The determination of whether the pattern of trait evolution observed in a comparative analysis of species data is due to adaptation to current environments, to phylogenetic inertia, or to both of these forces requires that one control for the effects of either force when making an assessment of the evolutionary role of the other. Orzack and Sober (2001) developed the method of controlled comparisons to make such assessments; their implementation of the method focussed on a discretely varying trait. Here, we show that the method of controlled comparisons can be viewed as a meta-method, which can be implemented in many ways. We discuss which recent methods for the comparative analysis of continuously distributed traits can generate controlled comparisons and can thereby be used to properly assess whether current adaptation and/or phylogenetic inertia have influenced a trait's evolution. The implementation of controlled comparisons is illustrated by an analysis of sex-ratio data for fig wasps. This analysis suggests that current adaptation and phylogenetic inertia influence this trait.
Although the validity of some adaptive hypotheses can be assessed without knowledge of the history of the trait (Reeve and Sherman 1993; Sober and Orzack 2003), it is now generally understood that assessment of most adaptive hypotheses must account for phylogenetic inertia (e.g., see Martins 2000). This recognition is the basis of modern work on methods for the comparative analysis of species data (Ridley 1983; Felsenstein 1985; Harvey and Pagel 1991).
Much less understood is that the study of phylogenetic inertia must account for adaptation (Orzack and Sober 2001), since related species may resemble each other because they have similar focal adaptations (Grafen 1989). Phylogenetic inertia refers to the fact that a trait may not be perfectly adapted to its current environment because of its evolutionary history. Just as physical inertia is the ability of a mass to oppose changes in its velocity, phylogenetic inertia is the tendency of a trait to resist a current adaptive force. Phylogenetic inertia is thus always measured relative to an adaptive hypothesis, and in this regard it is distinct from a phylogenetic “effect,” which is simply a description of an association between phylogeny and pattern of trait evolution. A phylogenetic effect may be due to either phylogenetic inertia or similar focal adaptations in related taxa (see Derricksen and Ricklefs 1988). Phylogenetic inertia can have a multitude of causes, which we refer to as “phylogenetic constraints.” These include variational constraints and constraints resulting from selection pressures that conflict with current or past selection for the focal adaptation.
Most claims about the evolutionary roles played by adaptation and phylogenetic inertia have regarded them as exclusive explanations for a given trait. In addition, phylogenetic inertia is now often taken to be a null hypothesis, that is, as a hypothesis having explanatory precedence over an adaptive hypothesis. However, as we now describe, we believe that otherwise well-accepted understanding of the biology of trait evolution indicates that a role for phylogenetic inertia in a given trait's evolution need not rule out a role for current adaptation, and a role for current adaptation need not rule out a role for phylogenetic inertia.
The wing morphology of insects is a possible example of the simultaneous influence of current adaptation and of phylogenetic inertia. Most orders of insects differ from one another in their wing shapes and venation patterns. Yet, most families in a given order are characterized by wings with similar shapes and venation patterns (e.g., Arnett 2000). Consider two hypotheses for this pattern of ordinal differentiation and familial conservatism (Hansen and Houle 2004): (1) the differences among orders represent different adaptive optima for, say, flight performance; and (2) the wing morphologies are conserved among families within orders due to constraints on development, variation, or behavior. These two hypotheses are logically independent of each other. Both can be true, both can be false, and one can be true while the other is false. This logical independence does not necessarily entail statistical independence. For example, stabilizing selection can lead to canalization of a trait and thus to constraints on variation, which increase phylogenetic inertia. It follows that species that fit the adaptive hypothesis will likely also show a greater influence of phylogenetic inertia and vice versa. As a result, a fit between the predicted and observed trait values cannot be taken as evidence of adaptation unless one controls for phylogenetic inertia by, for example, assessing whether the adaptive prediction fits in taxa with high phylogenetic inertia and in those with low phylogenetic inertia. Similarly, a phylogenetic effect cannot be taken as evidence for phylogenetic inertia unless one controls for fit to the adaptive model. The idea of controlled comparisons as developed by Orzack and Sober (2001) is to assess the adaptive and inertial hypotheses jointly, in such a way that a test for each controls for the effect of the other.
Although phylogenetic inertia is essential as a concept to the endeavor of comparative analysis, it has been a stepchild in this effort. For example, Felsenstein's (1985) influential method of independent contrasts does not allow one to assess the extent to which phylogenetic inertia influences trait values. Instead, a particular pattern of inertia is assumed a priori and then used to control for phylogeny when assessing an adaptive hypothesis. If the assumed pattern of inertia is wrong, the method is statistically inefficient and thereby possibly misleading (Martins et al. 2002).
