Multiple Hypothesis Testing Under Dependence

Abstract

We have examined various aspects of multiple hypothesis testing under dependence. Traditional algorithms designed to control the error arising from multiplicity become severely conservative when the hypotheses exhibit positive dependence, resulting in a loss of power. There is a paucity of literature explicating the behaviour of traditional algorithms when the hypotheses are dependent. In the realm of multiple testing, a popular multiplicity correction is the Bonferroni correction, which is perhaps the oldest classical approach for controlling the Family-Wise Error Rate (FWER) at a desired level, regardless of dependence among hypotheses. However, under the global null and equicorrelated normal model, the actual FWER of the Bonferroni procedure is bounded above by a line connecting 0 and the desired level (alpha), making it overly conservative. We have also examined the actual FWER of the Bonferroni method in a nearly independent setup and shown that it approximates the FWER under independence when the correlations are smaller than the order of log n. In the second part, we explored the connection between multiple testing problems and classification theory. The Bayes rule, which is optimal for traditional classification problems, is also optimal for multiple testing problems under certain mild assumptions. However, the test statistic derived from the Bayes rule is challenging to simplify under dependence, limiting its practical application. We have simplified the optimal test statistic under a Gaussian model and, through extensive simulations and demonstrated that the performance of the Oracle rule significantly surpasses that of traditional approaches like the Benjamini- Hochberg (BH) FDR controlling procedure. Finally, we addressed the problem of estimating the proportion of null hypotheses under dependence. We have shown that the estimator proposed by Benjamini and Hochberg converges to 1 under independence. Additionally, simulations have been conducted to evaluate the performance of this estimator under various dependent structures, including m- dependent and block-dependent structures.