The most unusual, and thus irreplaceable, functions performed by species in three different species-rich ecosystems are fulfilled by only the rare species in these ecosystems.

This paper addresses the following question: how does one relate the biological and behavioral characteristics of animals to habitat characteristics of the locations at which they are found? Ecologists often assemble data on species composition at different localities, habitat descriptions of these localities, and biological or behavioral traits of the species. These data tables are usually analyzed two by two: species composition against habitat characteristics, or against behavioral data, using such methods as canonical analysis. We propose a solution to the problem of estimating the parameters describing the relationship between habitat characteristics and biology or behavior, and of testing the statistical significance of these parameters; this problem is referred to as the fourth-corner problem, from its matrix formulation. In other words, the fourth-corner method offers a way of analyzing the relationships between the supplementary variables associated with the rows and columns of a binary (presence or absence) data table. The test case that motivated this study concerns a coral reef fish assemblage (280 species). Biological and behavioral characteristics of the species were used as supplementary variables for the rows, and characteristics of the environment for the columns. Parameters of the association between habitat characteristics (distance from beach, water depth, and substrate variables) and biological and behavioral traits of the species (feeding habits, ecological niche categories, size classes, egg types, activity rhythms) were estimated and tested for significance using permutations. Permutations can be performed in different ways, corresponding to different ecological hypotheses. Results were compared to predictions made independently by reef fish ecologists, in order to assess the method as well as the pertinence of the variables subjected to the analysis. The new method is shown to be applicable to a wide class of ecological problems.

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