PHYLOGENETIC ANALYSIS OF THE EVOLUTIONARY CORRELATION USING LIKELIHOOD

Many evolutionary processes can lead to a change in the correlation between continuous characters over time or on different branches of a phylogenetic tree. Shifts in genetic or functional constraint, in the selective regime, or in some combination thereof can influence both the evolution of continuous traits and their relation to each other. These changes can often be mapped on a phylogenetic tree to examine their influence on multivariate phenotypic diversification. We propose a new likelihood method to fit multiple evolutionary rate matrices (also called evolutionary variance–covariance matrices) to species data for two or more continuous characters and a phylogeny. The evolutionary rate matrix is a matrix containing the evolutionary rates for individual characters on its diagonal, and the covariances between characters (of which the evolutionary correlations are a function) elsewhere. To illustrate our approach, we apply the method to an empirical dataset consisting of two features of feeding morphology sampled from 28 centrarchid fish species, as well as to data generated via phylogenetic numerical simulations. We find that the method has appropriate type I error, power, and parameter estimation. The approach presented herein is the first to allow for the explicit testing of how and when the evolutionary covariances between characters have changed in the history of a group.     

Neopolyploidy and diversification in Heuchera grossulariifolia

Newly formed polyploid lineages must contend with several obstacles to avoid extinction, including minority cytotype exclusion, competition, and inbreeding depression. If polyploidization results in immediate divergence of phenotypic characters these hurdles may be reduced and establishment made more likely. In addition, if polyploidization alters the phenotypic and genotypic associations between traits, that is, the P and G matrices, polyploids may be able to explore novel evolutionary paths, facilitating their divergence and successful establishment. Here, we report results from a study of the perennial plant Heuchera grossulariifolia in which the phenotypic divergence and changes in phenotypic and genotypic covariance matrices caused by neopolyploidization have been estimated. Our results reveal that polyploidization causes immediate divergence for traits relevant to establishment and results in significant changes in the structure of the phenotypic covariance matrix. In contrast, our results do not provide evidence that polyploidization results in immediate and substantial shifts in the genetic covariance matrix.     

The mouse skull as a source of morphometric data for phylogeny inference

Brownian motion has been a model widely used for describing phenotypic evolution of continuous characters under random drift. Evolution of traits evolving under weak stabilizing selection, together with drift, can also be modeled by the Ornstein-Uhlenbeck process, in which a population moves at random on an adaptive peak under the influence of drift with selection returning the population towards the optimum. Obviously, reliability of an evolutionary model stands or falls with the extent to which the underlying assumptions are supported or violated. Another potential problem of continuous characters as a source of data for phylogeny inference is the correlation between them. To assess whether the Brownian motion model or the Ornstein-Uhlenbeck model are suitable for modeling the evolution of continuous cranial and dental characters and to what extent these characters are correlated with one another, 11 measurements encompassing various aspects of the mouse skull morphology were collected and subjected to a comparative analysis using the generalized least squares method. It could be shown that only about one-half of the characters evolved according to the Brownian motion model or the Ornstein-Uhlenbeck model. Moreover, about 44% of the correlation coefficients exceeded 0.8, suggesting a need for removing at least phenotypic covariances from the data prior to a phylogenetic analysis. Finally, ancestral states of the characters under study were estimated with the generalized least square method. There has been a general trend towards enlarging the overall size of the skull and increasing the braincase volume in the species of the genus Mus.     

Evolution of sexual dimorphism in phenotypic covariance structure in Phymata

Sexual dimorphism is a consequence of both sex‐specific selection and potential constraints imposed by a shared genetic architecture underlying sexually homologous traits. However, genetic architecture is expected to evolve to mitigate these constraints, allowing the sexes to approach their respective optimal mean phenotype. In addition, sex‐specific selection is expected to generate sexual dimorphism of trait covariance structure (e.g., the phenotypic covariance matrix, P ), but previous empirical work has not fully addressed this prediction. We compared patterns of phenotypic divergence, for three traits in seven taxa in the insect genus Phymata (Reduviidae), to ask whether sexual dimorphism in P is common and whether its magnitude relates to the extent of sexual dimorphism in trait means. We found that sexual dimorphism in both mean and covariance structure was pervasive but also that the multivariate distance between sex‐specific means was correlated with sex differences in the leading eigenvector of P , while accounting for uncertainty in phylogenetic relationships. Collectively, our findings suggest that sexual dimorphism in covariance structure may be a common but underappreciated feature of dioecious populations.     

