Evaluating Individual Player Contributions in Basketball

Transcription

1 Section on Statistics in Sorts JSM 2010 Evaluating Individual layer Contributions in Basetball Jarad Niemi Abstract This aer introduces novel methodology for evaluating layer effectiveness in basetball. We model searately each layer s offensive and defensive contributions to their team and rovide estimates that adjust for all layers on the court. These offensive and defensive contributions are naturally combined to rovide a lus-minus statistic for each layer. We build a hierarchical model that estimates the underlying distributions for layer contributions and shrins layer estimates toward average with more shrinage given to layers who lay less. We illustrate the methodology using data from the NBA regular season. Based on this season s data, we find the best layer in the league to be au Gasol who rovides his team with a 10.9 ± 4.3 oint differential comared to an average layer. The best offensive layer is Steve Nash who rovides his team with an additional 11.3 ± 3.0 oints er game and the best defensive layer is Raja Bell who reduces his oonents scoring by 12.7 ± 7.3 oints er game. Key Words: Bayesian analysis, Gibbs samling, hierarchical model, lus-minus statistic, offensive rating, defensive rating 1. Introduction In team sorts around the world, statistics are maintained to measure individual layer erformance Kubato et al., 2007). These statistics are meant to give coaches, analysts, and sectators an understanding of a layer s contribution to his team. For examle, in basetball we have field goal ercentage, offensive rebounds, and assists that indicate a layer s offensive rowess and steals, blocs, and defense rebounds that indicate the strength of a layer defensively. Analysts quicly notice short-comings of these statistics, e.g. a risy layer might have a high number of assists er game, but an even higher number of turnovers er game. To remedy this, more statistics are tyically added such as assistto-turnover ratio. As more statistics are added, box scores quicly become hard to interret. For this reason, attemts have been made to tae combinations of box score statistics to rovide meaningful summaries of layer contribution. Examles are Dave Heeren s NBA efficiency rating Heeren, 1988), Dave Berri s layer value Berri, 1999), and John Hollinger s layer efficiency rating Hollinger, 2005). These methods succeed in roviding a one number summary of a layer s contribution, but, as ointed out in Winston 2009) and references therein), often have unfortunate imlications, e.g. that the worst shooter in the NBA should shoot more to increase their rating. Since tyical box score statistics cannot cature the inherent comlexities involved in sorts, we consider the lus-minus statistic. The lus-minus statistic is simly how many oints a layer s team outscores their oonents by when that layer is laying. Better layers allow their team to score more oints while forcing their oonents to score fewer and therefore their lus-minus statistic will generally be larger. The main concern with the lus-minus statistic is that it does not account for who is on the court at a given time, but adjusted lus-minus does. As imlemented by Winston 2009), the adjusted lusminus statistic numerically solves simultaneously for all layers lus-minus statistics that minimize the squared error in observed data minus redictions. The adjusted lus-minus University of California, Santa Barbara, Statistics & Alied robability, Santa Barbara, CA

