Statistical Analysis of Basketball Shooting in a Controlled (Practice)

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When I watch basketball on television, it is a common occurrence to have an announcer state that some player has the hot-hand. This raises the question: Are Bernoulli trials an adequate model for the outcomes of successive shots in basketball? This paper addresses this question in a controlled (practice) setting. A large simulation study examines the power of the tests that have appeared in the literature as well as tests motivated by the work of Larkey, Smith, and Kadane (LSK). Three test statistics for the null hypothesis of Bernoulli trials have been considered in the literature; one of these, the runs test, is effective at detecting one-step autocorrelation, but poor at detecting nonstationariy. A second test is essentially equivalent to the runs test, and the third is shown to be worthless. The LSK-motivated tests are shown to be effective at detecting nonstationarity. Finally, a case study of 2,000 shots by a single player is analyzed. For this player, the model of Bernoulli trials is inadequate. KEY WORDS: Bernoulli trials, the hot-hand, power, simulation study, case study. 1

1 Introduction
In this paper I consider a statistical analysis of basketball shooting in a controlled (practice) setting, with special interest in the hot-hand. In Section 2, I review and critically examine the two seminal papers on this topic: Gilovich, Vallone, and Tversky (GVT) [5], and Tversky and Gilovich (TG1) [10]. A simulation study of power is presented in Section 3. Finally, in Section 4, a case study of 2,000 trials is analyzed. In GVT and TG1, three additional topics appear which are beyond the scope of this paper: 1. Modeling game free throw shooting,

2. Modeling game shooting, and
3. Opinions and misconceptions of fans.
Readers interested in the first of these topics should refer to Wardrop [12] for a further analysis of the free throw data from the papers.
Several researchers have considered the second topic; the interested reader is referred to Larkey, Smith, and Kadane (LSK) [7], Tversky and Gilovich (TG2) [11], Hooke [6] and Forthofer [4]. For related work in baseball, see Albright [1], Albert [2], Stern and Morris [9] and Stern [8]. Topic 3 is considered briefly in Section 2 of this paper.

Finally, readers interested in statistical research in sports are referred to Bennett [3]. The chapters on basketball and baseball should prove to be of special interest to readers of this paper. 2 Review of Literature

GVT appeared in 1985 in a “psychology journal.” Four years later the same research was restructured as TG1 and appeared in a “statistics journal.” For the most part the papers present identical analyses and interpretations of the data, with the earlier paper generally providing more detail. Twenty-six members of the Cornell University varsity and junior varsity basketball teams generated the data that are examined. The players are labeled M1 (for male one) through M14, and F1 through F12. Each player provided two sequences of shot attempts: the shooting data and the prediction data. I will begin with an examination of the shooting data.

The plan was for each player to provide a sequence of 100 shots, but three of the players, M4 (90 shots), M7 (75), and M8 (50), fell short of the target number.
Twenty-six null hypotheses are tested; namely, for each player the null hypothesis is that his or her shots satisfy the assumptions of Bernoulli trials. Below is one summary of the data obtained by M9. Previous Current Shot

Shot S F Total
S 38 15 53
F 16 30 46
The researchers describe these data in two ways. First, note that M9 made 72 percent of his shots after a hit, but only 35 percent after a miss; a difference of 37 percentage points. Second, the researchers compute the serial correlation and obtain 0.37.

The researchers analyze each player’s data with three test statistics. The first two are a test of the serial correlation and the runs test. They summarize their...
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