A comparison of phenotypic variation and covariation patterns and the role of phylogeny, ecology, and ontogeny during cranial evolution of new world monkeys

Similarity of genetic and phenotypic variation patterns among populations is important for making quantitative inferences about past evolutionary forces acting to differentiate populations and for evaluating the evolution of relationships among traits in response to new functional and developmental relationships. Here, phenotypic co variance and correlation structure is compared among Platyrrhine Neotropical primates. Comparisons range from among species within a genus to the superfamily level. Matrix correlation followed by Mantel's test and vector correlation among responses to random natural selection vectors (random skewers) were used to compare correlation and variance/covariance matrices of 39 skull traits. Sampling errors involved in matrix estimates were taken into account in comparisons using matrix repeatability to set upper limits for each pairwise comparison. Results indicate that covariance structure is not strictly constant but that the amount of variance pattern divergence observed among taxa is generally low and not associated with taxonomic distance. Specific instances of divergence are identified. There is no correlation between the amount of divergence in covariance patterns among the 16 genera and their phylogenetic distance derived from a conjoint analysis of four already published nuclear gene datasets. In contrast, there is a significant correlation between phylogenetic distance and morphological distance (Mahalanobis distance among genus centroids). This result indicates that while the phenotypic means were evolving during the last 30 millions years of New World monkey evolution, phenotypic covariance structures of Neotropical primate skulls have remained relatively consistent. Neotropical primates can be divided into four major groups based on their feeding habits (fruit-leaves, seed-fruits, insect-fruits, and gum-insect-fruits). Differences in phenotypic covariance structure are correlated with differences in feeding habits, indicating that to some extent changes in interrelationships among skull traits are associated with changes in feeding habits. Finally, common patterns and levels of morphological integration are found among Platyrrhine primates, suggesting that functional/developmental integration could be one major factor keeping covariance structure relatively stable during evolutionary diversification of South American monkeys.     

Covariance structure in the skull of Catarrhini: a case of pattern stasis and magnitude evolution

The study of the genetic variance/covariance matrix (G-matrix) is a recent and fruitful approach in evolutionary biology, providing a window of investigating for the evolution of complex characters. Although G-matrix studies were originally conducted for microevolutionary timescales, they could be extrapolated to macroevolution as long as the G-matrix remains relatively constant, or proportional, along the period of interest. A promising approach to investigating the constancy of G-matrices is to compare their phenotypic counterparts (P-matrices) in a large group of related species; if significant similarity is found among several taxa, it is very likely that the underlying G-matrices are also equivalent. Here we study the similarity of covariance and correlation structure in a broad sample of Old World monkeys and apes (Catarrhini). We made phylogenetically structured comparisons of correlation and covariance matrices derived from 39 skull traits, ranging from between species to the superfamily level. We also compared the overall magnitude of integration between skull traits (r²) for all Catarrhini genera. Our results show that P-matrices were not strictly constant among catarrhines, but the amount of divergence observed among taxa was generally low. There was significant and positive correlation between the amount of divergence in correlation and covariance patterns among the 30 genera and their phylogenetic distances derived from a recently proposed phylogenetic hypothesis. Our data demonstrate that the P-matrices remained relatively similar along the evolutionary history of catarrhines, and comparisons with the G-matrix available for a New World monkey genus (Saguinus) suggests that the same holds for all anthropoids. The magnitude of integration, in contrast, varied considerably among genera, indicating that evolution of the magnitude, rather than the pattern of inter-trait correlations, might have played an important role in the diversification of the catarrhine skull.     

Comparative methods with sampling error and within-species variation: contrasts revisited and revised

Comparative methods analyses have usually assumed that the species phenotypes are the true means for those species. In most analyses, the actual values used are means of samples of modest size. The covariances of contrasts then involve both the covariance of evolutionary changes and a fraction of the within-species phenotypic covariance, the fraction depending on the sample size for that species. Ives et al. have shown how to analyze data in this case when the within-species phenotypic covariances are known. The present model allows them to be unknown and to be estimated from the data. A multivariate normal statistical model is used for multiple characters in samples of finite size from species related by a known phylogeny, under the usual Brownian motion model of change and with equal within-species phenotypic covariances. Contrasts in each character can be obtained both between individuals within a species and between species. Each contrast can be taken for all of the characters. These sets of contrasts, each the same contrast taken for different characters, are independent. The within-set covariances are unequal and depend on the unknown true covariance matrices. An expectation-maximization algorithm is derived for making a reduced maximum likelihood estimate of the covariances of evolutionary change and the within-species phenotypic covariances. It is available in the Contrast program of the PHYLIP package. Computer simulations show that the covariances are biased when the finiteness of sample size is not taken into account and that using the present model corrects the bias. Sampling variation reduces the power of inference of covariation in evolution of different characters. An extension of this method to incorporate estimates of additive genetic covariances from a simple genetic experiment is also discussed.     

