Average Power and λ-power in Multiple Testing Scenarios when the Benjamini-Hochberg False Discovery Rate Procedure is Used
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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 average power and the λ-power, which are the expected true positive fraction and the probability that the true positive fraction exceeds λ, respectively. We show that the average power converges as the number of simultaneous tests tends to infinity, to a limit that is nearly equivalent to the power introduced independently by JungSH:2005 and by LiuP:2007. Furthermore, we prove a CLT which allows asymptotic approximation to the λ-power. Moreover, we prove SLLNs and CLTs for all quantities connected to the BH-FDR procedure: the positive call fraction, true positive fraction, and false discovery fraction, with full characterization of almost sure limits and limits in distribution. We discuss ramifications of the CLT for the false discovery fraction, introducing a procedure which allows tighter control of the false discovery fraction than the BH-FDR that can be used at the design and analysis steps. We conduct a large simulation study covering a fairly substantial portion of the space of possible inputs. We show its application in design of a biomarker study, a micro-array experiment and a GWAS study.
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