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

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Pith's one-line read Four controlled basketball shooting experiments, including the classic study behind the hot hand fallacy, are too underpowered to detect realistic levels of streakiness, and only one shooter in the classic data is robustly non-random.

desk verdict A genuinely new asymptotic theory for run-based permutation tests of randomness, with a careful but calibration-dependent claim that existing hot hand experiments are underpowered. read the letter →

arxiv 1908.01406 v6 pith:F3WCIVGX submitted 2019-08-04 econ.EM stat.AP

David M. Ritzwoller , Joseph P. Romano This is my paper · ORCID
classification econ.EMstat.AP MSC 62G1062G0962M0262E2062P20
keywords hothandfallacyBernoullisequencespermutationtestsstreakshootingMarkovchainalternativeslocalasymptoticpowermultipletestingsmall-samplebias
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a formal statistical framework for testing whether a collection of Bernoulli sequences is truly random or streaky, then applies it to the four controlled basketball shooting experiments that have shaped the hot hand debate. It shows that permutation tests comparing success rates after streaks of makes with overall success rates, or with success rates after streaks of misses, are the only tests with exact finite-sample type I error control, and it derives their asymptotic distributions and local power against a class of Markov-chain alternatives. The central empirical conclusion is that all four experiments lack adequate power to detect realistic deviations from randomness calibrated from NBA shooting variation. One shooter in the classic 1985 experiment is robustly streaky after multiple-testing corrections, but the evidence is confined to that shooter. The paper therefore argues that the existing experiments cannot settle whether the hot hand is real or whether people systematically overestimate it, and that a direct test requires larger samples and belief measurements on the same scale as the streaky parameters.

What carries the argument

The load-bearing object is a parsimonious class of Markov-chain streaky alternatives: each shooter is either random or streaky, with a proportion $\zeta$ of streaky shooters, and a streaky shooter increases the chance of a make (and of a miss) by $\epsilon$ after a run of $m$ consecutive makes (or misses). Against this class the paper studies permutation tests based on the plug-in statistics $\hat{P}_{n,k}(X_i) - \hat{p}_{n,i}$ (success rate after $k$ consecutive makes minus overall success rate) and $\hat{D}_{n,k}(X_i)$ (success rate after $k$ consecutive makes minus success rate after $k$ consecutive misses), averaged over shooters for joint tests. An exact finite-sample theorem shows that permutation tests are the only tests with exact type I error control, avoiding the small-sample bias of the asymptotic approximations; asymptotic results then give the limiting permutation distribution and a closed-form local power approximation: for $m=k=1$, the required total sample size satisfies $ns \approx \big((z_{1-\alpha} - z_{1-\beta})/(2\zeta\epsilon)\big)^2$, with the general power given by $1 - \Phi(z_{1-\alpha} - \phi_T(k,m,h)\zeta)$. This machinery turns power analysis from a heavy simulation into an analytic calculation and is what allows the paper to assess all four experiments on one scale.

What would settle it

Simulate the classic experiment's design, about 26 shooters taking roughly 100 shots each, under the paper's own Markov-chain alternative with $\epsilon=0.038$, $\zeta=0.5$, and $m=3$, and run the same stratified permutation test; if more than 80% of simulated datasets reject randomness, the claim that these experiments cannot detect realistic streakiness is wrong.

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Extended reading notes

Core claim

The paper's central claim is that a definitive empirical statement about the hot hand fallacy cannot be made from the four controlled shooting experiments currently available. The theoretical part establishes exact finite-sample permutation tests for the null hypothesis that shot outcomes are i.i.d. Bernoulli, shows that these are the only tests with exact type I error control, and characterizes their asymptotic power against a class of Markov-chain alternatives whose two parameters, $\epsilon$ (size of the streak effect) and $\zeta$ (share of streaky shooters), are calibrated from the distribution of NBA field-goal percentages. Applied to the data, the tests reject randomness for exactly one shooter in the classic experiment after controlling for multiple testing, and that shooter's sequence is genuinely extreme; the evidence against randomness is otherwise absent. Because all four experiments would detect the benchmark alternatives only with low probability, the paper concludes that the existing data cannot resolve whether shooting is streaky or whether people overestimate streakiness, and that substantially larger experiments are required.

