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.