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In multiple hypothesis testing, the volume of data, defined as the number ofreplications per null times the total number of nulls, usually defines theamount of resource required. On the other hand, power is an important measureof performance for multiple testing. Due to practical constraints, the numberof replications per null may not be large enough in terms of the differencebetween false and true nulls. For the case where the population fraction offalse nulls is constant, we show that, as the difference between false and truenulls becomes increasingly subtle while the number of replications per nullcannot increase fast enough, (1) in order to have enough chance that the datato be collected will yield some trustworthy findings, as measured by aconditional version of the positive false discovery rate (pFDR), the volume ofdata has to grow at a rate much faster than in the case where the number ofreplications per null can be large enough, and (2) in order to control the pFDRasymptotically, power has to decay to 0 in a rate highly sensitive to rejectioncriterion and there is no asymptotically most powerful procedures among thosethat control the pFDR asymptotically at the same level.

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