2 Section on Statistics in Sorts JSM 2010 statistic has exactly the same interretation of the original lus-minus statistic, but accounts for who is on the court with whom. In a similar fashion, Rosenbaum 2004) casts the estimation of an adjusted lus-minus into a linear regression framewor. The regression taes into account the difference in oints scored while taing into account the layers contemoraneously on-court and which side is laying at home. This analysis is restricted to layers who lay more than 250 minutes across two seasons where a reference layer relaces those laying less than 250 minutes across the seasons. The analysis rovides estimates of lus-minus for each layer, but the offensive and defensive ratings have high uncertainty. We extend this adjusted lus-minus idea to a model based framewor where lay-bylay data are analyzed for offensive lay and defensive lay searately and then combined to rovide an overall layer contribution estimate. In addition, we include all layers in the analysis, but roose a model that shrins layer estimates toward the average with the amount of shrinage being inversely roortional to the amount of time layed. The aer continues as follows. Section 2 introduces the lay-by-lay data used for the analysis throughout. In section 3, we introduce the model and method of arameter estimation. Section 4 rovides analyses based on the lay-by-lay data. Section 5 briefly comares the method to a coule of others, suggests ossibilities for extending the model, and discusses alicability to other team sorts. 2. NBA Season The data for this analysis come from a single regular NBA season which comrises 82 games for each of the 30 teams. On average there are 20 substitution times which means there are around 20 sets of 10 not necessarily unique) layers that can score oints or get scored on. Between each substitution, we record the five layers for the away team, the five layers for the home team, the amount of time those layers are on the court and the number of oints that are scored by the away and home teams during that time. These lay-by-lay data contain all the nown 10-layer combinations. These combinations are clear whenever the on-court layers and substitutions are clearly identified. Unfortunately, the online data ublished by ESN.com and NBA.com do not state substitutions that occur during quarter breas which results in layer uncertainty. In some cases, layers on court can be inferred from resulting on-court actions, e.g. shots taen and fouls committed. When the layers cannot be inferred, the observations are removed and the resulting lay-by-lay data is ublished at htt://basetballgee.com. These data are modified so that free throws are awarded to the set of layers in the game before the foul, that resulted in the free throws, was committed. This adjustment is required due to a nuance in the way substitutions are handled during free throws which would assign oints to layers who have just entered the game and had no role in the foul being committed) and also would result in scores with zero elased time. Table 1 rovides an examle of the tye of data used for this analysis. In total, we have 32,000 observations from the NBA regular season and 443 layers. 3. Individual layer Contributions The method described here considers data on a squad, set of five layers, basis. At all times two squads are concurrently on the court: a home squad and an away squad for the home and away teams, resectively. The goal here is to extract from these squads the contribution for each individual layer after accounting for the layers on-court teammates and oonents. 4915

3 Section on Statistics in Sorts JSM 2010 Table 1: Examle of three observations of the NBA season roviding the names of the 5 layers on the court for the home and away teams, the number of seconds the layers were on the court, and the number of oints scored by both the away and home squads. Observation # Away squad Eddie House Eddie House Eddie House Kendric erins Kevin Garnett Kevin Garnett Marquis Daniels Marquis Daniels aul ierce Rasheed Wallace aul ierce Rajon Rondo Ray Allen Rasheed Wallace Ray Allen Home squad Anthony arer J.J. Hicson Anderson Varejao Daniel Gibson Jamario Moon Anthony arer LeBron James LeBron James Daniel Gibson Shaquille O Neal Mo Williams Jamario Moon Zydrunas Ilgausas Zydrunas Ilgausas LeBron James Seconds Away oints Home oints Data generating model We jointly consider the oints scored by the home and away squads. Let y ih y ia ) be the number of oints scored by home away) squad i, i {1, 2,..., n} where there were a total of n observations over the course of a season. We assume y i = y ih, y ia ) ind o y ih Ht i =1 θ h i δ a i o y ia At i =1 θ a i δ h i 1) where oy λ) reresents the oisson distribution with mean λ, H A) is the average number of oints er minute for the home away) team, h i a i ) is an indicator of whether layer is on home away) squad i, i.e. h i a i ) = 1 if yes, and 0 if no, t i is the time in minutes between substitutions, and is the number of layers in the NBA. This aroach combines the standard generalized linear model McCullagh and Nelder, 1999, Ch. 6) with a oisson rocess Lawler, 1995, Ch. 3). The relationshi with the generalized linear model is aarent if, for examle, we tae the logarithm of the home oisson mean and set θ j = exβ j ) where loght i ) becomes the offset. To understand the interretation of this model, first consider the case where θ i = δ i = 1 i,, i.e. all layers are equal both offensively and defensively. In this case, home away) squad i is exected to score Ht i At i ) oints. The value for H A) is the emirical average oints er minute for all home away) teams over the course of the season. Since t i is the number of minutes of the game layed by squad i, the exected oints scored by the home away) squad is roortional to the amount of time layed. In this model, the measure of layer s offensive defensive) contribution is θ δ ). If all other arameters are constant, then the multilicative increase in offensive defensive) scoring while layer is in the game comared to the average layer is θ δ ). Therefore if θ > 1 δ < 1) that layer s contribution to the team has a ositive offensive defensive) effect whereas if θ < 1 δ < 1) that layer s contribution has a negative effect. 4916

