Sharper Confidence Intervals for Hochberg- and Hommel-Related Multiple Tests Based On an Extended Simes Inequality

Abstract

This article concerns recent developments related to the now classical multiple-testing procedures (MTPs) of Holm, Hochberg, and Hommel based on marginal p-values. For a long time, the derivation of simultaneous confidence intervals (SCIs) corresponding to these MTPs was considered to be a difficult problem, but solutions were published in 2008 for Holm's MTP, and in 2012 for Hochberg's and Hommel's MTPs. These SCIs turned out to be as simple and easily implemented as the MTPs themselves, and to be remarkably similar. However, they also turned out to have the property/limitation, shared with other powerful stepwise MTPs, that no confidence assertions sharper than rejection assertions are possible unless all null hypotheses are rejected. A possibility is then to construct related families of MTPs that do not have this limitation but are somewhat less powerful, so users may choose among various such trade-off MTPs. It is shown in this article how an extended Simes inequality can be used to construct Hochberg- and Hommel-related MTPs of this kind that: (i) are more powerful than corresponding trade-off MTPs proposed previously, and (ii) lead to SCIs that are sharper than the ones proposed previously. Corresponding Holm-related MTPs and SCIs are considered for completeness and comparisons.