Statistica Sinica 20 (2010), 1227-1238

PROCEDURES CONTROLLING THE $k$-FDR USING

BIVARIATE DISTRIBUTIONS OF THE NULL $p$-VALUES

Sanat K. Sarkar and Wenge Guo

Temple University and New Jersey Institute of Technology

Abstract: Procedures controlling error rates measuring at least $k$ false rejections, instead of at least one, are often desired while testing a large number of hypotheses. The $k$-FWER, probability of at least $k$ false rejections, is such an error rate that has been introduced, 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 the $k$-dimensional joint distributions of the null $p$-values and assuming MTP$_2$ (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 MTP$_2$, and develop alternative procedures based only on the bivariate distributions of the null $p$-values.

Key words and phrases: Arbitrary dependence, average power, clumpy dependence, generalized FDR, multiple hypothesis testing, positive regression dependence on subset, stepwise procedure.