Annals of Statistics

Control of Directional Errors with Stagewise Multiple Test Procedures

Juliet Popper Shaffer

Full-text: Open access


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 $T_i$.

Article information

Ann. Statist., Volume 8, Number 6 (1980), 1342-1347.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F99: None of the above, but in this section
Secondary: 62J15: Paired and multiple comparisons 62F05: Asymptotic properties of tests

Hypothesis testing multiple tests multiple inference simultaneous inference stagewise multiple tests experimentwise error rate directional errors


Shaffer, Juliet Popper. Control of Directional Errors with Stagewise Multiple Test Procedures. Ann. Statist. 8 (1980), no. 6, 1342--1347. doi:10.1214/aos/1176345205.

Export citation