Directional multivariate tests rejecting null and negative effects in all variables

Abstract

This paper suggests two directional multivariate tests that aim at establishing superiority of a treatment over a control in at least one of several endpoints that are assumed to have a multivariate normal distribution. One of these tests is a one-sided, scale-invariant version of the classical Hotelling T²-test. The other is based on a summary score with weights derived from the data. Both tests overcome an important shortcoming of previous "one-sided" multivariate suggestions, namely that the null hypothesis was restricted to a single point in the multidimensional parameter space. The derivation of the tests is supplemented by simulations investigating their performance and by the application in an osteoporosis trial.

Use of multivariate distance measures for high‐dimensional data in tests for difference, superiority, equivalence and non‐inferiority

“…Our proposal shares this problem with other one‐sided test versions such as the ALR test of Tang et al. (1993), one‐sided applications of O'Brien's (1984) test versions or the proposal of Glimm and Läuter (2010). Even in a multiple test procedure with local tests for each component of $$\theta $$, it would be difficult to achieve a strict type I error control over the whole parameter space outside the positive orthant without dramatic power losses in situations where group differences in the desired direction occur only in some of the variables.…”

“…(2009). Most of these multivariate tests are two‐sided, but directed tests are available, too, such as the ALR test of Tang, Geller and Pocock (1993) or one‐sided tests with one score, where special problems with monotonicity may occur as reported by Glimm and Läuter (2010).…”

An evaluation of methods for testing hypotheses relating to two endpoints in a single clinical trial

“…Another approach uses the unrestricted alternative θ ≠ 0 but constructs tests to have high power against specific alternatives such as θ 1 = ⋯ = θ p > 0 . When the alternative is unrestricted, rejection of H 0 might occur due to data supporting an alternative other than that for which high power has been set , but modifications to avoid this problem can be made .…”