In this paper, we describe the extent to which the major comparative approaches to the study of adaptation of a continuously distributed trait allow one to make controlled comparisons. Specifically, we assess whether these methods properly control for adaptation when testing for phylogenetic inertia, and if they properly control for phylogenetic inertia when testing for adaptation.
Preliminaries
Method of Controlled Comparisons
To motivate our application of the controlled comparison method to continuously distributed traits, we review Orzack and Sober's (2001) analysis of a trait whose evolutionary dynamics can be described by a two-state continuous-time Markov process. We frame the adaptive hypothesis in terms of the prediction of an optimality model; this is a natural but not necessary choice.
Consider a trait that has two states, 0 and 1. Let u be the probability per unit time of change from state 0 to state 1, and let v be the probability of a change from state 1 to state 0. We assume that u and v remain constant. One can show that the conditional probabilities relating descendant (yd) and ancestral trait values (ya) are:
where t is the elapsed time between ancestor and descendant.
To see how the ancestral state affects the descendant state, consider the conditional expectation of the descendant given the ancestral state,
This expectation is a weighted average of the equilibrium trait value predicted by the adaptive hypothesis, u/(u + v), and the ancestral trait value, ya. The weighting depends on the term e−(u+v)t, which can be interpreted as the proportion of influence of the ancestral trait value. The larger the value of u + v, the more rapidly the influence of the ancestral trait is lost and the adaptive equilibrium achieved. One could use phylogenetic data to estimate u and v, for instance, by maximum likelihood (Pagel 1994). Given such estimates, one chooses one of the four controlled comparison outcomes shown in Table 1. Note that phylogenetic inertia is always assessed relative to a particular elapsed time.
For the two-state Markov process, the time it takes for half of the ancestral contribution to the trait to be lost is
This “half-life” (Hansen 1997) depends only on the overall rate of change and not on its direction. A half-life can be defined for any evolutionary model. No matter what model is chosen, the judgement of whether t1/2 is small or large is always made relative to the length of the relevant phylogeny. For example, for a given value of t1/2, one could judge the wing shape of Drosophila melanogaster to be strongly influenced by phylogenetic inertia if the phylogeny used in the analysis traces back only to the ancestral drosophilid, but judge it to be weakly influenced if the phylogeny traces back to the ancestral dipteran.
Controlling for Phylogenetic Inertia in the Study of Adaptation
When testing an adaptive hypothesis for traits x and y, one can control for phylogenetic inertia by comparing their predicted relationship with the relationship estimated by the method of generalized least squares (GLS; Grafen 1989; Martins and Hansen 1996a, 1997; Garland and Ives 2000; Rohlf 2001). Using GLS, one obtains an optimal estimator of the parameter vector β of a general linear model of the form
where y is the vector of response-trait values, X is the matrix of predictor-trait values, and V is the variance matrix of the error vector, e. The minimum-variance unbiased linear estimator of the parameter vector is To control for the influence of phylogenetic inertia on this estimator, we need to specify the phylogenetic variances and covariances in V; these depend on the evolutionary process and can take many forms (Hansen and Martins 1996). For example, for the two-state Markov process discussed above, the phylogenetic covariance for species i and j is where tij is the phylogenetic distance between the species and Var[ya] is the variance in the state of their most recent common ancestor (which may vary with time if the evolutionary process has not reached its stationary distribution).
The method of estimated GLS can be used to estimate any free model parameters in V (Martins and Hansen 1997; Butler et al. 2000). When these estimates depend on the other model parameters, they can be derived iteratively or all parameters can be optimized jointly by maximum likelihood.
Thus, one of the controlled comparisons, the assessment of current adaptation with phylogenetic inertia controlled for, can be accomplished by the use of either GLS or appropriate model-based likelihood methods, given a specific model for the evolutionary process.
Controlling for Adaptation in the Study of Phylogenetic Inertia
As noted above, the assessment of phylogenetic inertia requires a control for adaptation to the current environment. In the linear-model formalism (eq. 4), the predicted effects of current adaptation are captured as fixed effects in the Xβ term. If related species have similar focal adaptations they will have similar predictor-trait values in X; these will generate a phylogenetic effect in the response-trait values in y. After removing the effect of X, a phylogenetic effect among the residuals, e, is evidence for phylogenetic inertia. We note that many tests and estimation procedures for phylogenetic effects use raw data, rather than residuals (after current adaptation has been controlled for). Such estimates do not reveal the extent to which selective or variational constraints impede or preclude the evolution of a particular adaptation.
