Control of Directional Errors with Stagewise Multiple Test Procedures

This paper is concerned with jointly testing the hypotheses θᵢ = θᵢ₀, i = 1, ⋯, s, using tests based on independent statistics Tᵢ with distributions P(Tᵢ ≤ t) = Fᵢ(t, θᵢ) nonincreasing in θᵢ. Holm proposed a sequentially rejective test procedure, applicable to this problem, for which, for fixed α (0 ≤ α ≤ 1), the probability that the joint conclusion contains no false rejections is ≥ 1 - α for all possible values of the θᵢ. Suppose, however, that if the hypothesis θᵢ = θᵢ₀ is rejected, it is desired to conclude not only that θᵢ ≠ θᵢ₀ but also either that it is greater than θᵢ₀ or smaller than θᵢ₀. Usually one then requires a probability ≥ 1 - α 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 Tᵢ. 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 Tᵢ.