Strong approximations of level exceedences related to multiple hypothesis testing

TL;DR

This paper establishes conditions under which the common assumption of independence in multiple t-tests is valid in high-dimensional, low-sample-size scenarios, by linking dependent test statistics to independent ones through strong approximation.

Contribution

It provides a strong approximation framework that justifies the independence assumption in multiple t-tests under specific high-dimensional, low-sample-size conditions.

Findings

Independence assumption is valid if sample size diverges faster than the logarithm of the number of tests.

Derived a strong approximation linking dependent and independent test processes.

Applicable to genomics and similar fields with high-dimensional data.

Abstract

Particularly in genomics, but also in other fields, it has become commonplace to undertake highly multiple Student's $t$-tests based on relatively small sample sizes. The literature on this topic is continually expanding, but the main approaches used to control the family-wise error rate and false discovery rate are still based on the assumption that the tests are independent. The independence condition is known to be false at the level of the joint distributions of the test statistics, but that does not necessarily mean, for the small significance levels involved in highly multiple hypothesis testing, that the assumption leads to major errors. In this paper, we give conditions under which the assumption of independence is valid. Specifically, we derive a strong approximation that closely links the level exceedences of a dependent ``studentized process'' to those of a process of independent random variables. Via this connection, it can be seen that in high-dimensional, low sample-size cases, provided the sample size diverges faster than the logarithm of the number of tests, the assumption of independent $t$-tests is often justified.