Some Step-down Procedures Controlling the False Discovery Rate under Dependence.

Benjamini and Hochberg (1995) proposed the false discovery rate (FDR) as an alternative to the familywise error rate (FWER) in multiple testing problems. Since then, researchers have been increasingly interested in developing methodologies for controlling the FDR under different model assumptions. In a later paper, Benjamini and Yekutieli (2001) developed a conservative step-up procedure controlling the FDR without relying on the assumption that the test statistics are independent.In this paper, we develop a new step-down procedure aiming to control the FDR. It incorporates dependence information as in the FWER controlling step-down procedure given by Westfall and Young (1993). This new procedure has three versions: lFDR, eFDR and hFDR. Using simulations of independent and dependent data, we observe that the lFDR is too optimistic for controlling the FDR; the hFDR is very conservative; and the eFDR a) seems to control the FDR for the hypotheses of interest, and b) suggests the number of false null hypotheses. The most conservative procedure, hFDR, is proved to control the FDR for finite samples under the subset pivotality condition and under the assumption that joint distribution of statistics from true nulls is independent of the joint distribution of statistics from false nulls.