Dray S., Legendre P. (2008). Testing the species traits-environment relationships: the fourth-corner problem revisited. Ecology, 89:3400--3412.

Functional ecology aims at determining the relationships between species traits and environmental variables in order to better understand biological processes in ecosystems. From a methodological point of view, this biological objective calls for a method linking three data matrix tables: a table L with abundance or presence-absence values for species at a series of sites, a table R with variables describing the environmental conditions of the sites, and a table Q containing traits (e.g., morphological or behavioral attributes) of the species. Ten years ago, the fourth-corner method was proposed to measure and test the relationships between species traits and environmental variables using tables R, L, and Q simultaneously. In practice, this method is rarely used. The major reasons for this lack of interest are the restriction of the original method and program to presence-absence data in L and to the analysis of a single trait and a single environmental variable at a time. Moreover, ecologists often have problems in choosing a permutation model among the four originally proposed. In this paper, we revisit the fourth-corner method and propose improvements to the original approach. First, we present an extension to measure the link between species traits and environmental variables when the ecological community is described by abundance data. A new multivariate fourth-corner statistic is also proposed. Then, using numerical simulations, we discuss and evaluate the existing testing procedures. A new two-step testing procedure is presented. We hope that these elements will help ecologists use the best possible methodology to analyze this type of ecological problem.

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