Distance, dissimilarity, and mean–variance ratios in ordination

Summary Recent advances employing generalized linear models in ecology have led ecological statisticians to argue strongly for the need to re‐evaluate or re‐invent commonly used multivariate analyses in ecology to explicitly address the mean–variance ratios of the underlying species data. This paper re‐evaluates previous work on the importance of aligning distance‐ or dissimilarity‐based methods with the mean–variance ratios of underlying data and extends the analyses to additional simulated data designed to test the specific assertions of model‐based advocates with respect to ordination analysis. Similar analyses are conducted on an observed community dataset where the observed species abundances exhibit a quadratic variance‐to‐mean ratio. The results emphasize two general points with respect to (i) the sensitivity of ordination to species abundance distributions and (ii) the inherently hierarchical nature of community analysis. Previously observed problematic results are shown to result from extremely high between‐community variability in sample unit abundance, and are independent of the underlying species mean–variance ratios. Results for simulated data as well as for observed coral reef community data show similar results. Multivariate analyses of the newly simulated data from two communities along a simulated gradient show excellent results despite the underlying species negative binomial distributions and putatively problematic mean–variance ratios. Problems identified in previously observed results reflect a misunderstanding of the hierarchical nature of ecological communities. Mechanisms operating at the individual species level can manifest at the community level as a different ecological (and statistical) phenomenon. Specifically, uncorrelated or divergent changes in species abundances within communities lead inevitably to dispersion effects at the community level. The hierarchical nature of the analysis is independent of the specifics of the species mean–variance ratios, and would obtain whether species abundances are distributed with uniform, normal, or negative‐binomial distributions. The results presented here demonstrate that suitable ordination of the data accurately represents the ecological response at the community level. However, paired ordinations on non‐standardized and standardized data can elucidate the underlying species‐level changes in abundance as the mechanism driving the change in community dispersion.