Gaining power in multiple testing of interval hypotheses via conditionalization

TL;DR

This paper introduces a new conditionalization method to improve multiple testing procedures when null p-values are inflated, enhancing power while controlling error rates.

Contribution

The paper proposes a novel filtering and correction approach for multiple testing that is valid under independence and improves power in scenarios with inflated p-values.

Findings

Significant power gains in multiple testing scenarios with inflated p-values.

Validity of the procedure under independence of p-values.

Conditions for FWER control with the Bonferroni method.

Abstract

In this paper we introduce a novel procedure for improving multiple testing procedures (MTPs) under scenarios when the null hypothesis $p$-values tend to be stochastically larger than standard uniform (referred to as 'inflated'). An important class of problems for which this occurs are tests of interval hypotheses. The new procedure starts with a set of $p$-values and discards those with values above a certain pre-selected threshold while the rest are corrected (scaled-up) by the value of the threshold. Subsequently, a chosen family-wise error rate (FWER) or false discovery rate (FDR) MTP is applied to the set of corrected $p$-values only. We prove the general validity of this procedure under independence of $p$-values, and for the special case of the Bonferroni method we formulate several sufficient conditions for the control of the FWER. It is demonstrated that this 'filtering' of $p$-values can yield considerable gains of power under scenarios with inflated null hypotheses $p$-values.