The multinomial diversity model: linking Shannon diversity to multiple predictors

The multinomial diversity model, MDM, is a new method for relating Shannon diversity to complex environmental, spatial, and temporal predictors. It is based on a parameterized formulation of Shannon entropy and diversity, and a novel link between entropy and the log-likelihood of the multinomial model. The MDM relates diversity to the predictors by minimizing the entropy of the estimated species values. Model effects can be expressed as changes in entropy. Entropy can be partitioned within and between sites, species, and models, and changes in entropy can be attributed to model predictors. All entropies translate into diversity for meaningful ecological interpretation. This greatly enhances our capacity to model complex data sets, and yet also provide simple interpretations. By formulating diversity as a statistical model and working in terms of entropy, diversity is simplified both conceptually and analytically, and diversity analyses are extended beyond traditional simple hierarchies of alpha, beta, gamma, and measures of turnover. The MDM inherits the properties of generalized linear models, and thus proven methods can be used for model selection and graphical and numerical interpretation. A weighted version of the Shannon diversity model is proposed in order to extend the MDM to non-Shannon diversities. Two example analyses, based on simulated and field data, illustrate the theoretical concepts and the analytical methods.