Journal of Quantitative Analysis in Sports Volume 7, Issue 2 2011 Article 5 Stratified Odds Ratios for Evaluating NBA Players Based on their Plus/Minus Statistics Douglas M. Okamoto, Data to Information to Knowledge Recommended Citation: Okamoto, Douglas M. (2011) "Stratified Odds Ratios for Evaluating NBA Players Based on their Plus/Minus Statistics," Journal of Quantitative Analysis in Sports: Vol. 7: Iss. 2, Article 5. Available at: http://www.bepress.com/jqas/vol7/iss2/5 DOI: 10.2202/1559-0410.1320 2011 American Statistical Association. All rights reserved.
Stratified Odds Ratios for Evaluating NBA Players Based on their Plus/Minus Statistics Douglas M. Okamoto Abstract In this paper, I estimate adjusted odds ratios by fitting stratified logistic regression models to binary response variables, games won or lost, with plus/minus statistics as explanatory variables. Adapted from ice hockey, the plus/minus statistic credits an NBA player one or more points whenever his team scores while he is on the basketball court. Conversely, the player is debited minus one or more points whenever the opposing team scores. Throughout the NBA season, the league s better players are likely to have positive plus/minus statistics as reported by Yahoo!Sports and 82games.com. Crude or unadjusted odds ratios estimate the relative probabilities of a player having a positive plus/minus in a win, versus a negative plus/minus in a loss. Home and away games are twin strata with teams playing 41 home games and 41 road games during an 82-game regular season. Stratum-specific odds ratios vary because some players perform better at home than on the road and vice versa. In order to adjust for home court advantage, stratified odds ratios and their 95 percent confidence intervals are estimated for each of the Los Angeles Lakers during the 2009 2010 regular season. KEYWORDS: plus/minus statistic, odds ratio, logistic regression model
Okamoto: Stratified Odds Ratios for Evaluating NBA Players 1. Introduction Adapted from ice hockey, the plus/minus statistic credits an NBA player one or more points whenever his team scores while he is on the basketball court. Conversely, the player is debited one or more points whenever the opposing team scores. At the end of the game, a player s pluses and minuses are totaled to get his plus/minus (+/-) statistic. The Lakers won the Pacific Division of the Western Conference with a 57-25 win/loss record during the 2009-2010 regular season. In Figure 1, blue circles indicate Laker wins and red circles indicate Laker losses. Each dot corresponds to an ordered pair (x, y), with the x-coordinate equal to Kobe Bryant s +/- and the y-coordinate equal to the winning or losing margin. When Kobe is a plus, {X > 0}, a minus, {X < 0}; when the Lakers win, {Y>0}, they lose, {Y<0}. 50 40 LA Laker Winning or Losing Margin 30 20 10 0-30 -20-10 0 10 20 30 40 50-10 -20-30 Kobe Bryant Plus/Minus Win Loss Figure 1. Scatterplot of Los Angeles Laker Winning or Losing Margin vs. Kobe Bryant Plus/Minus: 2009 2010 NBA Regular Season (73 Games) Kobe was a plus in 48 of 51 Laker wins (NE quadrant) and 5 of 22 losses (SE quadrant). He was a minus in 3 Laker wins (NW quadrant), and 17 losses (SW quadrant). Not shown in the scatterplot are 9 Laker games (6 wins, 3 losses) in which Kobe did not play. Published by Berkeley Electronic Press, 2011 1
Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5 2. Odds Ratios The relative odds of the Lakers winning or losing when Kobe was a plus or minus are defined in terms of the conditional probabilities of Y given X, where the joint probability distribution of two binary random variables is as follows: a) {X=1} if Kobe was a plus, {X=0} if Kobe was a minus; and b) {Y=1} if the Lakers won, {Y=0} if the Lakers lost. The odds ratio (OR) is the relative probability of a Laker win when Kobe is a plus divided by the relative probability of a Laker win when he is a minus: equals [(48)(17)]/[(5)(3)] = 54 or 54:1, calculated from the cross product of four cell counts in the following two-by-two contingency table. Table 1. Laker Wins/Losses vs. Kobe Bryant Plus/Minus Minus Plus Totals Win 3 48 51 Loss 17 5 22 Totals 20 53 73 The Los Angeles Lakers were 54 