Global permutation tests for multivariate ordinal data: alternatives, test statistics, and the null dilemma

We discuss two-sample global permutation tests for sets of multivariate ordinal data in possibly high-dimensional setups, motivated by the analysis of data collected by means of the World Health Organisation’s International Classification of Functioning, Disability and Health. The tests do not require any modelling of the multivariate dependence structure. Specifically, we consider testing for marginal inhomogeneity and direction-independent marginal order. Max-T test statistics are known to lead to good power against alternatives with few strong individual effects. We propose test statistics that can be seen as their counterparts for alternatives with many weak individual effects. Permutation tests are valid only if the two multivariate distributions are identical under the null hypothesis. By means of simulations, we examine the practical impact of violations of this exchangeability condition. Our simulations suggest that theoretically invalid permutation tests can still be ‘practically valid’. In particular, they suggest that the degree of the permutation procedure’s failure may be considered as a function of the difference in group-specific covariance matrices, the proportion between group sizes, the number of variables in the set, the test statistic used, and the number of levels per variable.