Load-bearing premise

The underpowered conclusion rests on the assumption that realistic streakiness is no stronger than the amount implied by the spread of NBA shooting percentages; if the real hot-hand effect is much larger, some of the experiments would have enough power.

Editorial extensions

If this is right

  • If the power conclusion is right, a failure to reject randomness in the existing experiments is not evidence that basketball shooting is random; it is evidence only that the studies were too small.
  • The sample-size formula gives future experimenters a direct target: for the paper's benchmark alternative, the number of shooters times shots per shooter must be roughly $(z_{1-\alpha} - z_{1-\beta})^2 / (4\zeta^2\epsilon^2)$, which is far larger than any of the four experiments.
  • The robust rejection for one shooter means that at least one player in the classic data shot in a way that is very unlikely under randomness, so the claim that nobody has a hot hand is not supported even by the experiment that founded the fallacy.
  • A direct test of the fallacy requires measuring observers' probabilistic expectations of a make after a streak on the same scale as $\bar\theta^P_k$ or $\bar\theta^D_k$, not the hypothetical survey questions used so far.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If real streakiness is as large as the single robust shooter's estimated $\theta_D \approx 0.38$, some of the experiments would have had adequate power, so the underpowered conclusion should be read as conditional on the calibrated $\epsilon$ range rather than as a universal statement.
  • The same analytic power framework could be applied to other streak literatures, such as mutual-fund performance persistence or weak-form market efficiency, where small samples and null results may be underpowered in the same way.
  • A natural next step is to estimate $\epsilon$ and $\zeta$ directly from large shot-level NBA tracking data instead of calibrating them from cross-player variation, which would replace the paper's modeling judgment with measured parameters.
  • The proposed belief-elicitation design, asking observers to state the probability of the next make before each shot with proper scoring, could be piloted side-by-side with the old hypothetical surveys to see whether framing alone explains the gap between stated and revealed beliefs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper develops asymptotic theory for a class of permutation tests of randomness of Bernoulli sequences, using test statistics that compare the proportion of successes after k consecutive successes with either the overall success proportion or the proportion after k consecutive failures. The main theoretical results characterize the asymptotic and permutation distributions under the null, under stationary alternatives, and under a class of Markov-chain "streaky" alternatives, and yield local asymptotic power approximations. The paper then applies these tools to four controlled basketball shooting experiments. It finds that the GVT data contain one shooter (Shooter 109) whose sequence is significantly non-random even after multiple-testing correction, but that the aggregate evidence against randomness is concentrated in that shooter. It argues that all four experiments are underpowered against conservative Markov-chain alternatives calibrated to the cross-player dispersion of NBA field-goal percentages, and concludes that substantially larger datasets are needed to measure streakiness in basketball shooting.

Significance. If the results hold, the paper makes a valuable contribution to the long-running hot-hand debate and to the statistics of testing Bernoulli sequences. The distinction between individual, joint, and simultaneous tests is important, and the analytic power approximations are a practical advance that substantially reduces the computational cost of power calculations. The replication package and the placement of proofs in Online Appendix K are strengths, as is the simulation evidence at n=100 in Figures 2 and 3, which supports the accuracy of the power approximation in the low-power region. The paper's central empirical conclusion, that the existing controlled shooting experiments cannot resolve the hot-hand question, is consequential for behavioral economics. However, as detailed below, the uniqueness theorem for permutation tests is false as stated, and the "realistic" effect-size calibration is a modeling judgment that is load-bearing for the underpowered claim.

