Biologists are frequently interested in the evolutionary correlations between continuously distributed characters measured from species related by a phylogenetic tree. These correlations can arise by a number of causes. Evolutionary correlations can arise by natural selection. For example, an “evolutionary correlation,” that is, a correlation between the evolutionary changes in two characters, can arise if the traits are under correlated selection or evolving along a ridge in the adaptive landscape (Felsenstein 1988; Arnold et al. 2001; Martins et al. 2002; Jones et al. 2003), if the traits are evolving in response to selection toward randomly moving optima, in which the movement of each optimum is by correlated Brownian motion (Hansen and Martins 1996; Revell and Harmon 2008), or if the traits are functionally integrated to perform an ecological task (Walker 2007; Collar et al. 2008). Evolutionary correlations can also arise by genetic drift. For example, an evolutionary correlation between two characters will arise under drift if the characters themselves are genetically correlated (Lande 1979; Arnold et al. 2001; Revell and Harmon 2008).
However, the evolutionary correlation between characters can also change over time as the adaptive, functional, and genetic relationships between characters evolve. For example, selective regimes are expected to change as the phenotype and external environment change through time and among phylogenetic lineages. In fact, many hypotheses about adaptive phenotypic evolution specifically predict changes in the evolutionary correlation between traits. These include hypotheses pertaining to the origin of a novel trait or function that provides a lineage access to a new adaptive zone (so-called “key innovations”; Miller 1949; Galis 2000), the mechanical decoupling of morphological features (Liem 1973; Vermeij 1973; Lauder 1981), and shifts in the selective regime following the invasion of a novel habitat, ecological niche, or geographical area (Grant 1972; Schluter 1988).
Additionally, genetic constraints evolve (Roff 1997; Steppan et al. 2002; Jones et al. 2003; Bégin and Roff 2004) and so must the tendency toward a particular evolutionary correlation by drift. Persistent changes in the genetic covariances between characters can be induced by deterministic factors, such as a change in the regime of correlational selection (e.g., Jones et al. 2007; Revell 2007), as well as by stochastic forces, such as gene duplication or a population bottleneck (e.g., Whitlock et al. 2002; Revell and Harmon 2008). Under drift, a persistent change in the genetic correlation between characters is also expected to induce a persistent change in their evolutionary correlation (Revell and Harmon 2008).
A classic example of a hypothesis that predicts a change in the evolutionary correlation between characters is Liem's (1973) hypothesis that the origin of a novel pharyngeal jaw form in cichlid fishes contributed to the exceptional morphological and ecological diversity found in that group. Liem (1973) hypothesized that the specialization of the cichlid pharyngeal jaw on food processing freed the oral jaw to differentiate in different lineages to capture different kinds of prey. This hypothesis thus predicts that the evolutionary correlation between pharyngeal and oral jaws will be weaker in the cichlid radiation, where pharyngeal and oral jaws are functionally disassociated, than in other lineages of percoid fishes (Liem 1973; Lauder 1981; Hulsey et al. 2006).
Similarly, the genetic decoupling of serially repeated structures such as cilia, limbs, and teeth is hypothesized to have led to new multivariate patterns of morphospace occupation in clades as diverse as trochophores and rotifers (Strathmann et al. 1972; Vermeij 1973), theropod dinosaurs (including birds; Gatesy and Dial 1996; Hunter 1998), and mammals (e.g., Walker 1987; Stock 2001). A hypothesis of complete or partial genetic decoupling of these serial structures provides the testable prediction that the evolutionary correlation has decreased in the affected evolutionary lineages where characters have more freedom to evolve independently.
Although these hypotheses for multivariate phenotypic diversification imply a covariance structure among characters that changes over time, typical approaches for the analysis of continuous characters on a phylogeny ignore this possibility. To analyze the evolutionary correlation in a phylogenetic context we usually either: (1) obtain phylogenetically independent contrasts, using the method of Felsenstein (1985), and then analyze these contrasts using parametric regression or correlation techniques; or (2) use the method of phylogenetic generalized least squares (Grafen 1989; Martins and Hansen 1997; Rohlf 2001) to fit a bivariate or multiple regression model while controlling for the phylogenetic relationships among the taxa in the tree assuming a single, constant rate of Brownian motion evolution, or some modification thereof (Pagel 1999; e.g., Revell and Harrison 2008). The typical application of these approaches thus generally assumes both constant evolutionary rates for individual characters and invariant correlations between characters across the branches of the phylogenetic tree.