times more likely to have won when Kobe was plus in a win. Conversely, the Lakers were 54 times more likely to have lost when he was minus in a loss. Despite his 54:1 odds ratio, Kobe Bryant ranked second to Laker forward Pau Gasol whose [(40)(14)]/[(4)(2)] = 70 or 70:1 odds ratio calculated from Table 2 led the team in the 2009 2010 regular season. Table 2. Laker Wins/Losses vs. Pau Gasol Plus/Minus Minus Plus Totals Win 4 40 44 Loss 14 2 16 Totals 18 42 60 Odds ratios and 95 percent confidence intervals for Kobe Bryant, Pau Gasol and eight of their Laker teammates are shown in Figure 2. http://www.bepress.com/jqas/vol7/iss2/5 DOI: 10.2202/1559-0410.1320 2
Okamoto: Stratified Odds Ratios for Evaluating NBA Players Pau Gasol 70 Kobe Bryant 54 Ron Artest 31 Derek Fisher 27 Lamar Odom 16 Andrew Bynum 10 Luke Walton 7.8 Jordan Farmar 6.2 Shannon Brown 3.4 Sasha Vujacic 3.1 1.0 10.0 100.0 1000.0 10000.0 Figure 2. Odds Ratio Chart for the Los Angeles Lakers: 2009 2010 NBA Regular Season (82 Games) 3. Logistic Regression Models Fitting a logistic regression model to the logit transform of relative probabilities of the Los Angeles Lakers winning or losing, logit log Pr Y 1 / Pr{ Y 0} X where X is a Laker player s plus/minus statistic and α an intercept term, yields a maximum likelihood estimate of the logistic regression coefficient,, or a log odds ratio that is the antilogarithm of his odds ratio as calculated in Section 2, e.g., exp(3.996) = 54 or 54:1, for Kobe Bryant. Fitting a second logistic multiple regression model to the logit transform of relative probabilities of the Los Angeles Lakers winning or losing, Y Y 1 X 1 2 X 2 logit log Pr 1 / Pr{ 0} Published by Berkeley Electronic Press, 2011 3
Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5 where X 1 and X 2 are plus/minus statistics for two Laker players, say Kobe Bryant and Pau Gasol, yields maximum likelihood estimates of two logistic regression coefficients, 1 and 2. Taking antilogarithms of these two log odds ratios, exp(2.553) = 13 or 13:1, for Kobe Bryant and exp(3.733) = 42 or 42:1, for Pau Gasol. During the 2009 2010 regular seasons, Bryant and Gasol only played together in 56 of 82 Laker regular season games, winning 41 and losing 15 games. Table 3a. Kobe Bryant Plus/Minus vs. Pau Gasol Plus/Minus Laker Wins BRYANT+/- GASOL +/- Plus Minus Totals Plus 32 2 34 Zero 0 3 3 Minus 4 0 4 Totals 36 5 41 Table 3b. Kobe Bryant Plus/Minus vs. Pau Gasol Plus/Minus Laker Losses BRYANT+/- GASOL +/- Plus Minus Totals Plus 0 1 1 Zero 1 1 2 Minus 2 10 12 Totals 3 12 15 In Table 3a, there are 5 Laker wins in which Bryant is a minus, but Gasol is not; whereas, in 4 Laker wins Bryant is a plus and Gasol a minus. Similarly, in Table 3b there are 2 Laker losses in which Bryant is a plus and Gasol a minus; whereas, in 1 Laker loss Bryant is a minus and Gasol a plus. 4. Stratified Odds Ratios The relative odds of the Lakers winning a home game when Kobe was a plus equals [(30)(5)]/[(1)(1)] = 150 or 150:1, calculated from Table 4a. The relative odds of the Lakers winning an away game when Kobe was a plus equals [(18)(12)]/[(4)(2)] = 27 or 27:1, calculated from Table 4b. http://www.bepress.com/jqas/vol7/iss2/5 DOI: 10.2202/1559-0410.1320 4
Okamoto: Stratified Odds Ratios for Evaluating NBA Players Table 4a. Laker Wins/Losses vs. Kobe Bryant Plus/Minus Home Games Minus Plus Totals Win 1 30 31 Loss 5 1 6 Totals 6 31 37 Table 4b. Laker Wins/Losses vs. Kobe Bryant Plus/Minus Away Games Minus Plus Totals Win 2 18 20 Loss 12 4 16 Totals 14 22 36 Dividing Kobe Bryant s home odds ratio by his away odds ratio, the Lakers were five-and-a-half times more likely to have won at home than on the road when Kobe Bryant was a plus. Consequently, the estimation of stratified odds ratios or adjusted odds ratios that take into account stratification is a necessary refinement. Figure 3 shows the odds ratios for Kobe Bryant and eight of his Laker teammates, with home games represented in gold and away games represented in purple. Odds Ratio (Hom e and Away) Odds Ratio (Home and Away) 1000.0 1000.0 100.0 100.0 10.0 10.0 1.0 Sasha Shannon Jordan Andrew Lamar Derek Ron Kobe Pau 1.0 Sasha Vujacic Shanno Brow n Jordan Farmar Andrew By num Lamar Odom Fisher Derek Artest Ron Bry Kobe ant Gasol Pau Vujacic n Brown Farmar Bynum Odom Fisher Artest Bryant Gasol Home OR 2.0 2.2 2.7 6.0 34 62 999 150 18 Home OR 2.0 2.2 2.7 6.0 34 62 999 150 18 Away ay OR OR 5.1 5.1 6.1 6.1 11 11 13 9.2 14 17 27 999 Figure 3. Odds Ratio Chart for the Los Angeles Lakers: 2009 2010 NBA Regular Season (41 Home Games, 41 Away Games) Published by Berkeley Electronic Press, 2011 5
Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5 The logit transform of relative probabilities of the Lakers winning or losing as a function of a player s common log odds ratio for both home and away games is modeled as follows: 2 1 logit log Pr Y 1 / Pr{ Y 0} X or X where and are stratum-specific, nuisance parameters. Fitting a stratified logistic regression model yields a conditional likelihood estimate for the log odds ratio,, conditioned on sufficient statistics for the two nuisance parameters as in Mehta and Patel (1995). For example, Kobe Bryant s stratified odds ratio equals the antilogarithm of his log odds ratio, exp (3.695) = 40 or 40:1. Pau Gasol Kobe Bryant Ron Artest Derek Fisher Lamar Odom Andrew Bynum Jordan Farmar Luke Walton Shannon Brown Sasha Vujacic 13 8.6 6.2 3.7 3.6 5.6 50 40 29 21 0.1 1.0 10.0 100.0 1000.0 Figure 4. Stratified Odds Ratio Chart for the Los Angeles Lakers: 2009 2010 NBA Regular Season (41 Home Games, 41 Away Games) Pau Gasol led the Lakers during the 2009 2010 regular season with a stratified odds ratio, 50:1, which compared to Kobe Bryant s 40:1 stratified odds ratio means the Lakers were 25 percent more likely to win when he was plus (or lose when he was minus) than they were when Kobe was plus (or minus). http://www.bepress.com/jqas/vol7/iss2/5 DOI: 10.2202/1559-0410.1320 6
Okamoto: Stratified Odds Ratios for Evaluating NBA Players 5. 2010 NBA Playoffs The Lakers won their second straight NBA Championship with an overall record of 16 wins and 7 losses in the 2010 NBA Playoffs. In Figure 5, Kobe Bryant was plus in 14 of 16 Laker playoff wins (blue circles) and minus in 6 of 7 Laker playoff losses (red circles). He was neither plus nor minus in 1 playoff win (white circle), minus in another playoff win, and plus in 1 playoff loss. 30 25 20 LA Laker Winning or Losing Margin 15 10 5 0-20 -15-10 -5 0 5 10 15 20 25 30-5 -10-15 -20-25 Kobe Bryant Plus/Minus Win Loss Figure 5. Scatterplot of Los Angeles Laker Winning or Losing Margin vs. Kobe Bryant Plus/Minus: 2010 NBA Championship Playoffs (23 Games) If the single playoff game in which Kobe Bryant was neither plus nor minus is excluded, then his odds ratio was [(14)(6)/(1)(1)] = 84 or 84:1. The Lakers were 84 times more likely to have won when Kobe was plus than they were when he was minus in a win. Table 5. Laker Wins/Losses vs. Kobe Bryant Plus/Minus Minus Plus Totals Win 1 14 15 Loss 6 1 7 Totals 7 15 22 Published by Berkeley Electronic Press, 2011 7
Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5 K. Bryant 84 P. Gasol 39 L. Odom 39 J. Farmar 15 L. Walton 10 D. Fisher 5.5 S. Brown 4.4 R. Artest 3.0 A. Bynum 2.4 J. Powell 0.3 0.0 0.1 1.0 10.0 100.0 1000.0 10000.0 Figure 6. Odds Ratio Chart for the Los Angeles Lakers: 2010 NBA Championship Playoffs (23 Games) With his 84:1 odds ratio more than twice the odds ratio of any of his teammates, Kobe Bryant led the Los Angeles Lakers to their second consecutive NBA Championship and earned his second straight Larry O Brien trophy as the Finals MVP. His stratified odds ratio could not be calculated because he was minus in the single Laker playoff loss at home against the Boston Celtics, and plus in all 5 of their playoff wins away from home. Pau Gasol and Lamar Odom whose 39:1 odds ratio during the playoffs tied them for second, had stratified odds ratios of 23:1 and 29:1, respectively. Unlike Kobe Bryant, Pau Gasol and Lamar Odom were minuses in one of the 5 playoff games the Lakers won on the road. References Mehta C.R. and Patel N.R. (1995). Exact logistic regression: theory and examples, Statistics in Medicine, 14: pp. 2143-60. Rosenbaum, Dan T. (2004). Measuring how NBA players help their teams win, 82games.com, http://www.82games.com/comm30.htm. http://www.bepress.com/jqas/vol7/iss2/5 DOI: 10.2202/1559-0410.1320 8