Comparative Methods with Sampling Error and Within‐Species Variation: Contrasts Revisited and Revised

Abstract

Comparative methods analyses have usually assumed that the species phenotypes are the true means for those species. In most analyses, the actual values used are means of samples of modest size. The covariances of contrasts then involve both the covariance of evolutionary changes and a fraction of the within-species phenotypic covariance, the fraction depending on the sample size for that species. Ives et al. have shown how to analyze data in this case when the within-species phenotypic covariances are known. The present model allows them to be unknown and to be estimated from the data. A multivariate normal statistical model is used for multiple characters in samples of finite size from species related by a known phylogeny, under the usual Brownian motion model of change and with equal within-species phenotypic covariances. Contrasts in each character can be obtained both between individuals within a species and between species. Each contrast can be taken for all of the characters. These sets of contrasts, each the same contrast taken for different characters, are independent. The within-set covariances are unequal and depend on the unknown true covariance matrices. An expectation-maximization algorithm is derived for making a reduced maximum likelihood estimate of the covariances of evolutionary change and the within-species phenotypic covariances. It is available in the Contrast program of the PHYLIP package. Computer simulations show that the covariances are biased when the finiteness of sample size is not taken into account and that using the present model corrects the bias. Sampling variation reduces the power of inference of covariation in evolution of different characters. An extension of this method to incorporate estimates of additive genetic covariances from a simple genetic experiment is also discussed.