Multivariate two-sample permutation test with directional alternative for categorical data

This paper presents a distribution-free test, based on the permutation approach, on treatment effects with a multivariate categorical response variable. The motivating example is a typical case-control biomedical study, performed to investigate the effect of the treatment called "assisted motor activity" (AMA) on the health of comorbid patients affected by "low back pain" (LBP), "hypertension" and "diabetes". Specifically, the goal was to test whether the AMA determines an improvement in the functionality and the perceived health status of patients. Two independent samples (treated and control group) were compared according to 13 different binary or ordinal outcomes. The null hypothesis of the test consists in the equality in the distribution of the multivariate responses of the two groups, whereas under the alternative hypothesis, the health status of the treated patients is better. The approach proposed in this work is based on the Combined Permutation Test (CPT) method, which is suitable for analyzing multivariate categorical data in the presence of confounding factors. A stratification of the groups and intra-stratum permutation univariate two-sample tests are conducted to avoid the potential confounding effects. P-values from the partial tests are combined using the CPT approach to create a suitable test statistic for the overall problem.