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

Abstract

Summary1. 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. 2. 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. 3. 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. 4. 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. 5. 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. 6. 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.

Fourth-corner correlation is a score test statistic in a log-linear trait–environment model that is useful in permutation testing

“…This result was derived with saturated main effects (having free row and column parameters r i and c j ). Brown et al (2014) obtained a similar result from the log-linear model with T and E as main effects.…”

“…This paper shows that the fourth-corner correlation, heuristically developed by Legendre et al (1997) for examining trait-environment associations, has a close relationship with the Poisson log-linear model with interactions, which has recently been proposed as a model for trait-environment relationships (Brown et al 2014;Warton et al 2015b). The squared fourth-corner correlation is proportional to the score test statistic for testing the linear-by-linear interaction in the Poisson log-linear model with row and column main effects.…”

“…More recently, Pollock et al (2012), Jamil et al (2013 and Brown et al (2014) independently proposed model-based approaches that generalize the fourth-corner problem to multiple traits and environmental variables using generalized linear (mixed) models (GL(M)M) for vectorized Y, as in the bilinear regression approach of Gabriel (1998). Model-based approaches have great appeal (e.g.…”

“…Mainstream methods of variable selection are used to build parsimonious models. In these approaches, X becomes a matrix of (partial) regression coefficients estimating the direction and strength of the interaction between standardized traits and standardized environmental variables as well as their main effects on species distributions (Brown et al 2014).…”

“…One such, almost trivial, connection has been presented in the Appendix of Brown et al (2014). In there, for a nominal trait and a nominal environmental variable, the fourth-corner X is a contingency table obtained by merging columns and rows that belong to the same category of the trait and of the environmental variable, respectively, and the likelihood ratio test on interaction in a contingency table using a Poisson log-linear model is asymptotically equivalent with the usual chi-square test.…”