Extreme Shooters in the NBA

Transcription

1 Extreme Shooters in the NBA Manav Kant and Lisa R. Goldberg 1. Introduction December 24, 2018 It is widely perceived that star shooters in basketball have hot and cold streaks. This perception was first challenged by an influential article, Gilovich, Vallone and Tversky (1985), in which the authors argued that hot and cold streaks in basketball are a cognitive illusion and can be explained by mere chance [1]. They further stated that the breadth and persistence of the perception that something exceptional is going on illustrates just how poorly we understand randomness. Gilovich et al. mainly identified hot handedness in two ways: as an elevated probability of scoring after a streak of a fixed length, and as the tendency of a shooter to shoot in runs, or clusters, of hits and misses. Using these identifications, a number of subsequent studies came out with disparate answers to the hot hand question. Koehler and Conley (2003) determined that data from the NBA Long Distance Shootout contest did not provide evidence for hotness or for sequential dependencies [2]. More recently, however, Miller and Sanjurjo (2018) found that there is a subtle but substantial bias... in a standard measure of the conditional dependence of present outcomes on streaks of past outcomes in sequential data, and determined that upon correcting for the bias, the conclusions of prominent studies in the hot hand fallacy literature are reversed [3]. However, Daks, Desai and Goldberg (2018), accounting for the bias pointed out by Miller and Sanjurjo with a (nonparametric) permutation test, concluded that there was little evidence for hot handedness among star players of the Golden State Warriors team that won the NBA championship [4]. The brief summary given above is, of course, by no means a comprehensive history of the literature on hot handedness in basketball. Nevertheless, we hope it provides some insight into the continued debate on the subtleties in analyzing streak shooting. In this study, we explore a novel approach to the hot hand question by considering the possibility that the human perception of hot handedness may be more closely related to the percentage of shots made by a player in a game (relative to the expectations set by his season-long hit percentage) than to streaks of shots made throughout the game. We work with simple formulations of hot and cold that rely entirely on field goal percentages and number of shots taken, using data from the official NBA API on all field goals attempted in the , , and NBA seasons by top shooters. For any player, we identify extreme shooting in any particular game by determining the probability that the player would have scored more or an equivalent number of shots (out of those he took) than he actually did in the game, given that the probability of the player making any given shot was the player s season-long shooting percentage. The player is deemed an extreme shooter if certain high or low quantiles of the distribution of these probabilities, or single game p-values, throughout a season differ significantly from the corresponding quantiles of a distribution that assumes the player s single game p-values deviate modestly from what is expected based on his 1

2 season-long shot percentage. Hot and cold handedness, in our study, are merely two different cases of extreme shooting. A player can be hot handed, cold handed, or both. Our concept of "modest deviation" is formulated in terms of a mixture based on binomial distributions determined by the number of shots the player took in each game and his season-long shooting percentage. We determine that certain players including Klay Thompson, who is often considered the prime example of hot handed shooting display signs of hot and cold handedness in our extreme shooting formulation of the hot hand question. Our findings could influence the way that coaches select their lineups (possibly selecting hot handed players for matches against much better opponents) and how fans connect with the game of basketball (by connecting people s intuition with hard data). 2. Methodology 2.1. Motivating Example Consider a hypothetical player that takes 20 shots in each game and makes 50% of those shots throughout the course of the season. We don t expect him to make exactly 10 of 20 shots in each game, and we don t expect him to make all 20 shots in half the games and no shots in the other half. Our expectations lie somewhere between those two extremes. We might select a binomial distribution with 20 trials and mean 0.5 as a model of how we might expect his game averages to be distributed. If the game averages are unusually concentrated in the tails of the distribution, our player is an extreme shooter Identifying Extreme Shooting in a Single Game Generalizing from the previous example, a player s shooting percentage in game n, q n, is extreme if it is unusually high or low relative to his season average, q. A complication is that unlike our hypothetical player, a real player is bound to attempt different numbers of shots in different games. Thus, simply analyzing the player s hit percentage in all games would be an oversimplification of the problem. A player who makes 12 shots out of 16 is a more impressive shooter than one who makes 3 shots out of 4. This, however, would not be reflected in an analysis based purely on shot percentages. To put our observations on a more common footing, we compute the p-value, p n, of q n with respect to the binomial distribution B(s n, q), where s n is the number of shots attempted in game n. The binomial distribution serves as a model of a modest distribution around the season average, and the p-value is the probability of equalling or exceeding the observed value in the binomial distribution B(s n, q). A low p-value corresponds to a high game average and conversely since, ) * p " = $ % s " k ( q + (1 q) )*3+. +,) * - * In the previous example, if the player s season-long hit fraction was 0.5, the p-value for the former case (12 shots out of 16) would be , whereas that for the latter case (3 shots out of 4) would be This reflects the fact that it becomes more difficult to attain or exceed a given high shooting percentage as the number of attempted shots increases Evaluating a Player Over the Course of a Season 2

