Abstract:
Comparative studies of contemporary species shed light on the adaptive significance of phenotypic and genotypic variation through evidence that two or more quantitative traits have coevolved. The phylogenetic comparative method seeks this evidence in the slope of a linear regression, with the tree relating the species incorporated into the covariance matrix of the error term. In this work I resolve the longstanding issue of how inaccuracies in the phylogenetic tree influence subsequent regression estimates. Motivated by the generalized least-squares problem, I introduce a square root of the inverse covariance matrix that is readily interpretable in terms of the phylogenetic tree. Using this result, I show that small inaccuracies in the phylogeny translate into scale factors on exactly one linear contrast of the data; consequently, the impact of any such inaccuracy can be understood as an instance of weighted least squares. I will discuss the implications of this work on future comparative studies, including the development of robust statistical procedures.