Controlled Comparisons as a Meta-Method
In this section we discuss the extent to which existing comparative methods can generate controlled comparisons. Most of the methods are based on the assumption that evolution proceeds as if by a Brownian-motion process. We therefore start by describing the influences of current adaptation and phylogenetic inertia when two traits, x and y, are governed by a joint Brownian-motion process.
Imagine that at some time in the past the relationship between y and x was described by a variance matrix {{Vy, Vyx}, {Vyx, Vx}}, such that the ancestral regression is
Now imagine that the selective regime subsequently changed to its present state, such that current changes are binormally distributed with mean zero and variance matrix: where σy2 and σx2 are the variances and σyx is the covariance per unit time in the evolutionary change of the two traits, and t is the time since the species diverged (Felsenstein 1985). The regression coefficient, βc = σyx/σx2, is used to measure current adaptation of trait y to trait x. The total regression observed over the tip species is then where modulates the influence of the ancestral trait regression. Note that A(t) is not a constant, but depends on time back to the ancestral state. Its evolutionary half-life is Vx/σx2; note also that the variance of change in the response variable, σy2, affects neither the half-life nor phylogenetic inertia as measured by A(t).
Independent Contrasts and Controlled Comparisons
Felsenstein's (1985) method of independent contrasts is the most commonly used approach to correct for phylogenetic effects in comparative analyses. It does not allow one to make controlled comparisons, as it only assumes the pattern of phylogenetic inertia that is inherent in the Brownian-motion process, and provides no means for estimating the level of phylogenetic inertia. In the formalism given above, the method assumes that A(t) = 1.0, so that there is no discounting of the past or no distinction between ancestral and current adaptive components of trait variation. The evolutionary relationship between the traits is assumed to be the same throughout the phylogeny; therefore, a correlation or regression between the traits can be interpreted as an unweighted average of evolutionary changes distributed over the entire phylogeny.
Lynch's Phylogenetic Heritability and Pagel's Lambda
Lynch's (1991) concept of phylogenetic heritability was developed by applying a mixed-model pedigree analysis from quantitative genetics to comparative data. The phylogeny is regarded as a pedigree, and the ancestral component of a trait is regarded as the breeding values of its ancestors. Thus, every species' trait value has an ancestral component and a unique component. Given the unique components for all of the species in the phylogeny, one can generate an estimate of the influence of current adaptation, βc. The ancestral component for each species is treated as a random effect with an across-species covariance structure that is determined by the phylogeny. Any phylogenetic covariance structure could be used in this model, but Lynch emphasized a structure derived from a Brownian-motion process. The estimate of phylogenetic heritability, H2, is the fraction of among-species variation that is attributable to the ancestral component. In the formalism given above, H2 corresponds to the influence of the ancestral trait regression, A(t), at an unspecified time in the past. In a pedigree, this time would be one generation. In a phylogeny, it would be the time back to the nearest common ancestor a species shares with another species in the phylogeny. This suggests interpreting H2 for a phylogeny as an average value of A(t), where the average is taken over these times for all of the species pairs. If more than one trait is involved, the components of trait covariance can also be divided into ancestral and nonancestral components.
Lynch's (1991) method can generate controlled comparisons for adaptation and phylogenetic inertia because its estimate of the influence of current adaptation, βc, and its estimate of the influence of ancestral trait values, H2, both properly control for the effects of each other. It can even be used to estimate and compare βa and βc, although Housworth et al. (2004) noted that it is likely that only very large datasets will generate reliable estimates of these parameters. Reducing the number of estimates needed reduces the sample size needed for reliable estimates. Accordingly, these authors suggested that one assume that βa = βc, so that only an estimate of the total regression, β, would be available to test for current adaptation. In any case, an estimate of βa will usually be based on data only from extant species and, therefore, could not directly capture the effects of an ancient selective regime. It is simply a measure of the component of trait regression that is shared among species. It should also be noted than an estimate of βc is likely to be extremely unreliable if H2 is high. For these reasons, the controlled-comparison assessment of adaptation is best carried out with β (see Table 2).
Based on a suggestion by Pagel (1999), Freckleton et al. (2002) developed a method for analyzing phylogenetic effects in which the among-species variance matrix, V, of a trait is partitioned as σ2I + σ2(T − I)λ, where σ2 is the variance per unit time of change in the trait, I is the identity matrix, T is a matrix whose ijth entry is the phylogenetic branch length shared by species i and j, and λ is a measure of phylogenetic influence. It is straightforward to show that this partitioning is the same as in Lynch's (1991) partitioning, and that λ is identical to H2. Therefore, if one uses Freckleton et al.'s approach, the controlled comparisons shown in Table 2 can be implemented if one substitutes an estimate of λ for H2.
