Nonparametric MANOVA in Mann-Whitney effects

Multivariate analysis of variance (MANOVA) is a powerful and versatile method to infer and quantify main and interaction effects in metric multivariate multi-factor data. It is, however, neither robust against change in units nor a meaningful tool for ordinal data. Thus, we propose a novel nonparametric MANOVA. Contrary to existing rank-based procedures we infer hypotheses formulated in terms of meaningful Mann-Whitney-type effects in lieu of distribution functions. The tests are based on a quadratic form in multivariate rank effect estimators and critical values are obtained by the bootstrap. This newly developed procedure consequently provides asymptotically exact and consistent inference for general models such as the nonparametric Behrens-Fisher problem as well as multivariate one-, two-, and higher-way crossed layouts. Computer simulations in small samples confirm the reliability of the developed method for ordinal as well as metric data with covariance heterogeneity. Finally, an analysis of a real data example illustrates the applicability and correct interpretation of the results.