Resampling‐Based Empirical Bayes Multiple Testing Procedures for Controlling Generalized Tail Probability and Expected Value Error Rates: Focus on the False Discovery Rate and Simulation Study

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Abstract
This article proposes resampling‐based empirical Bayes multiple testing procedures for controlling a broad class of Type I error rates, defined as generalized tail probability (gTP) error rates, gTP (q,g) = Pr(g (Vn,Sn) > q), and generalized expected value (gEV) error rates, gEV (g) = E [g (Vn,Sn)], for arbitrary functions g (Vn,Sn) of the numbers of false positives Vn and true positives Sn. Of particular interest are error rates based on the proportion g (Vn,Sn) = Vn /(Vn + Sn) of Type I errors among the rejected hypotheses, such as the false discovery rate (FDR), FDR = E [Vn /(Vn + Sn)]. The proposed procedures offer several advantages over existing methods. They provide Type I error control for general data generating distributions, with arbitrary dependence structures among variables. Gains in power are achieved by deriving rejection regions based on guessed sets of true null hypotheses and null test statistics randomly sampled from joint distributions that account for the dependence structure of the data. The Type I error and power properties of an FDR‐controlling version of the resampling‐based empirical Bayes approach are investigated and compared to those of widely‐used FDR‐controlling linear step‐up procedures in a simulation study. The Type I error and power trade‐off achieved by the empirical Bayes procedures under a variety of testing scenarios allows this approach to be competitive with or outperform the Storey and Tibshirani (2003) linear step‐up procedure, as an alternative to the classical Benjamini and Hochberg (1995) procedure.

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