Matching species traits to environmental variables: A new three-table ordination method

This paper addresses the question of studying the joint structure of three data tables R, L and Q. In our motivating ecological example, the central table L is a sites-by-species table that contains the number of organisms of a set of species that occurs at a set of sites. At the margins of L are the sites-by-environment data table R and the species-by-trait data table Q. For relating the biological traits of organisms to the characteristics of the environment in which they live, we propose a statistical technique called RLQ analysis (R-mode linked to Q-mode), which consists in the general singular value decomposition of the triplet (R(t)D(I)LD(J)Q,D-q,D-p) where D-I, D-J, D-q, D-p are diagonal weight matrices, which are chosen in relation to the type of data that is being analyzed (quantitative, qualitative, etc.). In the special case where the central table is analysed by correspondence analysis, RLQ maximizes the covariance between linear combinations of columns of R and Q. An example in bird ecology illustrates the potential of this method for community ecologists.