The Problem of Inherited Maladaptation
Although a controlled-comparison test of adaptation can be achieved with a Brownian-motion-based model such as Lynch's (1991) method, this use engenders a serious interpretational problem. Consider a species in which trait y is maladapted to trait x. For example, y may be too small as compared to the optimal value in the current environment. The process of adaptation should result in an increase in the value of y, but no such systematic tendency for y to increase is consistent with a Brownian-motion process (or any martingale process), which only specifies that changes in y are correlated with changes in x—not with the value of x. In such a model the change in the trait over a given time interval, t, is normally distributed with mean zero and a variance that is proportional to t. Accordingly, the expected trait value of a species conditional on the trait value of an ancestor is:
This process is fundamentally different from the two-state Markov process discussed above, where the influence of the ancestor eventually disappears because this stochastic process is “mixing.” The Brownian-motion process is not mixing, and the influence of the ancestral trait is never lost. However, the ancestral component of among-species variation will decrease relative to the nonphylogenetic component because the latter increases linearly with the time the species have evolved independently of each other.This problem is most obvious in analyses with fixed predictor variables (categorical or otherwise). Brownian-motion-based methods are commonly used with such predictors, although these cannot be assumed to have followed a Brownian-motion process. The idea that seems to underlie this practice is that current adaptation and phylogenetic inertia are truly independent phenomena that can be added together in the model. In effect, the residuals are construed to evolve as a Brownian-motion process, while adaptation to a novel environment is construed to be instantaneous and consists of a phenotypic shift that is the same regardless of the starting trait value. Thus, if at time t a species has trait value yt in an environment with optimum θ1, but then inhabits a new environment with optimum θ2 (>θ1), yt will shift by a factor Δy (=θ2 − θ1), regardless of whether yt is optimal in the new environment. Again, this implies that whatever degree of maladaptation was present in the old environment will be present in the new environment (see Fig. 1). Even a species that is perfectly preadapted to the new environment would become maladapted. We believe this makes little biological sense.
We emphasize that the problem occurs even when a predictor trait is continuously distributed. When such a trait evolves as if by a Brownian-motion process, maladaptation will be inherited in the sense that there is no tendency to reduce any deviation from the optimal relationship between the predictor and response traits.
This problem of inherited maladaptation underscores the fact that if Brownian-motion-based methods are used to test adaptive hypotheses of the sort we consider here, these methods must be interpreted as descriptors of phylogenetic pattern and not as process models. If process models are desired, we need to look elsewhere.
Controlled Comparisons and the Ornstein-Uhlenbeck Process
Many, if not most, adaptive hypotheses are based on categorical predictor variables, as when adaptation to a few different environmental conditions is studied. The only comparative method that accounts for an adaptive hypothesis that specifies optima in discrete environments is the stabilizing-selection model of Hansen (1997). This method invokes the Ornstein-Uhlenbeck process as a model of long-term evolution in which a trait is subject to both selection toward an optimum and to constant stochastic noise (see also Hansen et al. 2000; Butler and King 2004; Martins et al. 2004). The strength of selection increases linearly with distance from the optimum. The conditional dependency of a descendant trait value yd on an ancestral trait value ya is
where θ is the optimum and α is a parameter that defines the strength of selection toward the optimum (Hansen 1997). The optimum plays the same role as the average character state at selective equilibrium, u/(u + v), does for the two-state Markov process (eq. 2), and α plays the same role as the overall rate of change, u + v, does for the two-state process. This similarity results from the fact that both of these processes are mixing and eventually will reach a stationary distribution where all traces of ancestry are removed. In both processes, there is an exponential rate of loss of the ancestral influence.The half-life for the Ornstein-Uhlenbeck process is
For given estimates of α and θ, the possible controlled-comparison outcomes for the Ornstein-Uhlenbeck-process model are shown in Table 3.The Ornstein-Uhlenbeck process has also been used to model the residuals from a statistical relationship without including the direct effect of α on the predicted trait optima (e.g., Garland et al. 1993; Butler et al. 2000; Martins et al. 2002). Although this is a reasonable procedure in the absence of historical information about the predictor variables, it amounts to assuming that all species have lived in their current environment since the root of the phylogeny and, thus, lacks an explicit microevolutionary justification.
