How to compare small multivariate samples using nonparametric tests

In the life sciences and other research fields, experiments are often conducted to determine responses of subjects to various treatments. Typically, such data are multivariate, where different variables may be measured on different scales that can be quantitative, ordinal, or mixed. To analyze these data, we present different nonparametric (rank-based) tests for multivariate observations in balanced and unbalanced one-way layouts. Previous work has led to the development of tests based on asymptotic theory, either for large numbers of samples or groups; however, most experiments comprise only small or moderate numbers of experimental units in each individual group or sample. Here, we investigate several tests based on small-sample approximations, and compare their performance in terms of α levels and power for different simulated situations, with continuous and discrete observations. For positively correlated responses, an approximation based on [Brunner, E., Dette, H., Munk, A., 1997. Box-type approximations in nonparametric factorial designs. J. Amer. Statist. Assoc. 92, 1494-1502] ANOVA-Type statistic performed best; for responses with negative correlations, in general, an approximation based on the Lawley-Hotelling type test performed best. We demonstrate the use of the tests based on the approximations for a plant pathology experiment.