4 Section on Statistics in Sorts JSM Hierarchical structure We assume indeendent Gamma riors with common shae and rate arameters for a layer s offensive and defensive contributions, i.e. θ ) = Gaα θ, α θ ) and δ ) = Gaα δ, α δ ) where x) = Gaα, β) is the gamma distribution roortional to x α 1 e xβ. Since E[X] = α/β, the exectation of both riors is 1. Intuitively this means the average NBA layer will have θ and δ equal to 1. In addition, this rior shrins layer contribution arameters toward the average with larger values of α θ and α δ resulting in more shrinage. We then build a hierarchical model by assuming the non-informative rior α θ, α δ ) = α θ )α δ ) = α 1 θ α 1 δ. This hierarchical structure allows the data to determine how much shrinage should be alied by learning the underlying distribution of the θs and δs. For examle, if the data indicate that all θs are in a small neighborhood around one, then α θ will be large leading to small variability in offensive layer contributions and therefore a large amount of shrinage. 3.2 arameter estimation The goal of these methods is to obtain the osterior distribution of the model arameters conditional on the observed data, i.e. θ, δ, α θ, α δ y) where θ = θ 1,..., θ ), δ = δ 1,..., δ ), and y = y 1, y 2,..., y n ). Using Bayes Rule and the conditional indeendence assumtions, we have [ n ] θ, δ, α θ, α δ y) α θ, α δ ) θ α θ )δ α δ ) y i θ, δ) 2) =1 where y i θ, δ) is given in equation 1). Since the osterior θ, δ, α θ, α δ y) is not available in an analytically tractable form, we aroximate the osterior via a Marov chain Monte Carlo MCMC) simulation aroach, namely a Metroolis-within-Gibbs algorithm Gelfand and Smith, 1990). Once converged, the MCMC rocedure rovides draws from the osterior distribution for all arameters. The two general stes in the algorithm are 1) jointly samling α θ and α δ and 2) cycling through all layers jointly samling θ and δ for that layer. Due to conditional indeendencies, the first ste is accomlished by using two univariate random wals to samle from α θ θ) and α δ δ) where the target distributions are and α θ θ) =1 α δ δ) =1 θ δ α θ 1 α δ 1 ex α θ ex α δ θ =1 δ =1 α α θ 1 Γα θ ) θ α α δ 1 δ Γα δ ). The second ste involves cycling through layers and jointly samling their offensive and defensive contribution arameters. The full conditional distributions for these arameters for layer, i.e. θ, δ y, θ, δ ) where x indicates the vector x with comonent removed, are indeendently Gamma distributed with udated hyer-arameters, i.e. Gaθ α θ, β θ ) Gaδ α δ, β δ ). The hyer-arameters are rovided in equation 3) and 4917

