Abstract
Consider the multiple testing problem of testingmnull hypothesesH1,…,Hm, among whichm0hypotheses are truly null. Given theP-values for each hypothesis, the question of interest is how to combine theP-values to find out which hypotheses are false nulls and possibly to make a statistical inference onm0. Benjamini and Hochberg proposed a classical procedure that can control the false discovery rate (FDR). The FDR control is a little bit unsatisfactory in that it only concerns the expectation of the false discovery proportion (FDP). The control of the actual random variable FDP has recently drawn much attention. For any level1−α, this paper proposes a procedure to construct an upper prediction bound (UPB) for the FDP for a fixed rejection region. When1−α=50%, our procedure is very close to the classical Benjamini and Hochberg procedure. Simultaneous UPBs for all rejection regions' FDPs and the upper confidence bound for the unknownm0are presented consequently. This new proposed procedure works for finite samples and hence avoids the slow convergence problem of the asymptotic theory.
Highlights
In this paper, we consider the problem of testing m null hypotheses H1, . . . , Hm, among which m0 hypotheses are truly null
The method of this paper applies to data where true null P -values are independent, or to slightly dependent data where the joint binomial dominant condition is satisfied
This assumption does not rely on any specification for the false null P -values
Summary
We consider the problem of testing m null hypotheses H1, . . . , Hm, among which m0 hypotheses are truly null. For a fixed rejection region 0, t of the P -values, we would like to find the 1 − α upper prediction bound UPB for the false discovery proportion Qt. As we mentioned, the distribution of Qt is unknown. We have used the fact that {Vt ≤ C1−α m0, t and Rt 0} is the same as the set {Rt 0}, which is obtained by noting that Vt must be zero when Rt 0 Following this proof, we can see that. The most conservative approach is to replace m0 with m, in which case we obtain a conservative 1 − α UPB for Qt. The independence assumption among true null P -values can be used to give a confidence inference for m0; we can find a better estimate of the UPB for Qt. For any given 0 < λ < 1, a 1 − α UCB for m0 is given by. Q1−α1,1−α2 t, λ is a conservative 1 − α α α1 α2 UPB for the false discovery proportion Qt
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