Global tests of association for multivariate ordinal data

Abstract

Global tests are in demand whenever it is of interest to draw inferential conclusions about sets of variables as a whole. The present thesis attempts to develop such tests for the case of multivariate ordinal data in possibly high-dimensional set-ups, and has primarily been motivated by research questions that arise from data collected by means of the 'International Classification of Functioning, Disability and Health'. The thesis essentially comprises two parts. In the first part two tests are discussed, each of which addresses one specific problem in the classical two-group scenario. Since both are permutation tests, their validity relies on the condition that, under the null hypothesis, the joint distribution of the variables in the set to be tested is the same in both groups. Extensive simulation studies on the basis of the tests proposed suggest, however, that violations of this condition, from the purely practical viewpoint, do not automatically lead to invalid tests. Rather, two-sample permutation tests' failure appears to depend on numerous parameters, such as the proportion between group sizes, the number of variables in the set of interest and, importantly, the test statistic used. In the second part two further tests are developed which both can be used to test for association, if desired after adjustment for certain covariates, between a set of ordinally scaled covariates and an outcome variable within the range of generalized linear models. The first test rests upon explicit assumptions on the distances between the covariates' categories, and is shown to be a proper generalization of the traditional Cochran-Armitage test to higher dimensions, covariate-adjusted scenarios and generalized linear model-specific outcomes. The second test in turn parametrizes these distances and thus keeps them flexible. Based on the tests' power properties, practical recommendations are provided on when to favour one or the other, and connections with the permutation tests from the first part of the thesis are pointed out. For illustration of the methods developed, data from two studies based on the 'International Classification of Functioning, Disability and Health' are analyzed. The results promise vast potential of the proposed tests in this data context and beyond.