We study a class of  tests of the randomness of  Bernoulli sequences and their application to analyses of the human tendency to perceive streaks of consecutive successes 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, under a set of general stationary processes, and under a Markov model of "streakiness" , which allow us to evaluate their local asymptotic power. The results are applied to evaluate tests of randomness 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 substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. Although multiple testing procedures reveal that one shooter in the experiment exhibits a shooting pattern significantly inconsistent with randomness - supplying strong evidence that basketball shooting is not random for all shooters all of the time - we find that the evidence against randomness  is limited to this shooter. Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness in Bernoulli sequences .
We study a class of permutation tests of the randomness of a collection of Bernoulli sequences and their application to analyses of the human tendency to perceive streaks of consecutive successes as overly representative of positive dependence - the hot hand fallacy. In particular, we study permutation tests of the null hypothesis 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 characterize the asymptotic distributions of these test statistics and their permutation distributions under randomness, under a set of general stationary processes, and under a class of Markov chain alternatives , which allow us to derive their local asymptotic power. The results are applied to evaluate the empirical support for the hot hand fallacy provided by four controlled basketball shooting experiments . We establish that substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. In one experiment for which we were able to obtain data, multiple testing procedures reveal that one shooter exhibits a shooting pattern significantly inconsistent with randomness - supplying strong evidence that basketball shooting is not random for all shooters all of the time . However, we find that the evidence against randomness in this experiment is limited to this shooter. Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness , and thereby constitute a direst test of the hot hand fallacy .
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