FDR Step-down and Step-up Procedures for the Correlated Case

Controlling the false discovery rate has been increasingly utilized in high dimensional screening studies where multiplicity is a problem. Most methods do not explicitly take the correlation between the data or the test statistics into account, with consequent loss of power. In this paper, we use least favorable configurations to obtain critical values for both step-down and step-up procedures, valid for both dependent and independent hypotheses. The concept of a "minimum critical value" (MCV) is introduced. For the step-down case with MCV = 0, our step-down procedure is the same as that of Troendle (2000). It is conjectured that, for a given MCV, there is no uniformly more powerful step-down FDR procedure. Empirical results suggest that, for maximizing power, the "optimum" MCV is a decreasing function of the number of false hypotheses. Various tables are given, with a special "condensed" table valid for numbers of hypotheses from 30 to 10,000 and ρ = .5 specifically designed for the case where few false hypotheses are anticipated or where a satisfactory outcome is the discovery of a few false hypotheses. Intermediate values for the latter table may be obtained by interpolation. An application to high dimensional genomic data is given.