Comparative methods for the analysis of continuous variables: geometric interpretations.

This study is concerned with statistical methods used for the analysis of comparative data (in which observations are not expected to be independent because they are sampled across phylogenetically related species). The phylogenetically independent contrasts (PIC), phylogenetic generalized least-squares (PGLS), and phylogenetic autocorrelation (PA) methods are compared. Although the independent contrasts are not orthogonal, they are independent if the data conform to the Brownian motion model of evolution on which they are based. It is shown that uncentered correlations and regressions through the origin using the PIC method are identical to those obtained using PGLS with an intercept included in the model. The PIC method is a special case of PGLS. Corrected standard errors are given for estimates of the ancestral states based on the PGLS approach. The treatment of trees with hard polytomies is discussed and is shown to be an algorithmic rather than a statistical problem. Some of the relationships among the methods are shown graphically using the multivariate space in which variables are represented as vectors with respect to OTUs used as coordinate axes. The maximum-likelihood estimate of the autoregressive parameter, p, has not been computed correctly in previous studies (an appendix with MATLAB code provides a corrected algorithm). The importance of the eigenvalues and eigenvectors of the connection matrix, W, for the distribution of p is discussed. The PA method is shown to have several problems that limit its usefulness in comparative studies. Although the PA method is a generalized least-squares procedure, it cannot be made equivalent to the PGLS method using a phylogenetic model.