Phylogenetic Autocorrelation and Other Methods that Split Phylogeny from Adaptation
Cheverud and Dow (1985) and Cheverud et al. (1985) proposed that an adaptive hypothesis could be tested by comparing observed and predicted species-specific components of a response trait y. The observed vector of species-specific components, yS, is derived from
where y is the vector of trait values, W is the phylogenetic connectivity matrix, and ρ is the scalar phylogenetic autocorrelation coefficient. An estimate of ρ is obtained by a regression of species data on a weighted average of other species. Any set of weights can be used in the connectivity matrix (Gittleman and Kot 1990). No set of weights based on an evolutionary model was proposed by Cheverud et al. (1985), but Martins and Hansen (1996b) and Rohlf (2001) subsequently derived weights based on a Brownian-motion process. Martins (1996), Diniz-Filho (2001) and Martins et al. (2002), studied the performance of the method in relation to different evolutionary processes in more detail.This method does not generate a controlled comparison for current adaptation because it explains as much trait evolution as possible through phylogenetic inertia before adaptation is invoked; such a role for phylogenetic inertia as a null hypothesis conflicts with the biology of trait evolution that we outlined above. If the estimate of ρ is used to control for phylogenetic inertia, the fit of the adaptive model will be based upon an incorrect estimate of inertia.
This example illustrates the general point that any method based on removing a phylogenetic effect from the data before testing for adaptation is incompatible with controlled comparisons. Another method that involves such a removal is Gittleman and Kot (1990) who extended Cheverud et al.'s (1985) model to allow exploration of the pattern of phylogenetic autocorrelation. However, the testing for adaptation is conducted as in the original model, and thus does not correctly control for phylogenetic inertia.
The phylogenetic-eigenvector regression (Diniz-Filho et al. 1998) is a method based on principle coordinate analysis for reducing a connectivity matrix containing linear phylogenetic distances to a few significant eigenvectors, which are taken to be a descriptor of phylogenetic effect. Diniz-Filho et al. did not discuss its use in studying adaptation, but if the method is used in manner similar to the autocorrelation methods, it will also inappropriately treat phylogenetic inertia as a null hypothesis.
Although these methods cannot be used presently to generate controlled comparisons, it is conceivable that they could be modified to do so. The first step would be to develop a way of regressing the species data jointly on the adaptive model and the phylogenetic effects.
Using Mean-Square Error as a Measure of Phylogenetic Effect
Blomberg et al. (2003) proposed measuring the strength of the phylogenetic effect in a comparative dataset by comparing the mean-square errors of phylogenetically uncorrected data and phylogenetically corrected data. Their model is:
where y is the vector of trait values, μ is the grand mean of the trait, e is an error vector, and T is a matrix of phylogenetic relationships for a given evolutionary model. In the case of a Brownian-motion process, its entries would be shared phylogenetic branch lengths. The mean-square error of a phylogenetically uncorrected model is MSE0 = e′e/(n − 1), and that of a model corrected by GLS is MSE = e′ T−1e/(n − 1). Blomberg et al.'s (2003) K-statistic is where c is a scaling factor computed as the expected value of MSE0/MSE. This scaling is necessary, as the value of the statistics depend on the shape of the tree. A value of K below 1.0 indicates that the uncorrected model fits better and a value above 1.0 indicates that the corrected model fits better.
Blomberg et al. (2003) only considered the error around an estimate of the grand mean, μ, but presumably, residuals from a more complicated regression or ANOVA model that accounts for adaptation could be obtained. However, there is no straightforward way of using an estimated K to control for phylogenetic inertia when fitting an adaptive model. The problem is that K does not correspond to an explicit parameter in the model and, therefore, an estimated K cannot be fit back into the analysis. One possibility is to use K or, more generally, R2 statistics as a criterion to fit parameters like the phylogenetic heritability or α in an Ornstein-Uhlenbeck model (Martins and Hansen 1997; Butler et al. 2000), but then we are back into the realm of parameterized models.
It should also be noted that the statistical properties of the use of R2 to fit parameters in the variance structure are poorly understood. The estimates are generally not maximum likelihood estimates, and although it may work in specific cases, such an approach should be used with caution. Maximum-likelihood estimators seem preferable.
Nested Analysis of Variance
The use of analysis of variance to decompose trait variation into phylogenetic and nonphylogenetic components was proposed by Clutton-Brock and Harvey (1977) and Stearns (1983) and later extended by Armbruster (1988) and Bell (1989), who presented methods to decompose trait covariances and trait relationships. In principle, an adaptive hypothesis can be tested at any taxonomic level at which such a decomposition is carried out. Therefore, this approach can be used to conduct controlled comparisons, as it is possible to study the interaction between adaptive predictions and taxonomic effects and to assign variation to different levels conditional on the fit of the adaptive hypothesis within levels (e.g., Jordano 1995). These methods are severely limited in that they cannot use phylogenetic information beyond a simple taxonomic hierarchy.