3 We look at whether a player shoots in an extreme manner by examining P, the collection of a player s single-game shooting percentage p-values realized over the course of a season. Consider first, the cumulative distribution function (CDF) associated with P, denoted F. Roughly speaking, an unexceptional distribution of game shooting percentages would drive F toward the CDF of the uniform distribution of the unit interval, denoted U, which is a 45 degree line. However, this is not exactly right, as a binomial distribution is discrete and the single-game p-values are constructed from different binomial distributions. Consequently, a more accurate null hypothesis is a mixture of discrete distributions. Let M be the CDF of a mixture of the p-value distributions of binomial distributions, where the weight of the p-value distribution of B(s, q) in the mixture is the fraction of games in the season in which the player attempted s shots. In what follows, we will use simulations of seasons based on the null hypothesis constructed above to determine which, if any, quantiles of F are exceptional and what these exceptional quantiles indicate about the distribution. For example, an unusually low 10% quantile would correspond to a preponderance of unusually high-percentage games (indicating hot handed shooting), and an unusually high 90% quantile would correspond to a preponderance of unusually low-percentage games (indicating cold handed shooting). In general, unusually low lower quantiles (quantiles below 50%) indicate hot handedness, and unusually high upper quantiles (quantiles above 50%) indicate cold handedness. Our simulation is based on 30,000 replications of a star player s season. In each replication, we draw a shooting percentage at random from each binomial distribution, B(s n, q) and note its quantile in the empirical distribution of single game p-values associated to the draws. Finally, for all Q such that Q is a multiple 5 on the interval [10, 90], we compute the p-value of the Q% quantile of the player s single game p-value distribution with respect to the distribution of Q% quantiles of the 30,000 season simulations. If Q > 50 and this Q% quantile p-value is greater than 0.9, then we deem the player cold handed. If Q < 50 and the Q% quantile p-value is less than 0.1, then we deem the player hot handed. For practical purposes, we define the Q% quantile of any distribution consisting of n values to be the value at the (Q/100)n th position when the values are placed in ascending order Criteria for Player Evaluation In order to reduce the effects of the noise that is present in small datasets, we set criteria for evaluation based on the number of games played in a season and the average number of field goals attempted per game by any player. We look at a player only if he played 60 or more games (out of a possible 82) and attempted an average of at least 14 field goals per game during the season being studied. By ensuring that each player played a reasonable number of games, we reduce the probability of making a Type I error; it is more likely that, by sheer chance, the lowest 3 p-values in a distribution consisting of 30 p-values are low enough to indicate hot handedness when there is none, than that the lowest 8 p-values in a distribution consisting of 80 p-values are as such. Our criterion on average number of field goals attempted accomplishes the same purpose, but for single game p-values rather than for the season-long distribution of p-values. Our use of a player s season-long shooting percentage to set expectations for his shot percentage in any given game introduces an element of look-ahead bias to our analysis. This is because the player s season-long hit percentage is determined from an aggregate of the data from all of the games he played, many of which may have been played after the game being studied. In essence, we measure the player s performance against a null distribution that is based on a value that was not 3

