Selecting an Optimal Rejection Region for Multiple Testing : A Decision-Theoretic Alternative to Fdr Control, with an Application to Microarrays

As a measure of error in testing multiple hypotheses, the decisive false discovery rate (dFDR), the ratio of the expected number of false discoveries to the expected total number of discoveries, has advantages over the false discovery rate (FDR) and positive FDR (pFDR). The dFDR can be optimized and often controlled using decision theory, and some previous estimators of the FDR can estimate the dFDR without assuming weak dependence or the randomness of hypothesis truth values. While it is suitable in frequentist analyses, the dFDR is also exactly equal to a posterior probability under the assumption of random truth values, even without independence.The test statistic space in which null hypotheses are rejected, called the rejection region, can be determined by a number of multiple testing procedures, including those controlling the family-wise error rate (FWER) or the FDR at a specified level. An alternate approach, which makes use of the dFDR, is to reject null hypotheses to maximize a desirability function, given the cost of each false discovery and the benefit of each true discovery. The focus of this method is on discoveries, unlike related approaches based on false nondiscoveries. A method is provided for achieving the highest possible desirability under general conditions, without relying on density estimation. A Bayesian treatment of differing costs and benefits associated with different tests is related to the weighted FDR when there is a common rejection region for all tests. An application to DNA microarrays of patients with different types of leukemia illustrates the proposed use of the dFDR in decision theory. Comparisons between more than two groups of patients do not call for changes in the results, as when the FWER is strictly controlled by adjusting -values for multiplicity