Complications and Extensions
For clarity we have framed our arguments in terms of a single predictor variable (trait x) and a single response variable (trait y). However, the prediction of the adaptive model for a trait can be based on any number of categorical or continuous predictor variables that can be combined in a general linear model. Standard model-selection criteria such as AIC (Burnham and Anderson 1998) can be used to choose among alternative adaptive scenarios (Butler and King 2004). Information about the time the species have evolved in particular environments can be incorporated into the models; this is crucial to the outcome of a test of whether the trait is significantly affected by phylogenetic inertia. Despite the importance of incorporating time, it has not been included in most comparative methods. One reason is that it is rare to have precise temporal information about the association of species and environments. However, different qualitative hypotheses about the history of association can be tested (Butler and King 2004).
In principle, controlled comparisons can also be implemented with multiple continuously distributed response traits; such an implementation could be based on the multivariate framework developed by Lynch (1991). None of the above methods can be used with categorical response traits. Martins and Hansen (1997) suggested using a generalized linear model (GLM) in such a context and showed how phylogenetically corrected parameter estimates could be obtained in a GLM framework, but a complete method for statistical testing has yet to be developed.
Although we have discussed the method of controlled comparisons as a way of testing binary hypotheses about adaptation and phylogenetic inertia, it can accommodate several hypotheses of either kind in a particular analysis. We illustrate the logic of testing several adaptive hypotheses and several inertial hypotheses in Table 4.
An Example of Controlled Comparisons
The method of controlled comparisons is motivated by the observation that adaptation and phylogenetic inertia can both influence a trait at a given time. As such, this method builds upon well-accepted understanding of trait evolution (see above). Unfortunately, this understanding appears not to have influenced the practice of biologists in that almost all published comparative analyses do not contain controlled comparisons. Accordingly, we generate controlled comparisons here for a continuously distributed trait so as to illustrate their implementation.
Our example concerns the sex ratios produced by a variety of fig-wasp species (for an overview of their ecology and evolution, see Compton 1996). For each of 15 species, Herre et al. (2001) measured the sex ratios produced by one, two, or three females (foundresses) and the distribution of foundress numbers. They also used molecular data to infer a phylogeny for the wasps (for further details, see Herre et al. 2001, p. 207).
Given these data, Herre et al. wished to assess the adaptive prediction that the magnitude of the deviation between the average observed sex ratio and the optimal sex ratio would be negatively related to the frequency of the foundress number (see Herre 1985, 1987). The rationale for this prediction is that the cost of adaptation is paid if the benefit of an optimal sex ratio is accrued often enough.
To test their prediction, Herre et al. (2001) measured the “uncorrected” regression among species values (phylogeny was ignored) and the “corrected” regression (when phylogeny was accounted for by the method of independent contrasts). They found a significant negative relationship for the single-foundress data but not for the two- and three-foundress data; in addition, the regression coefficients in the uncorrected and corrected analyses were not significantly different for each of the three foundress numbers (pp. 208–209). As a result, the authors concluded (p. 208) that “phylogenetic relationships give essentially no additional information concerning the evolution of [these sex ratios].” If this claim is meant to imply that there is evidence for adaptation after phylogeny is accounted for, then it reasonably applies to the single-foundress sex ratio. However, if this claim is meant to imply that the result shows that phylogenetic inertia does not affect this trait, it is incorrect. In fact, this test result reveals nothing about the effect of phylogenetic inertia because the expectation of the regression coefficient relating species' trait values is the same whether or not one accounts for phylogeny (Pagel 1993; Rohlf 2001). To this extent, statistical identity of the regression coefficients in the uncorrected and corrected analyses does not indicate that there is no influence of phylogenetic relationships on this trait. Instead, it simply means that the adaptive “signal” in the data is not changed by accounting for phylogeny.
We generated controlled comparisons for the single-foundress sex ratio data to jointly assess the influences of current adaptation and of phylogenetic inertia. We used Lynch's (1991) method as implemented by Housworth et al. (2004), despite its implication that foundress number evolves according to a neutral (Brownian-motion) process. The test for adaptation involved assessing whether the predicted adaptive trend matched the observed trend in the data (as estimated by β) and the test for phylogenetic inertia involved estimating the phylogenetic heritability (as estimated by H2). Point estimates of β and H2 and permutation-based confidence intervals for these estimates were derived using software provided by E. A. Housworth (see also http://compare.bio.indiana.edu/v46/comparehigh.html). The number of permutations for each estimate was 1000.
