Abstract

A common problem in statistics is how to combine results from many tests when the tests are not independent. Traditional approaches, like Bonferroni, are valid but often overly conservative and lose power. Recently, “heavy‑tailed” methods, such as the Cauchy combination test and the harmonic mean p‑value, have become increasingly popular in genetic applications as more efficient ways to aggregate signals and handle unknown dependence. In this talk, I will present a unified perspective on the validity and power of these methods under different dependence structures. I will explain when they behave like Bonferroni—offering little advantage—and when they can deliver clear power gains. In particular, I will show that when p‑values are pairwise asymptotically independent (such as pairwise normally distributed), heavy‑tailed combination tests are equivalent to the Bonferroni test as significance thresholds shrink. However, under stronger forms of tail dependence modeled via multivariate regularly varying copulas, they remain asymptotically valid and achieve substantial improvements. A key insight is that the tail index of the transformation distribution determines the trade‑off between validity and power, with γ = 1 maximizing power while preserving validity, whereas Bonferroni emerges as the degenerate case when γ → 0. These results offer both theoretical insights and practical guidance, and I will illustrate a few examples in genetic analysis.