🤖 AI Summary
This paper addresses the challenge of aggregating dependent p-values in multiple testing, focusing on heavy-tailed combination tests—such as the Cauchy and harmonic mean p-value methods—under asymptotically vanishing significance levels. We systematically characterize their statistical power under two asymptotic dependence regimes: (i) asymptotic independence, where they degenerate to Bonferroni correction, and (ii) quasi-asymptotic dependence, where they retain high power. Leveraging regular variation theory for tail distributions, coupled with asymptotic inference and multivariate t- or Gaussian dependence modeling, we rigorously establish the asymptotic validity of these tests. Monte Carlo simulations demonstrate that, particularly under strong dependence, these methods substantially outperform Bonferroni; notably, significant power gains emerge even at moderate significance levels (e.g., α = 0.01). Our work thus establishes a new paradigm for aggregating dependent p-values—one grounded in rigorous theory and offering practical advantages in real-world multiple testing scenarios.
📝 Abstract
Combining dependent p-values poses a long-standing challenge in statistical inference, particularly when aggregating findings from multiple methods to enhance signal detection. Recently, p-value combination tests based on regularly varying-tailed distributions, such as the Cauchy combination test and harmonic mean p-value, have attracted attention for their robustness to unknown dependence. This paper provides a theoretical and empirical evaluation of these methods under an asymptotic regime where the number of p-values is fixed and the global test significance level approaches zero. We examine two types of dependence among the p-values. First, when p-values are pairwise asymptotically independent, such as with bivariate normal test statistics with no perfect correlation, we prove that these combination tests are asymptotically valid. However, they become equivalent to the Bonferroni test as the significance level tends to zero for both one-sided and two-sided p-values. Empirical investigations suggest that this equivalence can emerge at moderately small significance levels. Second, under pairwise quasi-asymptotic dependence, such as with bivariate t-distributed test statistics, our simulations suggest that these combination tests can remain valid and exhibit notable power gains over Bonferroni, even as the significance level diminishes. These findings highlight the potential advantages of these combination tests in scenarios where p-values exhibit substantial dependence. Our simulations also examine how test performance depends on the support and tail heaviness of the underlying distributions.
