Distance-Based Tests for Homogeneity of Multivariate Dispersions
Article first published online: 20 OCT 2005
Volume 62, Issue 1, pages 245–253, March 2006
How to Cite
Anderson, M. J. (2006), Distance-Based Tests for Homogeneity of Multivariate Dispersions. Biometrics, 62: 245–253. doi: 10.1111/j.1541-0420.2005.00440.x
- Issue published online: 20 OCT 2005
- Article first published online: 20 OCT 2005
- Received January 2005. Revised June 2005. Accepted July 2005.
- Levene's test;
- Multivariate analysis;
- Permutation tests;
- Principal coordinates;
- Robust methods
Summary The traditional likelihood-based test for differences in multivariate dispersions is known to be sensitive to nonnormality. It is also impossible to use when the number of variables exceeds the number of observations. Many biological and ecological data sets have many variables, are highly skewed, and are zero-inflated. The traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non-Euclidean dissimilarity would be more appropriate. Distance-based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed here. They rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes. The tests are straightforward multivariate extensions of Levene's test, with P-values obtained either using the traditional F-distribution or using permutation of either least-squares or LAD residuals. Examples illustrate the utility of the approach, including the analysis of stabilizing selection in sparrows, biodiversity of New Zealand fish assemblages, and the response of Indonesian reef corals to an El Niño. Monte Carlo simulations from the real data sets show that the distance-based tests are robust and powerful for relevant alternative hypotheses of real differences in spread.