5 Section on Statistics in Sorts JSM 2010 derived in Aendix A. α θ = α θ + α δ = α δ + β θ = α θ + β δ = α δ + n [h i y ih + a i y ia ] n [h i y ia + a i y ib ] n h i Ht i n h i At i θ h i δa i + a i At i θ a i δh i + a i Ht i θ a i θ h i δh i δa i 3) To understand the interretation of these udated hyer-arameters, consider α θ = α δ = 0. Then α θ α δ ) is the oints scored by on) layer s team while layer is on the court and β θ β δ ) is the exected oints scored by on) layer s team while layer is on the court taing into account all the layers that are currently on the court and their current contribution arameters. Code was written in C and called from the statistical environment R R Develoment Core Team, 2009). Convergence was assessed using the method of Broos and Gelman, 1997) on multile chains initialized from disersed starting locations. This diagnostic is available using the function gelman.diag in the acage coda in R. To lace these multilicative arameters on a scale comatible with current lus-minus statistics, we multily the average NBA team oints er game by the arameter minus one. For the offensive contribution, this is equivalent to finding the exected number of extra oints a team scores if an average layer was relaced with this layer for an entire game. Similarly, the defensive contribution is the exected number of extra oints an oonent would score and the lus-minus contribution is the exected difference in oints between the layer s team and her oonent s team. 4. Analysis of the NBA regular season The NBA regular season was analyzed using 10 chains with initial values overdisersed relative to the osterior. Each chain had 1,000 burn-in and 1,000 inferential iterations and no lac of convergence was detected. All results resented combine the inferential iterations of all 10 chains. On average this season, the home team scored 2.11 oints er minute m), the away team scored 2.05 m, and 99.8 oints er game 48 minutes) er team. Figure 1 rovides osterior estimates for the hierarchical arameters and redictive distributions for layer offensive, defensive, and lus-minus contributions. The redictive distributions indicate that a layer contributing an extra 10 oints er 48 minutes is an incredible offensive layer while a defender reducing his oonent s scoring by 10 oints in 48 minutes is incredible defensively. These estimates aear in line with other lus-minus statistics that also have most layers in the range -10 to 10. Table 2 rovides estimates and credible intervals for the to 50 NBA layers in terms of lus-minus. The table has been sorted according to the lower 2.5%-tile of the lusminus credible interval. This analysis suggests that au Gasol was the most valuable NBA layer roviding a , 15.18) oint differential after adjusting for his teammates and oosing layers. This differential occurs because Gasol rovides , 10.65) 4918

6 Section on Statistics in Sorts JSM 2010 α θ y) α δ y) G[θ 1] y) G[δ 1] y) G[θ δ] y) Offense Defense lus Minus Figure 1: osterior distribution to row) for rior arameters and redictive distributions bottom row) for a new layer s offensive, defensive, and lusminus contributions to a team. 4919

7 Section on Statistics in Sorts JSM 2010 Table 2: Estimates and credible intervals for lus-minus, offensive, and defensive contributions for the to 50 NBA layers in the regular season. layer lus-minus Offensive Defensive 1 au Gasol ,15.18) ,10.65) ,-0.55) 2 Dwight Howard ,13.59) , 6.19) ,-3.91) 3 Vince Carter ,14.09) , 4.00) ,-6.24) 4 Andrei Kirileno ,14.35) , 6.42) ,-3.49) 5 Kobe Bryant ,12.65) , 6.96) ,-2.10) 6 Anderson Varejao ,13.10) , 4.93) ,-4.21) 7 Dere Fisher ,13.63) , 9.83) , 0.64) 8 Deron Williams ,12.23) , 9.36) , 0.85) 9 Matt Barnes ,12.91) , 6.88) ,-1.97) 10 aul Millsa ,12.58) , 8.64) ,-0.06) 11 LeBron James ,11.78) , 7.33) ,-0.83) 12 Matt Bonner ,14.83) , 7.56) ,-2.13) 13 Ron Artest ,11.91) , 5.68) ,-2.45) 14 Greg Oden ,19.75) ,10.80) ,-1.41) 15 Ray Allen ,11.13) , 5.04) ,-2.50) 16 Mie Bibby ,12.20) , 5.62) ,-2.35) 17 Manu Ginobili ,11.58) , 6.14) ,-1.41) 18 Chris Andersen ,12.90) ,13.33) , 5.25) 19 Kyle Korver ,15.56) ,11.62) , 1.85) 20 Jameer Nelson ,12.24) , 4.86) ,-2.94) 21 Kevin Durant ,10.24) , 6.49) ,-0.36) 22 Tim Duncan ,10.93) , 2.49) ,-4.55) 23 Kevin Garnett ,11.86) , 5.00) ,-2.60) 24 Jason Williams ,12.26) , 7.99) , 0.10) 25 Channing Frye ,11.56) ,13.51) , 6.08) 26 Al Horford ,10.51) , 6.22) ,-0.58) 27 Brandon Roy ,11.19) , 7.47) , 0.33) 28 Josh Smith ,10.67) , 6.85) ,-0.04) 29 Ryan Anderson ,14.89) , 6.58) ,-2.41) 30 Jared Dudley ,11.53) ,11.81) , 4.57) 31 Rashard Lewis ,10.69) , 5.47) ,-1.20) 32 Nic Collison ,11.68) , 2.32) ,-4.77) 33 Richard Jefferson , 9.86) , 1.77) ,-4.37) 34 Lamar Odom , 9.99) , 4.08) ,-2.15) 35 Kendric erins ,10.80) , 4.20) ,-2.33) 36 Rajon Rondo , 9.05) , 4.05) ,-1.30) 37 Wesley Matthews ,11.78) ,10.15) , 3.33) 38 Jason Kidd , 9.09) , 4.73) ,-0.75) 39 Carlos Boozer , 9.40) , 6.48) , 0.88) 40 Jamal Crawford , 9.60) , 7.06) , 1.38) 41 Dir Nowitzi , 9.10) , 6.43) , 1.08) 42 Steve Nash , 9.66) ,14.30) , 8.58) 43 Andrew Bynum ,10.37) , 4.83) ,-1.02) 44 Shawn Marion , 9.29) , 4.57) ,-0.79) 45 Andrew Bogut , 9.98) , 3.55) ,-2.25) 46 Joe Johnson , 8.83) , 5.77) , 0.75) 47 Rodrigue Beaubois ,14.81) ,11.05) , 2.83) 48 Leandro Barbosa ,14.66) ,17.11) , 8.98) 49 Nene Hilario , 8.69) , 8.18) , 3.26) 50 Dwyane Wade , 8.75) , 6.57) , 1.69) 4920

