Estimating false discovery proportion under arbitrary covariance dependence

Multiple hypothesis testing is a fundamental problem in high-dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any single-nucleotide polymorphisms (SNPs) are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In this article, we propose a novel method-based on principal factor approximation-that successfully subtracts the common dependence and weakens significantly the correlation structure, to deal with an arbitrary dependence structure. We derive an approximate expression for false discovery proportion (FDP) in large-scale multiple testing when a common threshold is used and provide a consistent estimate of realized FDP. This result has important applications in controlling false discovery rate and FDP. Our estimate of realized FDP compares favorably with Efron's approach, as demonstrated in the simulated examples. Our approach is further illustrated by some real data applications. We also propose a dependence-adjusted procedure that is more powerful than the fixed-threshold procedure. Supplementary material for this article is available online.