4 fixed at the time of the game, and that in fact, the game being studied played a role in determining. In order to reduce the effects of look-ahead bias on our analysis, we determined that, to be declared an extreme shooter, a player must not have a clear trend in his hit percentage throughout the season. For example, if a player who attempted 20 shots in each game (for the sake of simplicity) consistently shot at around 20% for the first half of the season and at around 80% for the second, then a season-long hit percentage of about 50% may be an incorrect expectation for either half of the season, as the player likely simply got better at hitting shots around halfway through the season. Then, neither a 20% hit percentage in a game at the beginning of the season nor an 80% hit percentage in a game at the end of the season would be considered extreme performances, as they would both correspond to the expected performances for their respective portions of the season. 3. Results We first applied our methodology to those players in the season that satisfied our player evaluation criteria (see section 2.4), and then analyzed the members of that group that also satisfied those criteria in the and seasons Season Results For the season, we found 4 players--klay Thompson, DeMar DeRozan, Gordon Hayward, and Jimmy Butler--to be hot handed, cold handed, or both (see Table I). Table I. Quantile p-values for DeMar DeRozan, Klay Thompson, Jimmy Butler, and Gordon Hayward in the season. See Table V for quantile p-values for all players in the season. In our analysis, we found that the quantile p-values for the 30%, 35%, and 40% quantiles of Klay Thompson s single game p-value distribution were 0.052, 0.065, and respectively (see Table I). These values give us reason to believe that Thompson is a hot handed shooter, as they indicate that his shooting follows a distribution that has a greater concentration of single game p-values at its lower end than does the null. As lower p-values result from higher shot percentages (see section 2.2), this implies that Thompson has a proclivity to get hot (with respect to the expectations set by his season-long hit fraction) in a disproportionately large number of games. A visualization of this empirical result is presented in Figure 1.1 (A). At the 30%, 35%, and 40% quantiles (demarcated by the dashed yellow lines), the CDF of Klay Thompson s single game p-value distribution is clearly to the left of the null distribution. However, at the 45% quantile and above, Thompson s distribution sticks fairly close to the null. Finally, looking at Figure 1.1 (B), we determined that Klay Thompson s shooting percentages throughout the season did not seem to follow any clear trend. Thus, look-ahead bias likely did not play a large role in the outcomes from our analysis. 4

5 (A) (B) Figure 1.1. The graph on the left displays the CDF of Klay Thompson s single game p-value distribution in the season (in red) as compared to that of the null distribution for Klay Thompson that season (in green). The graph on the right is a time series graph of Thompson s hit fraction in each game throughout the season. We further found DeMar DeRozan and Gordon Hayward to be cold handed (in contrast with Klay Thompson) as illustrated in Figure 1.2 and Figure 1.3. Figure 1.2. DeMar DeRozan s single game p-value distribution for the season as compared to the null distribution, and the time series graph of DeRozan s hit fractions throughout that season. 5

6 Figure 1.3. Gordon Hayward s single game p-value distribution for the season as compared to the null distribution, and the time series graph of Hayward s hit fractions throughout that season. Finally, we found Jimmy Butler to be a general extreme shooter (showing signs of hot and cold handedness) in the season, with 25%, 85%, and 90% quantile p-values of 0.054, 0.917, and respectively (see Table I). These values indicate that his shooting in the season followed a distribution in which single game p-values are concentrated at both tails. This can be seen in the graph of the CDF associated with Butler s single game p-value distribution and the time series graph of his crude shooting percentages, the latter of which makes apparent the variability associated with Butler s shooting (see Figure 1.4). Figure 1.4. Jimmy Butler s single game p-value distribution for the season as compared to the null distribution, and a time series graph of Butler s hit fractions throughout that season Season Results For the NBA season, we found evidence that one player--demarcus Cousins--was a cold handed shooter, as illustrated by Table II and Figure 2. 6

