Consilience of methods for phylogenetic analysis of variance

Simulation-based and permutation-based inferential methods are commonplace in phylogenetic comparative methods, especially as evolutionary data have become more complex and parametric methods more limited for their analysis. Both approaches simulate many random outcomes from a null model to empirically generate sampling distributions of statistics. Although simulation-based and permutation-based methods seem commensurate in purpose, results from analysis of variance (ANOVA) based on the distributions of random F-statistics produced by these methods can be quite different in practice. Differences could be from either the null-model process that generates variation across many simulations or random permutations of the data, or different estimation methods for linear model coefficients and statistics. Unfortunately, because the null-model process and coefficient estimation are intrinsically linked in phylogenetic ANOVA methods, the precise reason for methodological differences has not been fully considered. Here we show that the null-model processes of phylogenetic simulation and randomizing residuals in a permutation procedure are indeed commensurate, and that both also produce results consistent with parametric ANOVA, for cases where parametric ANOVA is possible. We also provide results that caution against using ordinary least-squares estimation along with phylogenetic simulation; a typical phylogenetic ANOVA implementation.