Inequalities for the false discovery rate (FDR) under dependence

Abstract

Inequalities are key tools to prove FDR control of a multiple test. The present paper studies upper and lower bounds for the FDR under various dependence structures of $p$-values, namely independence, reverse martingale dependence and positive regression dependence on the subset (PRDS) of true null hypotheses. The inequalities are based on exact finite sample formulas which are also of interest for independent uniformly distributed $p$-values under the null. As applications the asymptotic worst case FDR of step up and step down tests coming from an non-decreasing rejection curve is established. In addition, new step up tests are established and necessary conditions for the FDR control are discussed. The reverse martingale models yield sharper FDR results than the PRDS models. Already in certain multivariate normal dependence models the familywise error rate of the Benjamini Hochberg step up test can be different from the desired level $\alpha$. The second part of the paper is devoted to adaptive step up tests under dependence. The well-known Storey estimator is modified so that the corresponding step up test has finite sample control for various block wise dependent $p$-values. These results may be applied to dependent genome data. Within each chromosome the $p$-values may be reverse martingale dependent while the chromosomes are independent.