Some Results on False Discovery Rate in Stepwise multiple testing procedures

Abstract

The concept of false discovery rate (FDR) has been receiving increasing attention by researchers in multiple hypotheses testing. This paper produces some theoretical results on the FDR in the context of stepwise multiple testing procedures with dependent test statistics. It was recently shown by Benjamini and Yekutieli that the Benjamini–Hochberg step-up procedure controls the FDR when the test statistics are positively dependent in a certain sense. This paper strengthens their work by showing that the critical values of that procedure can be used in a much more general stepwise procedure under similar positive dependency. It is also shown that the FDR-controlling Benjamini–Liu step-down procedure originally developed for independent test statistics works even when the test statistics are positively dependent in some sense. An explicit expression for the FDR of a generalized stepwise procedure and an upper bound to the FDR of a step-down procedure are obtained in terms of probability distributions of ordered components of dependent random variables before establishing the main results.