A frequent goal in ecology is to understand the relationships among biological communities and their environment. Anderson and McCardle (2001) provided a nonparametric method, known as Permanova, which is often used for this purpose. Permanova represents a significant advance, allowing the use of linear models of environmental effects to explore and identify the similarities or dissimilarities among biological communities. A typical application of Permanova is to use the abundances of p species collected from n samples to compute the Bray-Curtis dissimilarity index among paired samples. Permanova then relates the n × n matrix of Bray-Curtis values (or some other metric) to the concomitant environmental measures. However, the identities of the individual species in the communities are not retained in the Permanova analysis. The negative binomial is often considered an appropriate probability distribution describing the abundances of a single species, and then relating the counts of that species to explanatory variables through negative binomial regression. This paper explores the use of linear models where the response — a multivariate n × p matrix of the counts of individuals of each species — is considered to follow a negative multinomial distribution.
Using simulated datasets and observed field data showed that analyses of communities using either Permanova or negative multinomial regression provide similar results. In both cases the analyses indicated differences in community structure that appear to be related to explanatory environmental variables. However, negative multinomial regression provides coefficient estimates (and their standard errors) for each of the p species. These coefficients allow further evaluation of the role of each species as it contributes to changes in community structure among samples collected under different environmental conditions. Negative multinomial regression does have the disadvantage that it requires an adequate number of observations. This is because the number of parameters to be estimated must be less than the number of available observations, as is the case in parametric linear models, i.e. n >>p. This constraint means that negative multinomial analysis of communities with a large number of distinct species requires large sample sizes. Permanova does not share this limitation. Nonetheless, negative multinomial linear models appear to be a useful complement to Permanova.