Unfortunately, the transformation used by Herre et al. (2001) to convert the molecular distances in their phylogeny to divergence times (needed for their independent-contrast analysis) is not available (S. West, pers. comm.). Accordingly, we performed our analysis using the molecular distances to avoid using an arbitrary transformation. To this extent, the exact relationship between our analysis and their analysis is unclear; however, we regard them as comparable. Lynch's (1991) method can be applied with molecular distances (E. A. Housworth, pers. comm.). Their use amounts to assuming that the response trait evolves at different rates in different parts of the phylogeny (resulting in unequal trait variances). Sex ratios were arc-sine transformed as described by Herre et al. (2001, p. 209), so that our analysis would conform to theirs as closely as possible.
Lynch's (1991) method generated a point estimate for β of −0.227. The associated permutation-based 95% confidence interval did not include zero. This negative estimate supports the adaptive prediction of Herre et al. (2001) that trait y (the deviation between average and optimal sex ratios) would be negatively related to trait x (the frequency of single-foundresses). The maximum-likelihood estimate of H2 was 0.938. This estimate suggests that phylogenetic relationships influence the evolution of the single-foundress sex ratio, although the permutation-based 95% confidence interval encompassed the entire range of allowable estimates (0.0–1.0) and 325 of 1000 estimates were less than 0.0 or greater than 1.0. This is due to the small number of taxa (15) in the analysis (E. A. Housworth, pers. comm.).
The best supported conclusion from these controlled comparisons is that both adaptation and phylogenetic inertia have influenced the evolution of this trait (cf., the upper-left-hand entry in Table 2). Such a conclusion is clearly best viewed as heuristic because of the small number of taxa involved and because of the use of Lynch's (1991) method, which has the assumption that the traits evolve according to a joint Brownian-motion process. The resulting problem of inherited maladaptation obscures a process-based interpretation of the results. We note that an analysis with more species could support any of the four outcomes shown in the table. No matter how one interprets the results of our analysis, our main point is simply to illustrate how the hypotheses of adaptation and phylogenetic inertia can be jointly assessed via the method of controlled comparisons.
Discussion
We have shown how several well-known comparative methods can generate the controlled comparisons that are necessary to correctly assess the roles of adaptation and phylogenetic inertia in trait evolution. The stabilizing-selection model of Hansen (1997) is the approach most similar to Orzack and Sober's (2001) original formulation. Both involve a mixing stochastic process in which the influence of the ancestral trait declines at an exponential rate. In the stabilizing-selection model, the parameter α controls this rate and is thus a direct measure of phylogenetic inertia. An estimate of α and an estimate of an adaptive optimum allows one to generate controlled comparisons. This approach is the simplest way of doing a controlled comparison with continuous response variables and fixed predictor variables.
If both the response and predictor variables are continuously distributed traits, it is more difficult to justify treating the predictor variables as fixed effects. In this instance, Lynch's (1991) phylogenetic mixed model seems to be the most straightforward way of implementing controlled comparisons. Here, both response and predictor variables are assumed to follow a joint multivariate Brownian-motion process, and a joint estimate of the regression parameter and of the phylogenetic heritability of the response trait allow one to make controlled comparisons. Alternatively, λ as defined by Pagel (1991) can be used in place of the phylogenetic heritability (see Freckleton et al. 2002, 2003).
Unfortunately, Lynch's and Pagel's methods, as well as Felsenstein's method of independent contrasts, suffer from a serious interpretational problem when used to test for adaptation because they are based on a Brownian-motion process. This process is inconsistent with any tendency for an evolutionary variable to adapt to the state of another variable, and it maintains whatever ancestral maladaptation was present, even if this implies that there is evolution away from the adaptive optimum. The methods must therefore be regarded as pattern-based approaches without an adaptation-based microevolutionary justification. As such they have the same status as the various phylogenetic autocorrelation methods, for which no process-based justifications have been proposed. A microevolutionary justification is also lacking for methods that treat trait residuals as evolving independently of the adaptive optima. This leaves us without any fully satisfactory comparative method for studying adaptation to a random continuous variable, and the development of such a method, for both continuous and categorical response variables, should be the focus of further research.
To our knowledge, almost all published comparative tests of adaptive models do not provide controlled comparisons, mostly because they do not control for adaptation when assessing phylogenetic inertia. For example, the method of independent contrasts assumes a pattern of phylogenetic inertia a priori and uses this to control for phylogeny when testing an adaptive hypothesis. This method does not allow one to control for adaptation when assessing an inertial hypothesis. Even methods based on phylogenetic autocorrelation, which attempt to assess the level of phylogenetic inertia, fail to generate controlled comparisons because they treat phylogenetic inertia as a null hypothesis.
