Phylogenetic Factor Analysis

Abstract

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 3-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.

Figure 1.

Processes driving columbine flower evolution inferred through PFA or LMBD. a) Loadings estimates from a factor PFA model. Purple circles represent traits positively associated with traits represented by other purple circles within a loading, and negatively associated with traits represented by green circles within a loading. Similarly, traits represented by green circles are positively associated with traits represented by green circles within a loading. Size represents the magnitude of the value of the loadings. Opacity represents the posterior probability that the sign of the given element is equal to the sign of the posterior mean. The greyed out cell represents a structural 0 introduced for identifiability reasons. The magnitude for anthocyanins and pollinator type is less relevant since those measurements are discrete. b) Correlation matrix estimate from a LMBD model. Red represents positive correlation, blue represents anticorrelation, and opacity represents the absolute difference in posterior probability of being greater than 0 and less than 0. Size of the circle represents the magnitude of the correlation. The PFA captures well two independent processes, while the LMBD groups these processes together.

Figure 2.

Processes driving transitions to placental reproduction inferred through PFAs. Loading estimates from the a) , b) and c) factor models. Loadings size, coloring and density follow those of Figure 1. Note that the magnitude for dichromatism, courtship behavior, ornamental display traits, sexual selection index, and superfetation is less relevant since those data are discrete. We include the two factor model for direct comparison to the results of Pollux et al. (2014). Loadings in the more probable and factor models do not support an association between matrotrophy index and gonopodium length nor body weights and lengths.

Figure 3.

Expected triggerfish fin shape given a range of a) first factor values and b) third factor values , holding all others constant. Purple dots estimate semilandmark locations. Green lines are interpolated to present a clearer outline of the fin shape. For the relation represented by the dorsal and anal fins go from more pointed to less pointed. For the relation represented by we see a rotation in the pectoral fin.

Figure 4.

Evolution of independent factors driving triggerfish fin morphology along inferred phylogeny. The colorings display contemporary and ancestral first and third factor values under a factor PFA model. For , green represents positive values and purple represents negative values. For , the scale is orange to blue. The Supplementary material available on Dryad contains plots for , and . Balistes polylepis and Balistes vetula have negative factor values for the first factor whereas the clade containing genus Rhinecanthus has positive factor values. In the third factor , the Balistes genus and the species Pseudobalistes fuscus have positive factor values whereas the genus Rhinecanthus has near 0 factor values. Conversely, the genus Xanthichthys has a negative factor value for , and has a near 0 value for . We display the posterior clade probabilities for probabilities 99%.

Figure 5.

Inferred ancestral fin shapes at the MRCA and , and of the expected substitution distance between the MRCA and two contemporaneous triggerfish species. In a), Xanthichthys mento has a flat dorsal and anal fin with a point, and a clockwise rotated pectoral fin relative to its ancestors. The dorsal and anal fins become rounder and the pectoral fin rotates counterclockwise moving backwards in time. In contrast, in b), Balistes capriscus has a broad pointed dorsal and anal fin, and a counterclockwise anal fin. The dorsal and anal fins become more pointed and then round out, while the pectoral fin rotates clockwise.