Data volume and power of multiple tests with small sample size per null

In multiple hypothesis testing, due to practical constraints, the number of repeated measurements for each null may not be large enough in terms of the subtle difference between false and true nulls. Under this circumstance, we show that (1) in order to have enough chance that the data to be collected will yield some trustworthy findings, as measured by a conditional version of the positive false discovery rate (pFDR), the minimum amount of data has to grow at a much faster rate than in the case where the number of repeated measurements for each null is large enough, and (2) in order to control the pFDR asymptotically as the difference between false and true nulls tends to 0, the power decays to 0 but the rate is sensitive to small changes in the rejection criteria and there is no asymptotically most powerful procedures among those that control the pFDR asymptotically at the same level.