A rank-based test for monotone trends in k-sample multivariate problems

In many practical applications, a multivariate outcome is measured on two or more groups of subjects. When these variables are analyzed separately, the information is not fully exhausted, as possible dependencies among the endpoints are not considered. Within parametric frameworks, a number of literature exists to test such multivariate outcomes. Also, a few methods have been proposed using a fully non parametric approach. It is of particular interest to test the hypothesis of no treatment effect against conjectured patterns that are associated with treatment efficacy. We proposed a rank-based test for detecting pre-specified alternative patterns across all endpoints. The test statistic we have derived is a multivariate generalization of a recent non parametric univariate test for alternative patterns. Since we do not require the continuity of the distribution functions, the newly proposed test is applied to data with ties, in particular, to multivariate-ordered categorical data. Moreover, the test is invariant under monotone transformations of the responses and the weights describing the alternative pattern. Finite sample performance of the proposed test is assessed through a simulation study and comparisons are made with the most popular test for one-sided multivariate problems. The application of the test statistic is demonstrated with an electroencephalogram dataset.