A METHOD FOR ASSESSING PHYLOGENETIC LEAST SQUARES MODELS FOR SHAPE AND OTHER HIGH‐DIMENSIONAL MULTIVARIATE DATA

Studies of evolutionary correlations commonly use phylogenetic regression (i.e., independent contrasts and phylogenetic generalized least squares) to assess trait covariation in a phylogenetic context. However, while this approach is appropriate for evaluating trends in one or a few traits, it is incapable of assessing patterns in highly multivariate data, as the large number of variables relative to sample size prohibits parametric test statistics from being computed. This poses serious limitations for comparative biologists, who must either simplify how they quantify phenotypic traits, or alter the biological hypotheses they wish to examine. In this article, I propose a new statistical procedure for performing ANOVA and regression models in a phylogenetic context that can accommodate high‐dimensional datasets. The approach is derived from the statistical equivalency between parametric methods using covariance matrices and methods based on distance matrices. Using simulations under Brownian motion, I show that the method displays appropriate Type I error rates and statistical power, whereas standard parametric procedures have decreasing power as data dimensionality increases. As such, the new procedure provides a useful means of assessing trait covariation across a set of taxa related by a phylogeny, enabling macroevolutionary biologists to test hypotheses of adaptation, and phenotypic change in high‐dimensional datasets.     

Phenotypic variation and covariation indicate high evolvability of acoustic communication in crickets

Studying the genetic architecture of sexual traits provides insight into the rate and direction at which traits can respond to selection. Traits associated with few loci and limited genetic and phenotypic constraints tend to evolve at high rates typically observed for secondary sexual characters. Here, we examined the genetic architecture of song traits and female song preferences in the field crickets Gryllus rubens and Gryllus texensis. Song and preference data were collected from both species and interspecific F1 and F2 hybrids. We first analysed phenotypic variation to examine interspecific differentiation and trait distributions in parental and hybrid generations. Then, the relative contribution of additive and additive‐dominance variation was estimated. Finally, phenotypic variance–covariance ( P ) matrices were estimated to evaluate the multivariate phenotype available for selection. Song traits and preferences had unimodal trait distributions, and hybrid offspring were intermediate with respect to the parents. We uncovered additive and dominance variation in song traits and preferences. For two song traits, we found evidence for X‐linked inheritance. On the one hand, the observed genetic architecture does not suggest rapid divergence, although sex linkage may have allowed for somewhat higher evolutionary rates. On the other hand, P matrices revealed that multivariate variation in song traits aligned with major dimensions in song preferences, suggesting a strong selection response. We also found strong covariance between the main traits that are sexually selected and traits that are not directly selected by females, providing an explanation for the striking multivariate divergence in male calling songs despite limited divergence in female preferences.     

Exploring patterns of interspecific variation in quantitative traits using sequential phylogenetic eigenvector regressions

A number of metrics have been developed for estimating phylogenetic signal in data and to evaluate correlated evolution, inferring broad-scale evolutionary and ecological processes. Here, we proposed an approach called phylogenetic signal-representation (PSR) curve, built upon phylogenetic eigenvector regression (PVR). In PVR, selected eigenvectors extracted from a phylogenetic distance matrix are used to model interspecific variation. In the PSR curve, sequential PVR models are fitted after successively increasing the number of eigenvectors and plotting their R(2) against the accumulated eigenvalues. We used simulations to show that a linear PSR curve is expected under Brownian motion and that its shape changes under alternative evolutionary models. The PSR area, expressing deviations from Brownian motion, is strongly correlated (r= 0.873; P < 0.01) with Blomberg's K-statistics, so nonlinear PSR curves reveal if traits are evolving at a slower or higher rate than expected by Brownian motion. The PSR area is also correlated with phylogenetic half-life under an Ornstein-Uhlenbeck process, suggesting how both methods describe the shape of the relationship between interspecific variation and time since divergence among species. The PSR curve provides an elegant exploratory method to understand deviations from Brownian motion, in terms of acceleration or deceleration of evolutionary rates occurring at large or small phylogenetic distances.     

