How does heterogeneity of multivariate dispersions affect ANOSIM, PERMANOVA, and the Mantel test?

Background/Question/Methods

ANOSIM, PERMANOVA and the Mantel test are all resemblance-based permutation methods widely used in biology and ecology. Primarily applied to analyze groups of sites based on the composition or relative abundances of species they contain (community data), they allow researchers to test the general null hypothesis of ‘no differences’ among a priori groups of multivariate (multi-species) samples. These tests, although generally directed to detect differences in the locations (centroids) of multivariate groups, are also potentially sensitive to heterogeneity of multivariate dispersions. It is unknown, however, just how these methods compare in terms of their relative sensitivity to differences in dispersion among groups. A simulation study was designed to examine the effects of heterogeneity of multivariate dispersions on the rejection rates of these tests and on a classical MANOVA test using Pillai’s trace. Increasing differences in dispersion among groups were simulated under scenarios of changing sample sizes, correlation structures, error distributions, numbers of variables and numbers of groups for balanced and unbalanced one-way designs. The power of these tests to detect real potential changes due to environmental impact or natural large-scale biogeographic changes in beta diversity was also compared empirically under simulations based on parameters derived from real ecological datasets.

Results/Conclusions

Overall, ANOSIM and the Mantel test behaved far more like ‘omnibus’ tests, being far more sensitive to heterogeneity in dispersions than either PERMANOVA or Pillai’s trace, with ANOSIM generally being more sensitive than Mantel. PERMANOVA and Pillai’s trace were much more focused on testing for differences in centroids alone, and were not easily perturbed by dispersion differences, especially for balanced designs. Furthermore, unlike Pillai’s trace, PERMANOVA was also robust to non-normal error structures and differences in correlation structure. The degree of sensitivity of any of these tests to heterogeneity depended heavily on whether the design was balanced or unbalanced, and for unbalanced designs it depended also on whether the groups with larger numbers of samples had greater or lesser dispersion than the other groups. For simulations based on real ecological datasets, the ANOSIM and Mantel tests were often conservative compared to PERMANOVA, which was generally, but not always, more powerful than the others to detect changes in community composition. Differences in the underlying construction of these test statistics result in important differences in the nature of the null hypothesis they are testing, their sensitivity to heterogeneity, and their power to detect important changes in ecological communities.