The University of Montana
Department of Mathematical Sciences

Technical report #14/2007

How to Compare Small Multivariate Samples Using Nonparametric Tests

Arne C. Bathke1, Solomon W. Harrar2, Laurence V. Madden3
Department of Statistics, University of Kentucky1
Department of Mathematical Sciences, University of Montana2
Department of Plant Pathology, Ohio State University3


In plant pathology, in particular, and plant science, in general, experiments are often conducted to determine disease and related responses of plants 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 propose 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 groups and samples. Here, we investigate several tests based on small-sample approximations, and compare their performance in terms of levels and power for different simulated situations, with continuous and discrete observations. For positively correlated responses, an approximation based on Brunner et al. (1997) 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.

Keywords: Rank Test, Small Samples, ANOVA-Type Test, Lawley-Hotelling Test, Bartlett-Nanda- Pillai Test.

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