Step-up and step-down procedures controlling the number and proportion of false positives

In multiple hypotheses testing, it is important to control the probability of rejecting “true” null hypotheses. A standard procedure has been to control the family-wise error rate (FWER), the probability of rejecting at least one true null hypothesis. For large numbers of hypotheses, using FWER can result in very low power for testing single hypotheses. Recently, powerful multiple step FDR procedures have been proposed which control the “False Discovery Rate” (expected proportion of Type I errors). More recently van der Laan, Dudoit and Pollard (2004) proposed controlling a generalized family-wise error rate k-FWER (also called gFWER(k)), defined as the probability of at least (k+1) Type I errors (k=0 for the usual FWER). Lehmann and Romano (2005) suggested both a single-step, and a step-down procedure for controlling the generalized family-wise error rate. They make no assumptions concerning the p-values of the individual tests. The step-down procedure is simple to apply, and cannot be improved without violation of control of the k-FWER. In this paper, by limiting the number of steps in step-down or step-up procedures, new procedures are developed to control k-FWER (and the proportion of false positives). Using data from the literature, the procedures are compared with those of Romano and Lehmann (2005), and, under the assumption of a multivariate normal distribution of the test statistics, show considerable improvement in the reduction of the number and proportion of false positives.