.000 Score Differential -3.563 13.648 -52.000 42.000
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Appendix D: Descriptive statistics for relative model 2008-2009 NBA season
Mean SD Minimum Maximum A2FGR 1.003 0.063 0.866 1.218 A3PR 1.000 0.093 0.703 1.205 AFTR 1.002 0.039 0.930 1.110 AOREBR 1.007 0.130 0.667 1.476 ADREBR 1.004 0.070 0.884 1.219 AASTR 1.009 0.114 0.765 1.407 ASTLR 1.010 0.137 0.633 1.352 ABLKR 1.027 0.256 0.355 1.878 ATOR 1.003 0.091 0.779 1.347 APFR 1.001 0.069 0.834 1.166 ACONR 1.000 0.047 0.870 1.156 A4DP 0.042 0.201 0.000 1.000 A3DP 0.342 0.475 0.000 1.000 A2DP 0.819 0.385 0.000 1.000 A1DPV1 0.140 0.347 0.000 1.000 A1DPV2 0.172 0.378 0.000 1.000 H2FGR 1.002 0.061 0.858 1.223 H3PR 1.001 0.092 0.680 1.198 HFTR 1.001 0.039 0.929 1.107 HOREBR 1.008 0.134 0.661 1.456 HDREBR 1.002 0.069 0.884 1.216 HASTR 1.003 0.112 0.754 1.408 HSTLR 1.012 0.135 0.629 1.331 HBLKR 1.030 0.251 0.386 1.870 HTOR 1.002 0.092 0.769 1.349 HPFR 1.002 0.068 0.841 1.175 HCONR 1.001 0.047 0.870 1.141 H4DP 0.016 0.126 0.000 1.000
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H3DP 0.266 0.442 0.000 1.000 H2DP 0.728 0.445 0.000 1.000 H1DPV1 0.011 0.103 0.000 1.000 H1DPV2 0.142 0.349 0.000 1.000 Score Differential -3.504 12.907 -48.000 45.000
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Appendix E: Descriptive statistics for possession model 2007-2008 NBA season
Mean SD Minimum Maximum APOSA 92.529 3.181 86.814 101.142 AOR 107.020 3.850 98.971 114.806 ADR 107.055 3.085 94.763 113.486 ACONA 8.220 0.343 7.194 9.096 A4DP 0.048 0.215 0.000 1.000 A3DP 0.378 0.485 0.000 1.000 A2DP 0.833 0.373 0.000 1.000 A1DPV1 0.132 0.339 0.000 1.000 A1DPV2 0.198 0.399 0.000 1.000 HPOSA 92.518 3.148 86.793 101.393 HOR 107.071 3.922 97.373 114.901 HDR 107.108 3.096 94.976 113.245 HCONA 8.214 0.336 7.200 9.078 H4DP 0.013 0.113 0.000 1.000 H3DP 0.270 0.444 0.000 1.000 H2DP 0.740 0.439 0.000 1.000 H1DPV1 0.008 0.086 0.000 1.000 H1DPV2 0.139 0.346 0.000 1.000 Score Differential -3.563 13.648 -52.000 42.000
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Appendix F: Descriptive statistics for possession model 2008-2009 NBA season
Mean SD Minimum Maximum APOSA 91.761 2.945 86.021 99.171 AOR 108.130 3.268 98.820 115.850 ADR 108.032 3.486 98.428 115.750 ACONA 8.298 0.362 7.188 9.368 A4DP 0.042 0.201 0.000 1.000 A3DP 0.342 0.475 0.000 1.000 A2DP 0.819 0.385 0.000 1.000 A1DPV1 0.140 0.347 0.000 1.000 A1DPV2 0.172 0.378 0.000 1.000 HPOSA 91.747 2.975 85.939 99.246 HOR 108.207 3.234 98.144 115.671 HDR 108.214 3.397 98.454 115.607 HCONA 8.305 0.361 7.179 9.300 H4DP 0.016 0.126 0.000 1.000 H3DP 0.266 0.442 0.000 1.000 H2DP 0.728 0.445 0.000 1.000 H1DPV1 0.011 0.103 0.000 1.000 H1DPV2 0.142 0.349 0.000 1.000 Score Differential -3.504 12.907 -48.000 45.000
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Appendix G: Descriptive statistics for play model 2007-2008 NBA season
