Average Power and $λ$-power in Multiple Testing Scenarios when the Benjamini-Hochberg False Discovery Rate Procedure is Used

Abstract: We discuss several approaches to defining power in studies designed around the Benjamini-Hochberg (BH) false discovery rate (FDR) procedure. We focus primarily on the \textit{average power} and the $\lambda$-\textit{power}, which are the expected true positive fraction and the probability that the true positive fraction exceeds $\lambda$, respectively. We prove results concerning strong consistency and asymptotic normality for the positive call fraction (PCF), the true positive fraction (TPF) and false discovery fraction (FDF). Convergence of their corresponding expected values, including a convergence result for the average power, follow as a corollaries. After reviewing what is known about convergence in distribution of the errors of the plugin procedure, (Genovese, 2004), we prove central limit theorems for fully empirical versions of the PCF, TPF, and FDF, using a result for stopped stochastic processes. The central limit theorem (CLT) for the TPF is used to obtain an approximate expression for the $\lambda$-power, while the CLT for the FDF is used to introduce an approximate procedure for determining a suitably small nominal FDR that results in a speicified bound on the FDF with stipulated high probability. The paper also contains the results of a large simulation study covering a fairly substantial portion of the space of possible inputs encountered in application of the results in the design of a biomarker study, a micro-array experiment and a GWAS study.