Inference with approximate local false discovery rates

Efron's two-group model is widely used in large scale multiple testing. This model assumes that test statistics are mutually independent, however in realistic settings they are typically dependent, and taking the dependence into account can boost power. The general two-group model takes the dependence between the test statistics into account. Optimal policies in the general two-group model require calculation, for each hypothesis, of the probability that it is a true null given all test statistics, denoted local false discovery rate (locFDR). Unfortunately, calculating locFDRs under realistic dependence structures can be computationally prohibitive. We propose calculating approximate locFDRs based on a properly defined N-neighborhood for each hypothesis. We prove that by thresholding the approximate locFDRs with a fixed threshold, the marginal false discovery rate is controlled for any dependence structure. Furthermore, we prove that this is the optimal procedure in a restricted class of decision rules, where decision for each hypothesis is only guided by its N-neighborhood. We show through extensive simulations that our proposed method achieves substantial power gains compared to alternative practical approaches, while maintaining conceptual simplicity and computational feasibility. We demonstrate the utility of our method on a genome wide association study of height.