Inference with approximate local false discovery rates

Efron's 2-group model is widely used in large-scale multiple testing. This model assumes that test statistics are drawn independently from a mixture of a null and a non-null distribution. The marginal local false discovery rate (locFDR) is the probability that the hypothesis is null given its test statistic. The procedure that rejects null hypotheses with marginal locFDRs below a fixed threshold maximizes power (the expected number of non-nulls rejected) while controlling the marginal false discovery rate in this model. However, in realistic settings the test statistics are dependent, and taking the dependence into account can boost power. Unfortunately, the resulting calculations are typically exponential in the number of hypotheses, which is impractical. Instead, we propose using $\textrm {locFDR}_N$, which is the probability that the hypothesis is null given the test statistics in its $N$-neighborhood. We prove that rejecting for small $\textrm {locFDR}_N$ is optimal in the restricted class where the decision for each hypothesis is only guided by its $N$-neighborhood, and that power increases with $N$. The computational complexity of computing the $\mathrm{ locFDR}_N$s increases with $N$, so the analyst should choose the largest $N$-neighborhood that is still computationally feasible. We show through extensive simulations that our proposed procedure can be substantially more powerful than alternative practical approaches, even with small $N$-neighborhoods. We demonstrate the utility of our method in a genome-wide association study of height.