A permutation approach for ranking of multivariate populations

The need to establish the relative superiority of each treatment/group when compared to all the others, that is ordering the effects with respect to the underlying populations, often occurs in many multivariate studies especially in the bio-medical field. Within the framework of multivariate stochastic ordering, the purpose of this work is to propose a nonparametric permutation-based solution for the problem of ranking of multivariate populations, i.e. estimating an ordering related to the possible stochastic dominance among several unknown multivariate distributions. The method is metric-free in the sense that it can be applied to any kind of response variables, i.e. continuous/binary or ordered categorical or mixed (some continuous/binary univariate components and some other ordered categorical), and it is valid also in case the sample sizes are lower than the number of responses. It will be theoretically argued and numerically proved that our method controls the risk of false ranking classification under the hypothesis of population homogeneity while under the alternatives we expect that the true rank can be estimated with satisfactory accuracy, especially for the ‘best’ populations. Finally, an application to a morphological analysis of primary bovine cerebellum cell cultures is proposed to highlight the practical relevance of the proposed methodology.