Bayesian Adjusted Plus/Minus
The plus/minus statistic in basketball is considered the gold standard for evaluating a player’s impact on the team. It answers the simple question: Is the team better or worse when this player is on the court?
The main issue is that the statistic struggles to filter out individual impact. A team might find that it outscores opponents by a large margin when a particular player is on the court but what if this player plays most of his minutes alongside the team’s best player, and against the opposing team’s bench unit? Given this information, the player’s plus/minus is no longer very reliable. Attempts have been made, however, to address some of these gray areas using Bayesian statistics. There are many different ways to apply Bayesian concepts; one of these involves using prior beliefs to update current knowledge.
With respect to plus/minus, the prior beliefs could be the players’ plus/minus per game over the last two years. This can be modeled as a normal distribution, where the mean, median, and mode are all equal. The prior beliefs, or distributions, are then used to adjust the current knowledge, which will be treated as the players’ plus/minus per game for this season as of December 2, 2018. The adjusted plus/minus values can then be compared to the players’ current values for this season to see how closely they match.
Before doing that however, players’ games are separated using k-means clustering based on the final scoring margin between the players’ team and the opposing team. This is done to control for the fact that the team’s performance as a whole is strongly correlated with individual plus/minus. Taking LeBron James as an example, the correlation between his individual plus/minus and his team’s plus/minus is shown for all games from the past two seasons.
The graph on the right shows games separated by cluster: cluster 1 — games lost by wide margin, cluster 2 — close games lost, cluster 3 — close games won, cluster 4 — games won by wide margin.
The posterior distribution for players’ plus/minus was calculated for each cluster. The “Posterior Mean” column of the table below displays the adjusted plus/minus values for each cluster of games.
Three more steps are needed to calculate the Bayes-adjusted plus/minus: multiply the posterior means by the number of games in each cluster; sum the products; and divide by the number of total games played. Below is the calculation for James, who played in 23 games this season as of 12/2/18.
The Bayes-adjusted plus/minus for LeBron James is 2.386. The table below shows the results for a few other players.
Discussion
As of December 2, 2018, players had at most 25 games under their belt but have played several more since, which means that their plus/minus now is likely more representative of their impact on the team than what was indicated two months ago. A Bayesian adjustment can serve as a prediction of what the player’s plus/minus is likely to be if he or she were to play more games.
While the Bayes estimated results are a bit off compared to the current values, they did a reasonable job of predicting the direction in which the player’s plus/minus was likely to move. This was not true for Russell Westbrook, whose plus/minus has fallen dramatically since 12/2/18, despite his Bayes-adjusted value predicting otherwise. The model shows potential for serving as a good estimator however given the case of Kyrie Irving, whose Bayes-adjusted plus/minus turned out to be greater than both his plus/minus over the last two seasons, as well as his plus/minus this season as of 12/2/18. The Bayes-adjusted plus/minus is generally expected to fall somewhere in between those two values but instead it surpassed both values in Irving’s case, mirroring his actual plus/minus as of 2/5/19.
While it might not be possible to completely isolate a player’s individual impact on the team, Bayesian statistics create a clearer picture by making use of prior information and knowledge.
