  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.
