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

TL;DR

This paper derives exact formulas for the false discovery proportion and rate in multiple hypothesis testing, applicable to various procedures and models, enhancing understanding of their behavior under different configurations.

Contribution

It provides the first explicit formulas for FDR in step-down procedures for any alternative distribution and offers new insights into least favorable configurations for FDR and FDP variance.

Findings

Explicit formulas for FDP and FDR for step-up and step-down procedures.

FDR formula valid for any alternative p-value distribution.

Analysis of least favorable configurations for FDR and FDP variance.

Abstract

In 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.