Analyzing environmental‐trait interactions in ecological communities with fourth‐corner latent variable models

In ecological community studies it is often of interest to study the effect of species related trait variables on abundances or presence‐absences. Specifically, the interest may lay in the interactions between environmental and trait variables. An increasingly popular approach for studying such interactions is to use the so‐called fourth‐corner model, which explicitly posits a regression model where the mean response of each species is a function of interactions between covariate and trait predictors (among other terms). On the other hand, many of the fourth‐corner models currently applied in the literature are too simplistic to properly account for variation in environmental and trait response and any residual covariation between species. To overcome this problem, we propose a fourth‐corner latent variable model which combines the following three features: latent variables to capture the correlation between species, fourth‐corner terms to account for environment‐trait interactions, and species‐specific random slopes for modeling excess heterogeneity between species in their environmental response. We perform an extensive numerical study comparing a variety of fourth‐corner models available in the literature which account for the aforementioned sources of variation to varying degrees. Simulation results demonstrate that the proposed fourth‐corner latent variable models performed well when testing for the fourth‐corner (interaction) coefficients, across both Type I error and power. By comparison, some models that do not full account for all relevant sources of variation suffer from inflated Type I error leading to potentially misleading inference. The proposed method is illustrated by an example on ground beetle data.