8 Section on Statistics in Sorts JSM 2010 additional oints for the Laers and restricts his oonents to , 6.39) fewer oints er game g). Since these intervals do not contain zero, Gasol s offensive, defensive, and lus-minus are all significantly better than average. The league MV was LeBron James who rovided an additional , 7.33) g for the Cavaliers while holding his oonents to , 6.04) fewer g resulting in an , 11.78) g differential. The overlaing lus-minus intervals for James and Gasol suggest that the two are not statistically significantly different from each other and an estimate of the robability that Gasol s lus-minus is greater than that for James is 84%. So although this analysis can distinguish layers from average, it cannot distinguish among the to layers in the league. In fact, Greg Oden may be the best lus-minus layer in the league owing mainly to his defense. Unfortunately injuries limited him to 21 games in the season and therefore his contributions have large uncertainties. The best offensive layer in the league when sorted by lower 2.5%-tile) is Steve Nash who rovides ,14.30) oints er game for the Suns, but also allows his oonents to score an additional , 8.58) oints er game. The best defensive layer is Raja Bell who limits his oonents to , 21.18) fewer oints er game, but reduces his team s scoring by , 19.78) oints er game. In the waning moments of a close-fought battle, these are two layers a coach would want to consider for strategic offense-defense substitutions. Comlete results for the through regular seasons are available at htt: // 5. Discussion The aroach resented in this article is a hierarchical model where layer contributions for both offense and defense affect the oints er minute scored by the away and home teams. The contribution arameters as well as their distributions are estimated using a Marov chain Monte Carlo aroach. Comared to other lus-minus statistics, the advantages of this hierarchical statistical modeling aroach are uncertainty quantification, arameter shrinage to the average layer, estimation of the underlying distribution of layer contributions, and searate assessment of a layer s offensive and defensive contribution to the team. In this article, the uncertainty of layer contribution arameters as well as their rior distribution are quantified. The arameters for the rior distribution have little uncertainty, i.e. the difference between Ga300, 300) and Ga400, 400) is minor on the scale of the multilicative effects we are estimating. In contrast, there is considerable uncertainty in layer contribution estimates esecially if interest centers on distinguishing individual layers. Using one season of data, the best layers can be clearly distinguished from average layers, but difficulty arises in searating the to layers. A next ste to decrease this uncertainty would be to create a dynamic model for the layer contribution arameters to allow for information to be shared across and, erhas, within seasons. An aealing asect of utilizing a rior distribution for the layer contribution arameters is shrining estimates toward average. In the full conditional distributions given in equation 3), the rior has the effect of adding α θ and α δ oints for a layer s team and his oonent s resectively. This has little effect on layers lie James who were on the court for 12,000 total oints, but has a large effect on layers lie Bell who were only on the court for 100 total oints. On one hand, this maes Bell s defensive ability that much more imressive since his defensive contribution was shrun to the average by a large amount. On the other hand, even Bell might have gotten lucy while he was on the court 4921