Mean SD Minimum Maximum AOPLA 105.233 3.494 99.085 114.276 ADPLA 105.208 3.906 98.704 115.518 AOPPP 0.941 0.041 0.836 1.037 ADPPP 0.942 0.028 0.837 1.004 ACONA 8.220 0.343 7.194 9.096 A4DP 0.048 0.215 0.000 1.000 A3DP 0.378 0.485 0.000 1.000 A2DP 0.833 0.373 0.000 1.000 A1DPV1 0.132 0.339 0.000 1.000 A1DPV2 0.198 0.399 0.000 1.000 HOPLA 105.188 3.412 99.028 114.330 HDPLA 105.196 3.896 98.645 115.876 HOPPP 0.942 0.042 0.822 1.036 HDPPP 0.942 0.028 0.837 0.999 HCONA 8.214 0.336 7.200 9.078 H4DP 0.013 0.113 0.000 1.000 H3DP 0.270 0.444 0.000 1.000 H2DP 0.740 0.439 0.000 1.000 H1DPV1 0.008 0.086 0.000 1.000 H1DPV2 0.139 0.346 0.000 1.000 Score Differential -3.563 13.648 -52.000 42.000
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Appendix H: Descriptive statistics for play model 2008-2009 NBA season
Mean SD Minimum Maximum AOPLA 104.341 3.582 97.241 114.362 ADPLA 104.347 3.699 97.041 115.469 AOPPP 0.951 0.033 0.866 1.018 ADPPP 0.950 0.031 0.872 1.012 ACONA 8.298 0.362 7.188 9.368 A4DP 0.042 0.201 0.000 1.000 A3DP 0.342 0.475 0.000 1.000 A2DP 0.819 0.385 0.000 1.000 A1DPV1 0.140 0.347 0.000 1.000 A1DPV2 0.172 0.378 0.000 1.000 HOPLA 104.327 3.578 97.119 114.000 HDPLA 104.340 3.748 97.248 115.056 HOPPP 0.952 0.032 0.860 1.017 HDPPP 0.952 0.030 0.870 1.011 HCONA 8.305 0.361 7.179 9.300 H4DP 0.016 0.126 0.000 1.000 H3DP 0.266 0.442 0.000 1.000 H2DP 0.728 0.445 0.000 1.000 H1DPV1 0.011 0.103 0.000 1.000 H1DPV2 0.142 0.349 0.000 1.000 Score Differential -3.504 12.907 -48.000 45.000
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Appendix I: Goodness-of-fit measures for null model 2007-2008 NBA season
Predicted Game Winner Percentage Observed Away Team Home Team Correct Game Winner Away Team 0 364 .0 Home Team 0 566 100.0 Overall Percentage 60.9
-2 Log Likelihood: 1245.027
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Appendix J: Goodness-of-fit measures for null model 2008-2009 NBA season
Predicted Game Winner Percentage Observed Away Team Home Team Correct Game Winner Away Team 0 352 .0 Home Team 0 578 100.0 Overall Percentage 62.2
-2 Log Likelihood: 1233.780
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Appendix K: Goodness-of-fit measures for null model 2007-2008 NBA season (streak)
Predicted Game Winner Percentage Observed Away Team Home Team Correct Game Winner Away Team 0 437 .0 Home Team 0 668 100.0 Overall Percentage 60.5
-2 Log Likelihood: 1483.207
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Appendix L: Goodness-of-fit measures for null model 2008-2009 NBA season (streak)
Predicted Game Winner Percentage Observed Away Team Home Team Correct Game Winner Away Team 0 430 .0 Home Team 0 675 100.0 Overall Percentage 61.1
-2 Log Likelihood: 1477.080
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