Validity and Power of Heavy-Tailed Combination Tests under Asymptotic Dependence

TL;DR

This paper develops a theoretical framework for heavy-tailed combination tests, like the Cauchy test, showing they remain valid under dependence and can be more powerful than Bonferroni when p-values are dependent.

Contribution

It introduces a unified model using multivariate regularly varying copulas to analyze the validity and power of heavy-tailed combination tests under dependence.

Findings

Combination tests with tail index γ ≤ 1 are asymptotically valid.

γ = 1 maximizes power while maintaining validity.

Tests with γ approaching 0 become overly conservative.

Abstract

Heavy-tailed combination tests, such as the Cauchy combination test and harmonic mean p-value method, are widely used for testing global null hypotheses by aggregating dependent p-values. However, their theoretical guarantees under general dependence structures remain limited. We develop a unified framework using multivariate regularly varying copulas to model the joint behavior of p-values near zero. Within this framework, we show that combination tests remain asymptotically valid when the transformation distribution has a tail index $\gamma \leq 1$, with $\gamma = 1$ maximizing power while preserving validity. The Bonferroni test emerges as a limiting case when $\gamma \to 0$ and becomes overly conservative under asymptotic dependence. Consequently, combination tests with $\gamma = 1$ achieve increasing asymptotic power gains over Bonferroni as p-values exhibit stronger lower-tail dependence and signals are not extremely sparse. Our results provide theoretical support for using truncated Cauchy or Pareto combination tests, offering a principled approach to enhance power while controlling false positives under complex dependence.