Oracle and adaptive compound decision rules for false discovery rate control

We develop a compound decision theory framework for multiple-testing problems and derive an oracle rule based on the z values that minimizes the false nondiscovery rate (FNR) subject to a constraint on the false discovery rate (FDR). We show that many commonly used multiple-testing procedures, which are p value-based, are inefficient, and propose an adaptive procedure based on the z values. The z value-based adaptive procedure asymptotically attains the performance of the z value oracle procedure and is more efficient than the conventional p value-based methods. We investigate the numerical performance of the adaptive procedure using both simulated and real data. In particular, we demonstrate our method in an analysis of the microarray data from a human immunodeficiency virus study that involves testing a large number of hypotheses simultaneously.