  Phylogenetic comparative methods explore the relationships between
quantitative traits adjusting for shared evolutionary history. This adjustment
often occurs through a Brownian diffusion process along the branches of the
phylogeny that generates model residuals or the traits themselves. For
high-dimensional traits, inferring all pair-wise correlations within the
multivariate diffusion is limiting. To circumvent this problem, we propose
phylogenetic factor analysis (PFA) that assumes a small unknown number of
independent evolutionary factors arise along the phylogeny and these factors
generate clusters of dependent traits. Set in a Bayesian framework, PFA
provides measures of uncertainty on the factor number and groupings, combines
both continuous and discrete traits, integrates over missing measurements and
incorporates phylogenetic uncertainty with the help of molecular sequences. We
develop Gibbs samplers based on dynamic programming to estimate the PFA
posterior distribution, over three-fold faster than for multivariate diffusion
and a further order-of-magnitude more efficiently in the presence of latent
traits. We further propose a novel marginal likelihood estimator for previously
impractical models with discrete data and find that PFA also provides a better
fit than multivariate diffusion in evolutionary questions in columbine flower
development, placental reproduction transitions and triggerfish fin
morphometry.