Phylogenetic analysis of correlation structure in stalk-eyed flies (Diasemopsis,Diopsidae)

Morphological divergence among species may be constrained by the pattern of genetic variances and covariances among traits within species. Assessing the existence of such a relationship in nature requires information on the stability of intraspecific correlation and covariance structure and the correspondence of this structure to the pattern of evolutionary divergence within a lineage. Here, we investigate these issues for nine morphological traits and 15 species of stalk-eyed flies in the genus Diasemopsis. Within-species matrices for these traits were generated from phenotypic data for all the Diasemopsis species and from genetic data for a single Diasemopsis species, D. dubia. The among-species pattern of divergence was assessed by calculating the evolutionary correlations for all pairwise combinations of the morphological traits along the phylogeny of these species. Comparisons of intraspecific matrices reveal significant similarity among all species in the phenotypic correlations matrices but not the covariance matrices. In addition, the differences in correlation structure that do exist among species are not related to their phylogenetic placement or change in the means of the traits. Comparisons of the phenotypic and phylogenetic matrices suggest a strong relationship between the pattern of evolutionary change among species and both the intraspecific correlation structure and the stability of this structure among species. The phenotypic and the phylogenetic matrices are significantly similar, and pairs of traits whose intraspecific correlations are more stable across taxa exhibit stronger coevolution on the phylogeny. These results suggest either the existence of strong constraints on the pattern of evolutionary change or a consistent pattern of correlated selection shaping both the phenotypic and phylogenetic matrices. The genetic correlation structure for D. dubia, however, does not correspond with patterns found in the phenotypic and phylogenetic data. Possible reasons for this disagreement are discussed.     

Joint reconstruction of divergence times and life-history evolution in placental mammals using a phylogenetic covariance model

Violation of the molecular clock has been amply documented, and is now routinely taken into account by molecular dating methods. Comparative analyses have revealed a systematic component in rate variation, relating it to the evolution of life-history traits, such as body size or generation time. Life-history evolution can be reconstructed using Brownian models. However, the resulting estimates are typically uncertain, and potentially sensitive to the underlying assumptions. As a way of obtaining more accurate ancestral trait and divergence time reconstructions, correlations between life-history traits and substitution rates could be used as an additional source of information. In this direction, a Bayesian framework for jointly reconstructing rates, traits, and dates was previously introduced. Here, we apply this model to a 17 protein-coding gene alignment for 73 placental taxa. Our analysis indicates that the coupling between molecules and life history can lead to a reevaluation of ancestral life-history profiles, in particular for groups displaying convergent evolution in body size. However, reconstructions are sensitive to fossil calibrations and to the Brownian assumption. Altogether, our analysis suggests that further integrating inference of rates and traits might be particularly useful for neontological macroevolutionary comparative studies.     

Multivariate Phenotypic Evolution: Divergent Acoustic Signals and Sexual Selection in <Emphasis Type="Italic">Gryllus</Emphasis> Field Crickets

Predicting the response to selection is at the core of evolutionary biology. Presently, thorough understanding of the effects of selection on the multivariate phenotype is lacking, in particular for behavioral traits. Here, we compared multivariate acoustic mating signals among seven field cricket species contrasting two selection regimes: (1) species producing songs with long trains of pulses for which preference functions for acoustic energy (chirp duty cycle) are linear and likely exert strong directional selection (‘trillers’); (2) species producing songs consisting of short chirps and for which preference functions for chirp duty cycle are concave and directional selection is likely weak or absent (‘chirpers’). We compared the phenotypic variance–covariance matrix (P) among species and uncovered two main patterns: First, surprisingly, pulse rate and chirp rate were positively correlated in six of seven species thus suggesting phenotypic coupling of timescales. Second, chirp rate and chirp duty cycle also covaried, but the direction of covariation differed between chirpers (positive) and trillers (negative). Multi-population Bayesian methods for matrix comparisons, Krzanowski’s subspace comparison and tensor analysis, revealed significant variation in P unrelated to phylogenetic distance, but strongly contrasting chirpers and trillers. We also found differences in the predicted selection response between chirpers and trillers. We thus report that variation in P is higher between than within selection regimes. Although effects from drift and shared ancestry cannot be fully excluded, these findings highlight a role for sexual selection in shaping patterns of phenotypic covariation that can ultimately affect the evolutionary trajectory of a multivariate mating signal.     

