REGRESSION PLANES TO IMPROVE THE PYTHAGOREAN PERCENTAGE A regression model using common baseball statistics to project offensive and defensive efficiency by Dennis Moy A thesis submitted in fulfillment of the requirements for the degree of honors in Statistics University of California - Berkeley 2006UNIVERSITY OF CALIFORNIA - BERKELEY ABSTRACT Prediction Planes for the Pythagorean Percentage by Dennis Moy Advisor: Professor David Aldous Department of Statistics In 1985, Bill James, arguably the most renowned analytical baseball statistician, devised a very simple, but effective formula that predicted a team’s winning percentage given its runs scored and runs allowed. Despite its remarkable accuracy, this model, coined Pythagorean expectation, was used primarily on seasons of the past rather than performance forecasts. This thesis develops prediction models for runs scored and runs allowed that will be converted by Pythagorean expectation to winning percentages. Data from the past twenty years were taken from four different sources of baseball statistics via the internet to produce 562 arrays that underwent computations through GRETL to create two different ordinary least-squares regression planes (offense and defense). The GRETL outputs yielded robust models that had strong positive R2 results with significant F-statistics from the Wald test that evaluated the planes’ goodness of fit, which with a potentially adjusted Pythagorean expectation, can now forecast future winning percentages. Armed with this knowledge and a little calculus, baseball executives can determine which talent is more valuable when building a successful team to maximize winning percentage.i TABLE OF CONTENTS List of Figures and Tables...................................................................................................... ii Acknowledgements................................................................................................................ iii Glossary................................................................................................................................... iv Introduction and Background............................................................................................... 1 Materials and Method............................................................................................................. 8 Data Analysis and Findings.................................................................................................. 10 Discussion of Results............................................................................................................ 20 Conclusion and Extensions ................................................................................................. 25 Appendix (Derivations)........................................................................................................ 28 Bibliography........................................................................................................................... 31ii LIST OF FIGURES AND TABLES Number Page Table 1: Summary Statistics from GRETL........................................................................ 10 Figure 1: Scatterplot of Runs (Adjusted) versus Year....................................................... 11 Figure 2: Scatterplot of OBP versus Year .......................................................................... 13 Figure 3: Scatterplot of SLG versus Year........................................................................... 13 Figure 4: Scatterplot of WHIP versus Year ....................................................................... 14 Figure 5: Scatterplot of DER versus Year.......................................................................... 14 Table 2: OLS Estimates of Runs Scored (Adjusted) versus OBP and SLG .................. 15 Figure 6: Fitted, Actual Plot of Runs Scored (Adjusted) versus OBP and SLG......... 16 Figure 7: Residuals for Runs Scored (Adjusted) Regression Model ............................. 17 Table 3: OLS Estimates of Runs Allowed (Adjusted) versus WHIP and DER............ 17 Figure 8: Fitted, Actual Plot of Runs Allowed (Adjusted) versus WHIP and DER..... 18 Figure 9: Residuals for Run Allowed (Adjusted) Regression Model............................... 19iii ACKNOWLEDGMENTS I would like to express my sincere appreciation to my parents first and foremost for sending me through school for the past 19 years and always making sure I was aiming high and giving my best efforts. Also, I want to thank Professor David Aldous for being a helpful advisor who allowed me to apply all that I have learned to a topic I love. Thanks to the immortal Team Savage, who provided a gateway each week for me to chase and realize the dream of an intramural softball championship. Also, I need to thank my two roommates, Kevin and Herman, for dealing with my complaints about doing this for a whole semester and the rest of my friends, for using my thesis as an excuse for not going out and enjoying my last semester in college. Thank you to Mr. Spellicy for being a great mentor and telling me to pursue what I enjoy. Most importantly, I need to thank Ms. Delfino for suggesting me to take Advanced Placement Statistics as a sophomore at Lowell High School, as that paved the road for my interest and major in statistics. And last but not least, I want to state my deep gratitude to Eva for keeping me on top of my thesis and helping me edit until the perfect version came into fruition.iv GLOSSARY Batting Average (AVG or BA). Total Hits / Total At-Bats. On-Base Percentage (OBP). (Hits + Walks + Hit-By-Pitches) / (At-Bats + Walks + Sacrifice Flies + Hit-By Pitches). Slugging Percentage (SLG). Total Bases / Total At-Bats. OPS. On-Base Percentage + Slugging Percentage. WHIP. (Total Walks + Total Hits) / Total Innings Pitched. Commonly understood as average number of base runners per inning. Defensive Efficiency Rating (DER). The ratio of team defensive outs recorded in defensive opportunities. 1 — [(Total number of hits — home runs allowed) / (All balls hit into play — home runs allowed)]. Earned Run Average. (9 * Total earned runs allowed) / (Innings pitched). The most common measure of a pitcher’s ability because of its simplicity, despite being inherently biased and flawed.