Multiple Testing Procedures for One- and Two-Way Classified Hypotheses

Abstract

Multiple testing literature contains ample research on controlling false discoveries for hypotheses classified according to one criterion, which we refer to as `one-way classified hypotheses'. However, one often encounters the scenario of `two-way classified hypotheses' where hypotheses can be partitioned into two sets of groups via two different criteria. Associated multiple testing procedures that incorporate such structural information are potentially more effective than their one-way classified or non-classified counterparts. To the best of our knowledge, very little research has been pursued in this direction. This dissertation proposes two types of multiple testing procedures for two-way classified hypotheses. In the first part, we propose a general methodology for controlling the false discovery rate (FDR) using the Benjamini-Hochberg (BH) procedure based on weighted p-values. The weights can be appropriately chosen to reflect one- or two-way classified structure of hypotheses, producing novel multiple testing procedures for two-way classified hypotheses. Newer results for one-way classified hypotheses have been obtained in this process. Our proposed procedures control the false discovery rate (FDR) non-asymptotically in their oracle forms under positive regression dependence on subset of null p-values (PRDS) and in their data-adaptive forms for independent p-values. Simulation studies demonstrate that our proposed procedures can be considerably more powerful than some contemporary methods in many instances and that our data-adaptive procedures can non-asymptotically control the FDR under certain dependent scenarios. The proposed two-way adaptive procedure is applied to a data set from microbial abundance study, for which it makes more discoveries than an existing method. In the second part, we propose a Local false discovery rate (Lfdr) based multiple testing procedure for two-way classified hypotheses. The procedure has been developed in its oracle form under a model based framework that isolates the effects due to two-way grouping from the significance of an individual hypothesis. Simulation studies show that our proposed procedure successfully controls the average proportion of false discoveries, and is more powerful than existing methods.