Exact calculations for false discovery proportion with application to least favorable configurations

Abstract

International audienceIn a context of multiple hypothesis testing, we provide several new exact calculations related to the false discovery proportion (FDP) of step-up and step-down procedures. For step-up procedures, we show that the number of erroneous rejections conditionally on the rejection number is simply a binomial variable, which leads to explicit computations of the c.d.f., the {$s$-th} moment and the mean of the FDP, the latter corresponding to the false discovery rate (FDR). For step-down procedures, we derive what is to our knowledge the first explicit formula for the FDR valid for any alternative c.d.f. of the $p$-values. We also derive explicit computations of the power for both step-up and step-down procedures. These formulas are ``explicit'' in the sense that they only involve the parameters of the model and the c.d.f. of the order statistics of i.i.d. uniform variables. The $p$-values are assumed either independent or coming from an equicorrelated multivariate normal model and an additional mixture model for the true/false hypotheses is used. This new approach is used to investigate new results which are of interest in their own right, related to least/most favorable configurations for the FDR and the variance of the FDP

Deriving and comparing the distribution for the number of false positives in single step methods to control -FWER

“…We thank the reviewer for doing so. Roquain and Villers (2011) derived similar theoretical results for the distribution of the false discovery proportion (FDP) for a variety of models for the p-value families. With similar calculations, they derive the probability for the number of rejections and the probability of the number of false rejections conditional on a number of rejections.…”

“…It was brought to our attention by one of the reviewers of this manuscript. The reviewer informed us that our main results were indeed available in Roquain and Villers (2011). We thank the reviewer for doing so.…”

“…With similar calculations, they derive the probability for the number of rejections and the probability of the number of false rejections conditional on a number of rejections. With some work, the reader of Roquain and Villers (2011) could arrive at the Eqs. (10)-(12) for the distribution of V .…”

“…The authors would like to thank the two anonymous reviewers for their comments, and especially for mentioning the article by Roquain and Villers (2011).…”

A Tight Prediction Interval for False Discovery Proportion under Dependence

“…For example, Ge et al [12,20] proposed a formula for the upper prediction bound of the FDP assuming that test statistics under true null hypotheses are independent and also proposed a permutation algorithm to obtain a simultaneous upper prediction band of the FDP. Under the assumption that p-values are independent or follow a conditional equicorrelated multivariate normal model, Roquain and Villers [21] provided exact calculations for the cumulative distribution function (CDF) and moments of FDP for the step-up and step-down procedures. Ghosal and Roy [22] proposed a nonparametric Bayesian procedure to obtain the posterior distribution of FDP under the intraclass or autoregressive correlation structure.…”