October 9, 2015 Statistical Analysis in Sports 2 Creating a Rating System § Measuring success in basketball § Possessions § Scoring efﬁciency § Turning statistics into a rating system § Expected winning percentages § Using the ratings § Predicting games § Predicting point spreads § Alternative rating methodologies § Elo § Correlated-Gaussian

Some background § Wins and losses! § Points scored and points allowed. § Seems trivial, but points and margin of victory are way more informative than wins and losses. § Points in a game are inﬂuenced by the quality of the two teams and how fast the game is played. § Points per possession, or 100 possession (Oliver, 2004). § Tempo-free statistics (Pomeroy, 2012). October 9, 2015 Statistical Analysis in Sports 4

The Impact of Tempo-Free Statistics § Kansas and Duke both score and average of 1.10 points per possession. § Kansas plays slow, whereas Duke likes to play fast. § Both teams play Missouri, which averages 0.95 points per possession. October 9, 2015 Statistical Analysis in Sports 5 Team Team Efﬁciency Missouri’s Efﬁciency Possessions Expected Score Kansas 1.10 0.95 60 66-57 Duke 1.10 0.95 74 81-70

Estimating Possessions § From Oliver (2004): § Points per possession can then be calculated by: § Off. PPP = Points Scored / Total Possessions § Def. PPP = Points Allowed / Total Possessions § Commonly multiplied by 100 to give us points per 100 possessions. October 9, 2015 Statistical Analysis in Sports 6

Specifying the Model § The parameters: § One offensive parameter per team (350) § One defensive parameter per team (350) § One intercept § Two home court parameters § Selecting the reference teams § One reference on offense and defense § Selected iteratively § Originally the last team alphabetically § Model rerun with reference team set to the team with the average offensive/defensive efﬁciency. October 9, 2015 Statistical Analysis in Sports 9

Specifying the Model § After each iteration, calculate each team’s offensive and defensive efﬁciency. § Offensive Efﬁciency = β 0 + β Team_Off § Calculate the mean offensive and defensive efﬁciency. § Determine which team is closest to the mean of each efﬁciency. § These are the new reference teams. § Estimate the model again with the updated reference teams. § Continue until the same teams are selected as the reference teams in consecutive runs. October 9, 2015 Statistical Analysis in Sports 10

Expected Winning Percentage § Pythagorean Win Expectation (James, 1983) § If teams win in proportion to their “quality”, n = 2. § n varies by sport, and reﬂects the role that chance plays in the outcome of games. § MLB: n = 1.83 § NHL: n = 2.15 § NFL: n = 2.37 § NBA: n = 16.5 October 9, 2015 Statistical Analysis in Sports 13

Expected Winning Percentage § Pythagenpat Win Percentage (Smyth & Heipp, 2009) § An adaptation of the Pythagorean rating where each team has their own exponent. § Based on the idea that points are more important in low-scoring games. § The more points that are scored, the higher ni will be (less chance). October 9, 2015 Statistical Analysis in Sports 14

Expected Winning Percentage § Linear/Logistic Combination Model (Kubatko, 2013) § Incorporates average margin of victory into the logit model. § Provides better predictions for extreme seasons and tends to be more stable over time. October 9, 2015 Statistical Analysis in Sports 15

Choosing the Optimal Exponents § Maximum Likelihood Estimation using the optim() function in R. § Calculate the pre-game adjusted efﬁciencies for each game from the 2002-03 season to the 2014-15 season. § Use optim to ﬁnd the exponents that minimize the binomial deviance, or log loss. October 9, 2015 Statistical Analysis in Sports 16

Exponent Results § Use these exponents to calculate the three ratings for each team. § To get a composite rating, I take the mean of the three ratings, weighted by Log Loss. October 9, 2015 Statistical Analysis in Sports 17 Method Exponent Log Loss Pythagorean 9.972 0.532 Pythagenpat 1.208 0.531 Linear/Logistic Combo -0.010 0.531

Predicting Game Winners § We can calculate a team’s probability of beating their opponent by using the Log5 formula (James, 1981). § This model generalizes to include the Bradley-Terry-Luce model commonly used in psychology, and the Rasch model in psychometrics (Long, 2013). § Kansas (0.9274) vs. Ohio State (0.9302): October 9, 2015 Statistical Analysis in Sports 20

