PERMANOVA, ANOSIM, and the Mantel test in the face of heterogeneous dispersions: What null hypothesis are you testing?

ANOSIM, PERMANOVA, and the Mantel test are all resemblance-based permutation methods widely used in ecology. Here, we report the results of the first simulation study, to our knowledge, specifically designed to examine the effects of heterogeneity of multivariate dispersions on the rejection rates of these tests and on a classical MANOVA test (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 environmental impacts or natural large-scale biogeographic gradients was also compared empirically under simulations based on parameters derived from real ecological data sets. Overall, ANOSIM and the Mantel test were very sensitive to heterogeneity in dispersions, with ANOSIM generally being more sensitive than the Mantel test. In contrast, PERMANOVA and Pillai's trace were largely unaffected by heterogeneity for balanced designs. PERMANOVA was also unaffected by differences in correlation structure, unlike Pillai's trace. For unbalanced designs, however, all of the tests were (1) too liberal when the smaller group had greater dispersion and (2) overly conservative when the larger group had greater dispersion, especially ANOSIM and the Mantel test. For simulations based on real ecological data sets, PERMANOVA was generally, but not always, more powerful than the others to detect changes in community structure, and the Mantel test was usually more powerful than ANOSIM. Both the error distributions and the resemblance measure affected results concerning power. 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. For balanced designs, PERMANOVA and PERMDISP can be used to rigorously identify location vs. dispersion effects, respectively, in the space of the chosen resemblance measure. ANOSIM and the Mantel test can be used as more omnibus'' tests, being sensitive to differences in location, dispersion or correlation structure among groups. Unfortunately, none of the tests (PERMANOVA, Mantel, or ANOSIM) behaved reliably for unbalanced designs in the face of heterogeneity.