Incorporating the Empirical Null Hypothesis into the Benjamini-Hochberg Procedure

For the problem of multiple testing, the Benjamini-Hochberg (B-H) procedure has become a very popular method in applications. We show how the B-H procedure can be interpreted as a test based on the spacings corresponding to the p-value distributions. This interpretation leads to the incorporation of the empirical null hypothesis, a term coined by Efron (2004). We develop a mixture modelling approach for the empirical null hypothesis for the B-H procedure and demonstrate some theoretical results regarding both finite-sample as well as asymptotic control of the false discovery rate. The methodology is illustrated with application to two high-throughput datasets as well as to simulated data.