Calculating a Weighted Point Spread § For each point spread method, compare the projected point spread to the actual margin of victory: § Calculate expected point spread for each game by averaging the three methods, weighted by RMSE. § We can also calculate win probabilities from the projected point spreads using logistic regression. October 9, 2015 Statistical Analysis in Sports 22 Method RMSE Weight GLS 10.774 0.331 Average Efﬁciencies 10.664 0.334 Net Efﬁciencies 10.651 0.335

In Game Win Probability § Adapted and expanded from Winston (2012) and Paine (2012). § Model assumes the margin of victory of a given game is ~N(PS, 10.612). § The mean and standard deviation of the distribution over the course of a game are given by: § StDev = 10.612 / sqrt(40 / minutes remaining) § Mean = (PS * (minutes remaining / 40)) + (Margin * (40 / minutes played)) § The win probability is given by the proportion of the distribution covering margins of victory that would result in the team winning. October 9, 2015 Statistical Analysis in Sports 25

Elo Ratings § Named after physics professor Arpad Elo. § Most widely used in chess and international soccer. § How it works: § Given a starting state of two teams, how is each team expected to perform? § How did the teams actually perform? § Update the ratings with this new information. October 9, 2015 Statistical Analysis in Sports 29

Calculating Elo § Long-run average rating of 1500. § All teams start out at a rating of 1300. § For a given game, Team A’s win probability is given by: § Team A’s rating is then updated using: October 9, 2015 Statistical Analysis in Sports 30

Adding Margin of Victory to Elo § Big wins and losses are more impressive and usually more informative, so: § Where the MOV Factor is given by: § Complicated but corrects for autocorrelation problems (favorites tend to win by more than they lose; Silver & Fischer-Baum, 2015). October 9, 2015 Statistical Analysis in Sports 31

Elo Ratings Pros/Cons § Pros: § Easy to calculate § Only need game scores (location can also be added in) § Track historical trends § Cons: § Ratings heavily dependent on performance in previous seasons (can be less accurate early in season) § Ratings are not retroactively adjusted to account for team’s being better/worst than expected. October 9, 2015 Statistical Analysis in Sports 32

Correlated-Gaussian Ratings § Basically a standardized average margin of victory. § Developed by Oliver (2004) to estimate a team’s expected winning percentage given their performance. § Not adjusted for strength of schedule, but can be. § The raw correlated-Gaussian rating can be used to estimate a team’s “luck”: § Luck = Win% - CorGaus% October 9, 2015 Statistical Analysis in Sports 33

References James, B. (1981). Baseball Abstracts. Lawrence, KS: Privately Printed. James, B. (1983). Baseball Abstracts. New York: Ballantine Books. Kubatko, J. (2013). Pythagoras of the hardwood [Web log post]. Retrieved from http://statitudes.com/blog/ 2013/09/09/pythagoras-of-the-hardwood/ Long, C. (2013). Baseball, chess, psychology, and psychometrics: Everyone uses the same damn rating system [Web log post]. Retrieved from http://angrystatistician.blogspot.com/2013/03/baseball-chess- psychology-and.html Oliver, D. (2004). Basketball on paper: Rules and tools for performance analysis. Dulles, Virginia: Potomac Books, Inc. Paine, N. (2012). Are NFL playoff outcomes getting more random? [Web log post]. Retrieved from http:// www.footballperspective.com/are-nﬂ-playoff-outcomes-getting-more-random/ Pomeroy, K. (2012, June 8). Ratings glossary [Web log post]. Retrieved from http://kenpom.com/blog/index.php/ weblog/entry/ratings_glossary Silver, N. & Fischer-Baum, R. (2015). How we calculate NBA Elo ratings [Web log post]. Retrieved from http:// ﬁvethirtyeight.com/features/how-we-calculate-nba-elo-ratings/ Smyth, D. & Heipp, B. (2009). Runs Per Win From Pythagenpat [Web log post]. Retrieved from http:// walksaber.blogspot.com/2009/01/runs-per-win-from-pythagenpat.html Winston, W. (2012). Mathletics: How gamblers, managers, and sports enthusiasts use mathematics in baseball, basketball, and football. Princeton, NJ: Princeton University Press. October 9, 2015 Statistical Analysis in Sports 34