Why the phylogenetic regression appears robust to tree misspecification

The phylogenetic comparative method uses estimates of evolutionary relationships to explicitly model the covariance structure of interspecific data. By accounting for common ancestry, the coevolution between 2 or more traits, as a response to one another or to environmental variables, can be studied without confounding similarities due to identity by descent. Because the true phylogeny is unknowable, an estimate must be used, introducing a source of error into phylogenetic comparative analysis that can be difficult to quantify. This manuscript aims to elucidate how tree misspecification is propagated through a comparative analysis. I focus on the phylogenetic regression under a Brownian motion model of evolution and consider the effect of local phylogenetic perturbations on the regression fit. Motivated by Felsenstein's method of independent contrasts, I derive a matrix square root of the phylogenetic covariance matrix that has an obvious phylogenetic interpretation. I use this result to transform the perturbed phylogenetic regression model into an ordinary linear regression in which one interpretable point has been affected. The simplicity of this formulation allows the contributions of data and phylogeny to be disentangled when studying the effect of tree misspecification. Consequentially, I find that branch length misspecification can be easily explained in terms of the reweighting of contrast scores between subtrees. An analytical consideration of this and other perturbations helps to explain why the phylogenetic regression appears generally to be robust to tree misspecification, and I am able to identify conditions under which the regression may not yield robust results. I discuss why soft polytomies do not meet these problematic conditions, leading to the conclusion that unresolved bifurcations should have only modest effects on the regression fit.