The use of a priori estimates of phylogenetic effects not only fails to remove obvious differences in selective regime from the estimates, but it also means that they cannot be used to correct for phylogenetic inertia when later evaluating the adaptive model. The statistical error in using an uncontrolled estimate of phylogenetic effect for this purpose can be likened to the common error of testing raw data for departures from normality or homoscedasticity and then inappropriately transforming them to correct for a “problem.” In a general linear model it is the residuals from the analysis and not the raw data that are assumed to be normally distributed and homoscedastic. Similarly, a phylogenetic effect in the raw data does not mean that a phylogenetic correction is appropriate. In addition, the absence of such an effect does not mean that a phylogenetic correction is unnecessary. There is a need for correction if and only if there is a phylogenetic effect after the parameters of the adaptive model have been fitted jointly with the parameters of the model of phylogenetic inertia.
This problem usually goes unnoticed in the literature. For example, in the recent review of phylogenetic effects by Freckleton et al. (2002), Pagel's (1999) λ was estimated for a number of comparative studies. The authors, however, do not mention whether λ is estimated on the raw data, on the residuals from, or simultaneously with whatever hypothesis was tested in the original study. Hence, it is not clear whether the original studies made appropriate corrections for phylogeny. Similarly, the review by Blomberg et al. (2003) concerned phylogenetic effects in raw data and did not tell us whether or what correction would be appropriate in the individual studies (as made clear by the authors). It is crucially important that estimates of phylogenetic inertia are not derived separately from the adaptive hypothesis being tested. Different values may easily be obtained for different parameters of the adaptive model, and if an a priori estimate of phylogenetic inertia is used, the control for phylogeny may be completely wrong.
Regardless of what method is used, a controlled comparison requires that the inertial parameter be estimated simultaneously with the parameters describing the adaptive hypothesis. This can be done by a joint likelihood approach. Alternatively, one can use an iterative scheme in which the adaptive hypothesis is fitted by GLS under the assumption that phylogenetic inertia is absent. Phylogenetic inertia is estimated from the residuals of this analysis and used to obtain a better estimate for the parameters of the adaptive model. This is then iterated until convergence is achieved (see Hansen 1997; Butler and King 2004).
In addition to the models we have reviewed, there are also several approaches that seek to assess the influence of phylogenetic inertia by only testing for the presence of a phylogenetic effect and do not directly estimate its magnitude (for review see Blomberg and Garland 2002). Obviously, these methods cannot be used to control for phylogenetic inertia, especially because statistical and biological significance need not coincide (see Martins 1996).
We still regard the study of phylogenetic effects in raw data to be worthwhile. Such a study is relatively uninformative about the need for phylogenetic correction when assessing adaptation, but it may be very useful for comparing the rates of evolution of different organisms or traits (e.g., Gittleman et al. 1996; Blomberg et al. 2003), even if the evolutionary mechanisms producing the phylogenetic effects are unclear.
Acknowledgments
We thank E. A. Herre, C. A. Machado, and S. A. West for supplying data; M. Björklund, E. Martins, and E. A. Housworth for critical advice and comments; K. Norberg for computational assistance; and E. Sober for comments and important early contributions to this project. This work was partially supported by National Science Foundation awards DEB 0344417, EIA 0220154, and SES 9906997 and National Institute of Aging award P01-AG0225000–01.
Literature Cited
Fig. 1.
A sample path of a stochastic trait model, yt = θi + et, where yt is the trait value at time t, θ1 and θ2 represent optima for two different environments, and et is the residual at time t, which is determined by a Brownian-motion process. The starting point of the sample path is θ1. A shift between environments occurs at time 50. Although y50 is close to the new optimum, θ2, y51 overshoots θ2 because the dynamics of the Brownian-motion process are such that the inherited maladaptation (= difference between yt and θi) must be transmitted from one time to the next
Table 1.
Controlled comparisons for the two-state model given observed estimates of u and v (see text for further details)
Table 2.
Controlled comparisons for Lynch's (1991) model given observed estimates of β and H2 (see text for further details)
Table 3.
Controlled comparisons for the Ornstein-Uhlenbeck model given observed values of θ and α (see text for further details)
Table 4.
Controlled comparisons for the Ornstein-Uhlenbeck model given estimates of θ and α and several hypotheses about adaptation and about phylogenetic inertia. Hypotheses H1, H2, and H3 predict different values, θ1, θ2, and θ3, for the optimum. Hy potheses I1, I2, and I3 predict different levels, α1, α2, and α3, of phylogenetic inertia. In general, these inertial hypotheses do not have to be restricted to different values of the α-parameter in the Ornstein-Uhlenbeck model, but may represent entirely different models of phylogenetic effect (including the Brownian-motion pro cess)