Existing methods have some flexibility to accommodate heterogeneity in the evolutionary rate (Garland and Ives 2000), and can be used, under some circumstances, to test for differences in the relationship between characters in different parts of the tree (Garland et al. 1993; Garland and Ives 2000). These methods are either based on independent contrasts (Felsenstein 1985), or on numerical simulation (Garland et al. 1993). Contrasts-based methods suffer the shortcoming that they require the unambiguous assignment of all sets of paired sister nodes to an evolutionary rate category (e.g., Garland and Ives 2000). Simulation-based methods suffer the shortcoming that they do not allow meaningful estimation of the parameters of the evolutionary process (e.g., the phylogenetic analysis of covariance of Garland et al. 1993). Alternative methods that use a generalized linear modeling approach to study the evolutionary rate of a single character (e.g., Martins 1994; Martins and Hansen 1997) could be plausibly adapted to estimate rate heterogeneity in multiple characters; however, these methods also rely on the calculation of independent contrasts (Martins 1994).
Herein, we propose a new method to test for shifts in the evolutionary correlation by using maximum likelihood to fit multiple evolutionary rate matrices, also called evolutionary variance–covariance matrices, to different branches of a phylogenetic tree. The evolutionary rate matrix contains, on its diagonal, the evolutionary variances or rates for individual characters, and, on its off-diagonal, the evolutionary covariances (Revell and Harmon 2008). Following Revell and Harmon (2008), we use the term “evolutionary rate matrix” because the matrix is a multivariate representation of the evolutionary rate (O’Meara et al. 2006), and (under Brownian motion) completely describes the distribution from which evolutionary changes are drawn. As the evolutionary correlation between two characters is a function of their evolutionary variances and covariance, as far as we know this method is the first in which a likelihood approach is used to estimate different evolutionary correlations in different parts of the tree.
The method is based on a Brownian motion model of continuous character evolution, in which evolutionary changes are drawn from a multivariate normal distribution with variances and covariances that are proportional to the time elapsed. Thus, our approach extends existing methods that also use a Brownian model to test for changes in the rate of evolution for a single character (O’Meara et al. 2006; Thomas et al. 2006) or concerted changes in all the elements of the multivariate evolutionary rate matrix (Revell and Harmon 2008). However, our method additionally tests for nonproportional shifts in the evolutionary variances and covariances (of which the evolutionary correlations are a function) among phylogenetic lineages.
A particular advantage of our approach is that it allows testing of any specific a priori hypothesis for rate matrix heterogeneity—regardless of whether that hypothesis consists of monophyletic sets of taxa (i.e., our approach is “noncensored,” in the sense of O’Meara et al. 2006). For example, we can use the reconstruction of a binary or discrete multistate characteristic on the tree to assign branches and even portions of branches to different rate matrix categories (O’Meara et al. 2006). These reconstructed traits can pertain to habitat, biogeographic region, ecology, or any other plausible influence on the evolutionary rate matrix. With appropriate a priori justification, we can go even further and assign internal branches and parts of branches of the tree to different rate matrix categories. For example, we might test a hypothesis in which the evolutionary rate matrix differed between the Pliocene and Pleistocene, or a hypothesis in which the rate matrix differed on the branches of the tree before and after a mass extinction event (O’Meara et al. 2006).
In the present study, we apply the likelihood method to an empirical dataset and phylogeny from centrarchid fishes to test an a priori hypothesis that the evolutionary correlation between two aspects of buccal morphology has changed as a consequence of a change in trophic habit. We also test the method for type I error and power using simulated data generated by one and two rate matrix Brownian motion simulations.