Devil in the details: how can we avoid potential pitfalls of CATS regression when our data do not follow a Poisson distribution?

Abstract

Background Community assembly by trait selection (CATS) allows for the detection of environmental filtering and estimation of the relative role of local and regional (meta-community-level) effects on community composition from trait and abundance data without using environmental data. It has been shown that Poisson regression of abundances against trait data results in the same parameter estimates. Abundance data do not necessarily follow a Poisson distribution, and in these cases, other generalized linear models should be fitted to obtain unbiased parameter estimates. Aims This paper discusses how the original algorithm for calculating the relative role of local and regional effects has to be modified if Poisson model is not appropriate. Results It can be shown that the use of the logarithm of regional relative abundances as an offset is appropriate only if a log-link function is applied. Otherwise, the link function should be applied to the product of local total abundance and regional relative abundances. Since this product may be outside the domain of the link function, the use of log-link is recommended, even if it is not the canonical link. An algorithm is also suggested for calculating the offset when data are zero-inflated. The relative role of local and regional effects is measured by Kullback-Leibler R2. The formula for this measure presented by Shipley (2014) is valid only if the abundances follow a Poisson distribution. Otherwise, slightly different formulas have to be applied. Beyond theoretical considerations, the proposed refinements are illustrated by numerical examples. CATS regression could be a useful tool for community ecologists, but it has to be slightly modified when abundance data do not follow a Poisson distribution. This paper gives detailed instructions on the necessary refinement.