Correspondence Analysis of Incidence and Abundance Data: Properties in Terms of a Unimodal Response Model

Correspondence analysis is commonly used by ecologists to analyze data on the incidence or abundance of species in samples. The first few axes are interpreted as latent variables and are presumed to relate to underlying environmental variables. In this paper correspondence analysis is shown to approximate the maximum likelihood solution of explicit unimodal response models in one latent variable. These models are logistic-linear for presence/absence data and loglinear for Poisson counts, with predictors that are quadratic in the latent variable. The approximation is best when the maxima and tolerances (widths) of the response curves are equal and the species' optima and the sample values of the latent variable are equally spaced. It is still fairly good for uniformly distributed optima and sample values, as shown by simulation. For the models extended to two latent variables, the approximation is often bad because of the horseshoe effect in correspondence analysis, but improves considerably in the simulations when this effect is removed as it is in detrended correspondence analysis.