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The positive false discovery rate (pFDR) is a useful overall measure oferrors for multiple hypothesis testing, especially when the underlying goal isto attain one or more discoveries. Control of pFDR critically depends on howmuch evidence is available from data to distinguish between false and truenulls. Oftentimes, as many aspects of the data distributions are unknown, onemay not be able to obtain strong enough evidence from the data for pFDRcontrol. This raises the question as to how much data are needed to attain atarget pFDR level. We study the asymptotics of the minimum number ofobservations per null for the pFDR control associated with multiple Studentizedtests and $F$ tests, especially when the differences between false nulls andtrue nulls are small. For Studentized tests, we consider tests on shifts orother parameters associated with normal and general distributions. For $F$tests, we also take into account the effect of the number of covariates inlinear regression. The results show that in determining the minimum sample sizeper null for pFDR control, higher order statistical properties of data areimportant, and the number of covariates is important in tests to detectregression effects.

Sample size and positive false discovery rate control for multiple testing