9 Section on Statistics in Sorts JSM 2010 and therefore a better estimate of his ability is to shrin it toward the average. Finally, many lus-minus statistics only rovide an estimate of the oint differential a layer creates when they are in the game relative to an average layer. In contrast, analyses using layer statistics often have misleading interretations, e.g. the worst shooter in the league should shoot more to increase their contribution. The aroach described here taes a middle ground where layers themselves are modeled rather then their statistics, but information about their offensive and defensive contributions can be inferred. References Berri, D. J. 1999), Who is most valuable? Measuring the layer s roduction of wins in the National Basetball Association, Managerial and Decision Economics, 20, Broos, S.. and Gelman, A. 1997), General methods for monitoring convergence of iterative simulations, Journal of Comutational and Grahical Statistics, 7, Gelfand, A. E. and Smith, A. F. M. 1990), Samling-based aroaches to calculating marginal densities, Journal of the American Statistical Association, 85, Heeren, D. 1988), The Basetball Abstract, rentice Hall, Englewood Cliffs, NJ. Hollinger, J. 2005), ro Basetball Forecast, otomac Boos Inc., Dulles, VA. Kubato, J., Oliver, D., elton, K., and Rosenbaum, D. 2007), A starting oint for analyzing basetball statistics, Journal of Quantitative Analysis in Sorts, 3, Lawler, G. F. 1995), Introduction to Stochastic rocesses, Chaman & Hall/CRC. McCullagh,. and Nelder, J. A. 1999), Generalized Linear Models, Chaman & Hall/CRC, 2nd edn. R Develoment Core Team 2009), R: A Language and Environment for Statistical Comuting, R Foundation for Statistical Comuting, Vienna, Austria, ISBN Rosenbaum, D. T. 2004), Measuring how NBA layers hel their teams win. htt: // Winston, W. 2009), Mathletics: How Gamblers, Managers, and Sorts Enthusiasts Use Mathematics in Baseball, Basetball, and Football, rinceton University ress. 4922

10 Section on Statistics in Sorts JSM 2010 A. Full conditional distributions Assume θ, δ iid Gaθ ; α θ, β θ )Gaδ, α δ, β δ ) and data observed according to equation 1). θ, δ y, θ, δ ) [ n o y ih; Ht i n Ht i n At i =1 =1 =1 θ h i δ a i θ a i δh i θ h i δ a i ) ) yih ex ) yia ex o Ht i At i θ α θ 1 ex θ β θ ) δ α δ 1 ex δ β δ ) n [ n [ θ h i θ a i ) δ a yih ] i ) δ h yia ] i ex θ ex θ n n θ α θ 1 ex θ β θ ) δ α δ 1 ex δ β δ ) n h θ i y ih +a i y ia n h δ i y ia +a i y ih ex θ ex δ n n θ α θ 1 ex θ β θ ) δ α δ 1 ex δ β δ ) θ α θ+ n [h i y ih +a i y ia] 1 δ α δ+ n [h i y ia +a i y ih] 1 y ia; At i =1 =1 h iht i =1 θ h i δ a i θ a i δh i a iat i h iht i h iat i θ a i δ h i ) ) θ h i δ a i δ θ a i δh i δ θ h i δ a i θ a i δh i n ex θ β θ + h iht i )] θ, δ) n a iat i n h iht i + a iat i + a iht i n ex δ β δ + h iat i θ h i δ a i θ a i δh i θ a i θ h i δh i δ a i θ h i θ a i δ a i δh i + a iat i + a iht i θ a i θ h i δh i δ a i 4923