Sequential tests controlling generalized familywise error rates

Abstract

Sequential methods are developed for conducting a large number of simultaneous tests while controlling the Type I and Type II generalized familywise error rates. Namely, for the chosen values of $\alpha$, $\beta$, $k$, and $m$, we derive simultaneous tests of $d$ individual hypotheses, based on sequentially collected data, that keep the probability of at least $k$ Type I errors not exceeding level $\alpha$ and the probability of at least $m$ Type II errors not greater than $\beta$. This generalization of the classical notions of familywise error rates allows substantial reduction of the expected sample size of the multiple testing procedure.