major comments (3)
  1. [Section 3.2, Theorem 3.2] Theorem 3.2 is false as stated. A simple counterexample for n=2 and alpha=0.05 is phi(0,0)=phi(1,1)=0.05, phi(0,1)=0.10, phi(1,0)=0. Under Bernoulli(p), E[phi] = (1-p)^2*0.05 + p(1-p)*(0.10+0) + p^2*0.05 = 0.05 for every p in (0,1), so phi has exact level alpha, but phi is not invariant under permutations because phi(0,1) differs from phi(1,0). The completeness of the binomial sufficient statistic yields only E[phi | sum X_j] = alpha, not permutation invariance. The claim that permutation tests are the only tests with exact type 1 error control therefore needs correction. The later confidence-bound statement in Section 5.4 only requires exactness of the permutation test itself, but the theorem and its surrounding discussion should be revised, for instance by proving uniqueness within a restricted class of tests or by replacing the "only" claim with a conditional-exactness result.
  2. [Section 5.3 and Equation (4.2)] The central underpowered conclusion is carried by the calibration of epsilon and zeta from the between-player distribution of NBA field-goal percentages. The parameter theta_D = 2*epsilon is a within-player swing in make probability, and cross-player dispersion in average field-goal percentage does not by itself bound that within-player swing. Shooter 109's estimated theta_D of 0.379 in Table 4 shows that larger within-player swings are observable in these data. The paper's own formula (4.2) makes the conditional nature explicit: for zeta=0.5 and the NBA Three-Point contest (ns approximately 5,600), the test based on D_1 reaches 80% power at epsilon approximately 0.033, which is below the paper's stated upper value of 0.038. The paper should either provide a within-player calibration from repeated-session data, or explicitly report the boundary of parameter values for which each experiment has adequate power, and temper the language that the chosen parameterization is a "conservative upper bound."
  3. [Section 4.2, Theorem 4.1] Theorem 4.1 contains the typo "as n to 0" where the intended statement is clearly "as n to infinity". More substantively, the theorem's variance expression for the P-statistic and the subsequent Remark 4.1 are used to justify the local power formula, and the simulation evidence in Figures 2 and 3 supports the approximation in the low-power region. However, the text in Section 4.3 also notes that the approximation overestimates power when the true power is close to 0.9. Because Figure 7 uses the same asymptotic approximation to conclude that some experiments have "reasonable power" for m=1 and m=2, the paper should state this high-power caveat prominently in Section 5.3 and indicate the direction of the potential bias.
minor comments (3)
  1. [Section 4.2, Theorem 4.1] The phrase "as n to 0" in statement (ii) should be corrected to "as n to infinity".
  2. [Section 4.3 and Figure 3] The overestimation of power near 0.9 is acknowledged in the text but should be stated as a limitation of the analytic approximation in the main results, not only in the simulation section.
  3. [Data Availability Statement] "Zenondo" should be "Zenodo" in the data availability statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the power analysis is conditional on externally calibrated effect sizes, not derived from the experimental outcomes it assesses.

full rationale

The paper's theoretical results (asymptotic distributions under the null, permutation distribution limits, and local power against Markov-chain alternatives) are derived from explicit i.i.d. and Markov assumptions; they are not obtained by assuming the conclusions of the empirical application. The streaky-alternative parameters ε ∈ {0.024, 0.038} and ζ ∈ {0.25, 0.5} are calibrated from the cross-player distribution of NBA field-goal percentages (Section 5.3, Figure 6), which is external to the GVT, Miller-Sanjurjo, Jagacinski, and Three-Point Contest datasets. The underpowered conclusion in Section 5.4 is therefore a conditional statement: given these modeling choices about realistic effect sizes, the four experiments have low power. That is a judgment about effect-size calibration, not a circular derivation, and the paper explicitly flags the conditionality of the claim. The GVT empirical findings, including the Shooter 109 result, come from direct permutation tests of the observed sequences and are not fitted inputs. The few self-citations (Lehmann and Romano 2005; Romano et al. 2011; Politis and Romano 1994) are standard mathematical results or textbook methods, and none is load-bearing in a way that forces the paper's conclusions. No equation or fitted parameter is renamed as a prediction, and no uniqueness theorem is imported to rule out alternatives. The main fragility—whether NBA cross-player dispersion in average shooting percentage bounds within-player streak swings—is a substantive modeling assumption and a potential correctness concern, but not a circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The core theory has no fitted constants. The only hand-selected numbers are the alternative-scenario parameters epsilon, zeta, and m used in the power analysis, calibrated from external NBA data rather than from the experiments being tested. No new physical or conceptual entities are postulated.

