Supplementary Materials include results of simulation experiments to investigate the impact of phylogenetic regression with model violations.

Modern comparative biology owes much to phylogenetic regression. At its
conception, this technique sparked a revolution that armed biologists with
phylogenetic comparative methods (PCMs) for disentangling evolutionary
correlations from those arising from hierarchical phylogenetic
relationships. Over the past few decades, the phylogenetic regression
framework has become a paradigm of modern comparative biology that has
been widely embraced as a remedy for shared ancestry. However, recent
evidence has sown doubt over the efficacy of phylogenetic regression, and
PCMs more generally, with the suggestion that many of these methods fail
to provide an adequate defense against unreplicated evolution—the primary
justification for using them in the first place. Importantly, some of the
most compelling examples of biological innovation in nature result from
abrupt lineage-specific evolutionary shifts, which current regression
models are largely ill-equipped to deal with. Here we explore a solution
to this problem by applying robust linear regression to comparative trait
data. We formally introduce robust phylogenetic regression to the PCM
toolkit with linear estimators that are less sensitive to model violations
than the standard least-squares estimator, while still retaining high
power to detect true trait associations. Our analyses also highlight an
ingenuity of the original algorithm for phylogenetic regression based on
independent contrasts, whereby robust estimators are particularly
effective. Collectively, we find that robust estimators hold promise for
improving tests of trait associations and offer a path forward in
scenarios where classical approaches may fail. Our study joins recent
arguments for increased vigilance against unreplicated evolution and a
better understanding of evolutionary model performance in challenging–yet
biologically important–settings.