Optimal FDR control in the two-group model
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The highly influential two group model in testing a large number of statistical hypothesis assumes that the test statistics come from a mixture of a high probability null distribution and a low probability alternative. Optimal control of the marginal false discovery rate (mFDR), in the sense that it provides maximal power (expected true discoveries) subject to mFDR control, is achieved by thresholding the local false discovery rate (locFDR) with a fixed threshold. In this paper we address the challenge of controlling the popular false discovery rate (FDR) rather than mFDR in the two group model. Since FDR is less conservative, this results in more rejections. We derive the optimal multiple testing (OMT) policy for this task, which turns out to be thresholding the locFDR with a threshold that is a function of the entire set of statistics. We show how to evaluate this threshold in time that is linear in the number of hypotheses, leading to an efficient algorithm for finding this policy. Thus, we can easily derive and apply the optimal procedure for problems with thousands of hypotheses. We show that for K=5000 hypotheses there can be significant power gain in OMT with FDR versus mFDR control.
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