free parameters (3)
  • epsilon (streakiness magnitude) = 0.024 and 0.038 used in power analysis
    Chosen from half the distance between the 33rd/66th and 25th/75th quantiles of NBA field goal percentages (Section 5.3). Hand-calibrated inputs to the power analysis, not fitted to the GVT data.
  • zeta (prevalence of streaky shooters) = 0.25 and 0.5 used in power analysis
    Chosen as a conservative upper bound based on the proportion of players with large estimated streakiness; a modeling judgment rather than an estimated parameter (Section 5.3).
  • m (streak length in alternative) = 3 benchmark, with values 1 to 4 examined
    The benchmark m=3 follows the emphasis in GVT and Miller-Sanjurjo on streaks of three; it is part of the alternative specification, not a fitted constant.
assumptions (5)
  • domain assumption Under H0, each sequence Xi is i.i.d. Bernoulli and the joint distribution is invariant under permutations (randomization hypothesis).
    Basis for permutation tests; introduced in Section 3.2.
  • domain assumption Shot outcomes across shooters are independent stationary Bernoulli processes, and under alternatives follow the class of Markov chain streaky processes in Section 4.1.
    Modeling assumption for the alternatives; specified in Sections 2 and 4.1.
  • domain assumption For the power analysis, p_i = 0.5 for all individuals.
    Section 4.1, justified by controlled shooting design where distances are set so shooting percentages are near 50%.
  • standard math The asymptotic results rely on standard CLTs for dependent sequences, including Rinott's Stein-method CLT and Ibragimov's results for alpha-mixing processes.
    Used in Theorems 3.3 and 3.4; proofs are in Online Appendix K.
  • domain assumption The NBA field goal percentage distribution is an appropriate external benchmark for calibrating realistic deviations from randomness.
    Section 5.3; if this benchmark is not representative, the underpowered conclusion is weakened.

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Pith. "Pith review of Uncertainty in the Hot Hand Fallacy: Detecting Streaky Alternatives to Random Bernoulli Sequences." pith.science (2026). https://pith.science/paper/F3WCIVGX

@misc{pith2026190801406,
  author       = {Pith},
  title        = {Pith review of: Uncertainty in the Hot Hand Fallacy: Detecting Streaky Alternatives to Random Bernoulli Sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3WCIVGX}},
  note         = {Machine review of arXiv:1908.01406}
}
read the original abstract

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 direct test of the hot hand fallacy.

Figures

Figures reproduced from arXiv: 1908.01406 by the authors.

Figure 1
Figure 1. Requisite Sample Size for Power of Tests of the Join [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Power Curve for Permutation Test Rejecting for Lar [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Power Contours for Permutation Test Rejecting for [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Shooter 109 Shooting Sequence and Permutation Dis [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]
Figure 5
Figure 5. Figure 5: Stratified Permutation Tests of H0 using D¯ k (X) and P¯ k (X) žŸ ¡¢£¤¥ 0.146 0.040 0.004 0.072 ¦§¨©ª«¬­® 0.021 0.053 0.125 0.103 ¯°±²³´µ¶ 0.368 0.126 0.013 0.129 ·¸¹º»¼½¾¿ 0.007 0.036 0.107 0.082 With Shooter 109 Without Shooter 109 Dk (X) -0.50 -0.25 0.00 0.25 -0.50 …
Figure 6
Figure 6. Figure 6: Distribution of Field Goal Shooting Percentage in [PITH_FULL_IMAGE:figures/full_fig_p033_6.png]
Figure 7
Figure 7. Figure 7: Power of Joint Tests of H0 for Four Controlled Basketball Shooting Experiments Zeta = 0.25 Zeta = 0.50 TUV WX YZ[ \]^ _`abc defgh ijklm nopqr stuvw xyz{| }~€ ‚ƒ„ † ‡ˆ‰Š‹ ŒŽ ‘’ • ™š›œ žŸ ¡¢£¤ ¥¦§¨ ©ª«¬ ­®¯° ±²³´ µ¶·¸ ¹º»¼ ½¾¿À ÁÂÃÄ ÅÆÇÈ ÉÊËÌ ÍÎÏÐ ÑÒÓÔ ÕÖ×Ø ÙÚÛÜ ÝÞ…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.