A general approach to construct powerful tests for intersections of one-sided null-hypotheses based on influence functions

TL;DR

This paper introduces a general Wald-type test for intersecting one-sided null hypotheses, leveraging influence functions to account for the joint behavior of p-values, resulting in a powerful testing procedure with controlled family-wise error.

Contribution

It develops a novel, influence function-based Wald test for multiple one-sided hypotheses that improves power by considering simultaneous p-value behavior without extra assumptions.

Findings

The test has attractive power properties.

It controls family-wise type 1 error.

It outperforms traditional methods like the minimum p-value test.

Abstract

Testing intersections of null-hypotheses is an integral part of closed testing procedures for assessing multiple null-hypotheses under family-wise type 1 error control. Popular intersection tests such as the minimum p-value test are based on marginal p-values and are typically evaluated conservatively by disregarding simultaneous behavior of the marginal p-values. We consider a general purpose Wald type test for testing intersections of one-sided null-hypotheses. The test is constructed on the basis of the simultaneous asymptotic behavior of the p values. The simultaneous asymptotic behavior is derived via influence functions of estimators using the so-called stacking approach. In particular, this approach does not require added assumptions on simultaneous behavior to be valid. The resulting test is shown to have attractive power properties and thus forms the basis of a powerful closed testing procedure for testing multiple one-sided hypotheses under family-wise type 1 error control.