A new non-parametric test for heterogeneity of multivariate dispersions
by
Marti J. Anderson
Department of Statistics, University of Auckland
Coauthors: Brian H. McArdle

Non-parametric multivariate tests for differences among groups in ANOVA designs have recently been developed. These methods are general and robust, using only a pair-wise distance matrix among objects (based on any distance or dissimilarity of choice), and obtaining P-values using permutations. The general null hypothesis tested, assuming exchangeability of objects, is that there are no differences among the groups. Although designed to test for differences in location, these tests are sensitive to differences in multivariate dispersions. A comparable test for heterogeneity of multivariate dispersions is needed in this context, both in order to separate out effects of dispersion from effects of location, and also as a test in its own right where the hypothesis of interest concerns potential differences in dispersion. It is desirable that such a test retains all of the robust and generalisable qualities of the non-parametric tests of location. We present here a new method that does precisely this. The test-statistic is a multivariate analogue to Levene's test, calculated from the distances of observations within groups to their group centroid. It relies on the use of principal coordinates, where distances are non-Euclidean. P-values are obtained using permutations. Some important issues, including the non-normality and non-independence of distances to centroids are in need of careful attention, and are solved by our approach. We will give some examples of the method for ecological data and hypotheses.

Date received: August 29, 2001