Statistical Analysis of Basketball Shooting in a Controlled (Practice)

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.
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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