Orthonormal transform to decompose the variance of a life-history trait across a phylogenetic tree

In recent years, there has been an increased interest in studying the variability of a quantitative life-history trait across a set of species sharing a common phylogeny. However, such studies have suffered from an insufficient development of statistical methods aimed at decomposing the trait variance with respect to the topological structure of the tree. Here we propose a new and generic approach that expresses the topological properties of the phylogenetic tree via an orthonormal basis, which is further used to decompose the trait variance. Such a decomposition provides a structure function, referred to as an "orthogram," which is relevant to characterize in both graphical and statistical aspects the dependence of trait values on the topology of the tree ("phylogenetic dependence"). We also propose four complementary test statistics to be computed from orthogram values that help to diagnose both the intensity and the nature of phylogenetic dependence. The relevance of the method is illustrated by the analysis of three phylogenetic data sets, drawn from the literature and typifying contrasted levels and aspects of phylogenetic dependence. Freely available routines which have been programmed in the R framework are also proposed.