# Ordination plots for model-based analysis of species composition data: Connecting two very different sets of methods

Traditional analysis of species composition data starts by computing a dissimilarity between each pair of samples. These dissimilarities are the basis for ordination plots, which explore patterns, and ANOVA-like inference. The more recent model-based analysis of species composition starts with an explicit probability model for the data. This provides a principled approach for inference but it has been difficult to produce an ordination plot. Often, a traditional ordination method is combined with a model-based analysis, but these make different assumptions about the data. I develop an approach to visualizing multivariate species composition that is consistent with a model-based analysis, in the sense that the plot and the analysis make the same assumptions. This approach retains the data distribution from a model-based analysis and makes no additional assumptions about the structure of the data.

The probability model is used to define a likelihood-based dissimilarity between each pair of observations. I derive these for Bernoulli (i.e., presence/absence) data, four models for count data, and Gaussian data. An ordination plot is constructed by multidimensional scaling. The approach can be extended to overlay model-based predictions of species composition on the ordination plot or compute a community-level residual. I illustrate these using nesting bird counts on Skokholm Island and tropical tree stem counts on Barro Colorado Island.