Uncertainty in the Hot Hand Fallacy: Detecting Streaky Alternatives to Random Bernoulli Sequences

Abstract

We study a class of tests of the randomness of Bernoulli sequences and their application to analyses of the human tendency to perceive streaks as overly representative of positive dependence-the hot hand fallacy. In particular, we study tests of randomness (i.e., that trials are i.i.d.) based on test statistics that compare the proportion of successes that directly follow k consecutive successes with either the overall proportion of successes or the proportion of successes that directly follow k consecutive failures. We derive the asymptotic distributions of these test statistics and their permutation distributions under randomness and under general models of streakiness, which allows us to evaluate their local asymptotic power. The results are applied to revisit tests of the hot hand fallacy implemented on data from a basketball shooting experiment, whose conclusions are disputed by Gilovich, Vallone, and Tversky (1985) and Miller and Sanjurjo (2018a). We establish that the tests are insufficiently powered to distinguish randomness from alternatives consistent with the variation in NBA shooting percentages. While multiple testing procedures reveal that one shooter can be inferred to exhibit shooting significantly inconsistent with randomness, we find that participants in a survey of basketball fans over-estimate an average player's streakiness, corroborating the empirical support for the hot hand fallacy.