Bound on FWER for correlated normal distribution

TL;DR

This paper derives an asymptotic upper bound on the family-wise error rate (FWER) for Bonferroni procedures in correlated normal hypothesis testing, showing that FWER decreases with positive correlation and highlighting the need for correlation correction.

Contribution

It provides a novel asymptotic bound on FWER for equicorrelated normal hypotheses, improving upon previous distribution-free bounds that overestimate error rates as hypotheses grow.

Findings

FWER asymptotically decreases as a convex function of correlation rho.

Bonferroni controls FWER at a level smaller than alpha under positive correlation.

Correlation correction is necessary for accurate error control in correlated tests.

Abstract

In this paper,our main focus is to obtain an asymptotic bound on the family wise error rate (FWER) for Bonferroni-type procedure in the simultaneous hypotheses testing problem when the observations corresponding to individual hypothesis are correlated. In particular, we have considered the sequence of null hypotheses H_{0i} : X_i follows N(0,1) , (i=1,2,....,n) and equicorrelated structure of the sequence (X_1,....,X_n). Distribution free bound on FWER under equicorrelated setup can be found in Tong(2014). But the upper bound provided in Tong(2014) is not a bounded quantity as the no. of hypotheses(n) gets larger and larger and as a result,FWER is highly overestimated for the choice of a particular distribution (e.g.- normal). In the equicorrelated normal setup, we have shown that FWER asymptotically is a convex function (as a function of correlation (rho)) and hence an upper bound on the FWER of Bonferroni-(alpha) procedure is alpha(1-\rho).This implies,Bonferroni's method actually controls the FWER at a much smaller level than the desired level of significance under the positively correlated case and necessitates a correlation correction.