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Phylogenetic comparative methods offer a suite of tools for studying trait evolution. However, most models inherently assume fixed trait values within species. Although some methods can incorporate error around species means, few are capable of accounting for variation driven by environmental or temporal gradients, such as trait responses to abiotic stress or ontogenetic trajectories. Such traits, often referred to as function-valued or infinite-dimensional, are typically expressed as reaction norms, dose-response curves, or time plots and are described by mathematical functions linking independent predictor variables to the trait of interest. Here, I introduce a method for extending ancestral state reconstruction to incorporate function-valued traits in a phylogenetic generalized least squares (PGLS) framework, as well as extensions of this method for testing phylogenetic signal, performing phylogenetic analysis of variance (ANOVA), and testing for correlated trait evolution using recently proposed multivariate PGLS methods. Statistical power of function-valued comparative methods is compared to univariate approaches using data simulations, and the assumptions and challenges of each are discussed in detail.

Phylogenetic Comparative Methods for Evaluating the Evolutionary History of Function-Valued Traits