Control of Directional Errors with Stagewise Multiple Test Procedures

Abstract

This paper is concerned with jointly testing the hypotheses $\theta_i = \theta_{i0}, i = 1, \cdots, s$, using tests based on independent statistics $T_i$ with distributions $P(T_i \leq t) = F_i(t, \theta_i)$ nonincreasing in $\theta_i$. Holm proposed a sequentially rejective test procedure, applicable to this problem, for which, for fixed $\alpha (0 \leq \alpha \leq 1)$, the probability that the joint conclusion contains no false rejections is $\geq 1 - \alpha$ for all possible values of the $\theta_i$. Suppose, however, that if the hypothesis $\theta_i = \theta_{i0}$ is rejected, it is desired to conclude not only that $\theta_i \neq \theta_{i0}$ but also either that it is greater than $\theta_{i0}$ or smaller than $\theta_{i0}$. Usually one then requires a probability $\geq 1 - \alpha$ that the joint conclusion contains neither false rejections nor false directional statements. This paper considers the use of Holm's nondirectional procedure for rejecting hypotheses, supplemented by decisions on direction based on the values of the $T_i$. It is shown that this procedure does not in general provide the required control over error probabilities, but that it does so under specified conditions on the distributions of the $T_i$.