Multiplicity, directional (type III) errors, and the null hypothesis

L. V. Jones and J. W. Tukey (2000) pointed out that the usual 2-sided, equal-tails null hypothesis test at level alpha can be reinterpreted as simultaneous tests of 2 directional inequality hypotheses, each at level alpha/2, and that the maximum probability of a Type I error is alpha/2 if the truth of the null hypothesis is considered impossible. This article points out that in multiple testing with familywise error rate controlled at alpha, the directional error rate (assuming all null hypotheses are false) is greater than alpha/2 and can be arbitrarily close to alpha. Single-step, step-down, and step-up procedures are analyzed, and other error rates, including the false discovery rate, are discussed. Implications for confidence interval estimation and hypothesis testing practices are considered.