Some Contributions to Multiple Hypotheses Testing under Dependence

Abstract

Large-scale multiple testing problems in various scientific disciplines often study correlated variables simultaneously. However, the existing literature lacks a study of the performances of FWER controlling procedures under dependence. This thesis concentrates mainly on FWER and generalized FWER controlling procedures under a correlated Gaussian sequence model framework. We establish upper bounds on Bonferroni FWER in the equicorrelated and non-negatively correlated non-asymptotic setup. We also derive similar upper bounds for the generalized FWER of the Lehmann-Romano procedure and propose an improved k-FWER controlling procedure. Towards this, we establish an inequality related to the probability that at least k out of n events occur, which extends and sharpens the classical ones. We have found that, under the non-negatively correlated setup, many classical procedures make zero rejections asymptotically as the number of hypotheses diverges. Specifically, we have shown that, under this setup, the Bonferroni and the Holm methods have zero FWER and power asymptotically. We have also established similar asymptotic zero results for the Hochberg and Hommel procedures under the equicorrelated setup. Finally, we consider the classical means-testing problem in an equicorrelated Gaussian and sequential framework. We focus on sequential test procedures that control the type I and type II familywise error probabilities at pre-specified levels. We establish that our proposed rules have the optimal expected sample sizes under every possible signal configuration asymptotically, as the two error probabilities vanish at arbitrary rates. The results in this thesis illuminate that dependence might be a blessing or a curse, subject to the type of dependence or the underlying paradigm. Several popular and widely used procedures fail to hold the FWER at a positive level asymptotically under positively correlated Gaussian frameworks. On the contrary, the expected sample size of the asymptotically optimal sequential multiple testing rule is a decreasing function in the common correlation under the equicorrelated framework. Thus, correlation plays a dual role in the classical fixed-sample size and the sequential paradigms.