FWER Goes to Zero for Correlated Normal

TL;DR

This paper investigates the asymptotic behavior of the Bonferroni method for controlling the familywise error rate in multiple testing, showing it tends to zero under certain correlation structures as the number of hypotheses increases.

Contribution

It proves that Bonferroni FWER approaches zero asymptotically in correlated normal models, extending to generalized error rates and arbitrary correlations.

Findings

Bonferroni FWER tends to zero with increasing hypotheses in equicorrelated normal models.

Results extend to generalized error rates and arbitrary correlation structures.

Provides theoretical insights into error control in large-scale multiple testing.

Abstract

Familywise error rate (FWER) has been a cornerstone in simultaneous inference for decades, and the classical Bonferroni method has been one of the most prominent frequentist approaches for controlling FWER. The present article studies the limiting behavior of Bonferroni FWER in a multiple testing problem as the number of hypotheses grows to infinity. We establish that in the equicorrelated normal setup with positive equicorrelation, Bonferroni FWER tends to zero asymptotically. We extend this result for generalized familywise error rates and to arbitrarily correlated setups.