Stepdown procedures controlling a generalized false discovery rate. Unpublished manuscript, available at http://astro.temple.edu/ sanat /reports.html

Abstract: Procedures controlling error rates measuring at least k false rejections, instead of at least one, can potentially increase the ability of a procedure to detect false null hypotheses in situations where one seeks to control k or more false rejections having tolerated a few of them. The k-FWER, the probability of at least k false rejections, is such an error rate that is recently introduced in the literature and procedures controlling it have been proposed. Recently, Sarkar (2007) introduced an alternative, less conservative notion of error rate, the k-FDR, generalizing the usual notion of false discovery rate (FDR), and proposed a procedure controlling it based on k-dimensional joint distributions of the null p-values and assuming the MTP2 (multivariate totally positive of order two) positive dependence among all the p-values. In this article, we assume a less restrictive form of positive dependence than the MTP2 and develop procedure based only on the bivariate, rather than the k-dimensional, distributions of the null p-values. Key words and phrases: Arbitrary dependence, average power, clumpy dependence, gene expression, generalized FDR, multiple hypothesis testing, positive regression dependence on subset, stepwise procedure.