7 Table II. Quantile p-values for DeMarcus Cousins single game p-value distribution in the season. See Table IV for quantile p-values for all players in the season. Figure 2. DeMarcus Cousins single game p-value distribution for the season as compared to the null distribution, and the time series graph of Cousins hit fractions throughout that season Season Results We found 4 players--russell Westbrook, Andrew Wiggins, Kemba Walker, and Klay Thompson--to be hot handed, cold handed, or both during the NBA season (see Table III). Westbrook in particular showed clear signs of hot handedness, with quantile p-values of 0.026, 0.087, and at his 10%, 15%, and 20% quantiles respectively (see Table III and Figure 3.1). Table III. Quantile p-values for Russell Westbrook, Andrew Wiggins, Kemba Walker, and Klay Thompson in the season. See Table VI for quantile p-values for all players in the season. 7

8 Figure 3.1. Russell Westbrook s single game p-value distribution for the season as compared to the null distribution, and the time series graph of Westbrook s hit fractions throughout that season. We also found evidence that both Andrew Wiggins and Kemba Walker were hot handed in the season (see Figure 3.2 and Figure 3.3). Figure 3.2. Andrew Wiggins single game p-value distribution for the season as compared to the null distribution, and the time series graph of Wiggins hit fractions throughout that season. 8

9 Figure 3.3. Kemba Walker s single game p-value distribution for the season as compared to the null distribution, and the time series graph of Walker s hit fractions throughout that season. Finally, we found evidence that Klay Thompson showed signs of both hot handedness and cold handedness in the season. This is evident in the graph of the CDF of his single game p- value distribution for that season--the graph seems to be to the left of the null distribution at the lower quantiles, and to the right of the null at the upper quantiles (see Figure 3.4). Figure 3.4. Klay Thompson s single game p-value distribution for the season as compared to the null distribution, and the time series graph of Thompson s hit fractions throughout that season. 4. Conclusion The perception of the hot hand in basketball is common among fans. However, many previous studies that focused on streak shooting within games largely did not find the statistical evidence for this phenomenon. 9

10 By looking at shot percentages throughout a season rather than shot streaks within a game, we determined that there is reason to believe that a number of NBA players are extreme shooters-- players that shoot according to single game p-value distributions that differ at the tails from what can be expected based on their season-long hit fractions and the number of field goals they attempted in each game. Some of these players, most visibly Klay Thompson, are widely perceived to be players that get hot in certain games, achieving results that normally seem far out of the range of possibilities. We did in fact find Thompson to be a hot-handed shooter in one season out of the three we studied and an extreme shooter in another. This indicates that our methodology may be better aligned with fan psychology than traditional streak shooting-based methods. 5. Acknowledgements We thank Alex Papanicolaou for providing us with shot data from the NBA API PHP Library. We are grateful to Nishant Desai and Josh Miller for their helpful comments in the initial stages of our study. 6. Resources Code hosted at: 10

11 References [1] Gilovich, Thomas Robert Vallone & Amos Tversky (1985). The hot hand hand in basketball: On the misperception of random sequences. Cognitive Psychology 17, [2] Koehler, J. J. & Conley, C. A. (2003) The hot hand myth in professional basketball. Journal of Sports & Exercise Psychology, 25(2), [3] Miller, J. & Sanjurjo, A. (2018). Surprised by the gambler s and hot hand fallacies? a truth in the law of small numbers. Econometrica 86(6), [4] Daks, A., Desai, N. and Goldberg, L.R. (2018). Do the Golden State Warriors have hot hands? The Mathematical Intelligencer 40(4),

12 Appendix Below are complete tables of quantile p-values for all players analyzed in the , , and NBA seasons. Table IV player quantile p-values. 12

13 Table V player quantile p-values. 13

14 Table VI player quantile p-values. 14