The fourth-corner solution - using predictive models to understand how species traits interact with the environment

An important problem encountered by ecologists in species distribution modelling (SDM) and in multivariate analysis is that of understanding why environmental responses differ across species, and how differences are mediated by functional traits. We describe a simple, generic approach to this problem - the core idea being to fit a predictive model for species abundance (or presence/absence) as a function of environmental variables, species traits and their interaction. We show that this method can be understood as a model-based approach to the fourth-corner problem - the problem of studying the environment-trait association using matrices of abundance or presence/absence data across species, environmental data across sites and trait data across species. The matrix of environment-trait interaction coefficients is the fourth corner. We illustrate that compared with existing approaches to the fourth-corner problem, the proposed model-based approach has advantages in interpretability and its capacity to perform model selection and make predictions. To illustrate the method we used a generalized linear model with a LASSO penalty, fitted to data sets from four different studies requiring different models, illustrating the flexibility of the proposed approach. Predictive performance of the model is compared with that of fitting SDMs separately to each species, and in each case, it is shown that the trait model, despite being much simpler, had comparable predictive performance, even significantly outperforming separate SDMs in some cases.