Can phylogenetic signal,character displacement,or random phenotypic drift explain the morphological variation in the genus Geonoma (Arecaceae)?

Plant clades may exhibit little or wide morphological variation as a result of (1) the retention of ancestral characteristics or phylogenetic signal, (2) character displacement, or (3) random phenotypic drift or convergence. Understanding the taxonomy and systematics of many plant lineages has been challenging due to continuous intra‐ and interspecific morphological variation. To assess which evolutionary hypothesis could explain the morphological diversity in the genus Geonoma (Arecaceae), we performed a Mantel test between phylogenetic and morphological distances of 54 taxa, and tested for phylogenetic signal using Blomberg's K‐statistic on continuous variables, and a randomization of character states. To obtain a phylogenetic (patristic) distance matrix for Geonoma, we constructed a molecular phylogeny of tribe Geonomateae using three nuclear DNA regions. A positive relationship between the patristic and a 26‐discrete‐character distance matrix (R² = 0.55, P < 0.001) supported the phylogenetic signal hypothesis. The randomization test showed that signal was present in 16 characters. No relationship was evident using a 17‐quantitative‐variable distance matrix (R² = 0.07, P = 0.13), supporting the random drift hypothesis or convergence, and all 17 K‐values were close to 0, suggesting less phylogenetic signal than under the Brownian model. If most morphological variables traditionally used to classify Geonoma evolved randomly or convergently, it might explain Geonoma's challenging taxonomy. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, 106 , 528–539.     

Resolving the paradox of stasis: models with stabilizing selection explain evolutionary divergence on all timescales

We tested the ability of six quantitative genetic models to explain the evolution of phenotypic means using an extensive database compiled by Gingerich. Our approach differs from past efforts in that we use explicit models of evolutionary process, with parameters estimated from contemporary populations, to analyze a large sample of divergence data on many different timescales. We show that one quantitative genetic model yields a good fit to data on phenotypic divergence across timescales ranging from a few generations to 10 million generations. The key feature of this model is a fitness optimum that moves within fixed limits. Conversely, a model of neutral evolution, models with a stationary optimum that undergoes Brownian or white noise motion, a model with a moving optimum, and a peak shift model all fail to account for the data on most or all timescales. We discuss our results within the framework of Simpson's concept of adaptive landscapes and zones. Our analysis suggests that the underlying process causing phenotypic stasis is adaptation to an optimum that moves within an adaptive zone with stable boundaries. We discuss the implication of our results for comparative studies and phylogeny inference based on phenotypic characters.     

Contemporary and historical evolutionary processes interact to shape patterns of within-lake phenotypic divergences in polyphenic pumpkinseed sunfish, Lepomis gibbosus

Historical and contemporary evolutionary processes can both contribute to patterns of phenotypic variation among populations of a species. Recent studies are revealing how interactions between historical and contemporary processes better explain observed patterns of phenotypic divergence than either process alone. Here, we investigate the roles of evolutionary history and adaptation to current environmental conditions in structuring phenotypic variation among polyphenic populations of sunfish inhabiting 12 postglacial lakes in eastern North America. The pumpkinseed sunfish polyphenism includes sympatric ecomorphs specialized for littoral or pelagic lake habitats. First, we use population genetic methods to test the evolutionary independence of within-lake phenotypic divergences of ecomorphs and to describe patterns of genetic structure among lake populations that clustered into three geographical groupings. We then used multivariate analysis of covariance (MANCOVA) to partition body shape variation (quantified with geometric morphometrics) among the effects of evolutionary history (reflecting phenotypic variation among genetic clusters), the shared phenotypic response of all populations to alternate habitats within lakes (reflecting adaptation to contemporary conditions), and unique phenotypic responses to habitats within lakes nested within genetic clusters. All effects had a significant influence on body form, but the effects of history and the interaction between history and contemporary habitat were larger than contemporary processes in structuring phenotypic variation. This highlights how divergence can be better understood against a known backdrop of evolutionary history.     

Evolutionary transition between bee pollination and hummingbird pollination in Salvia: Comparing means,variances and covariances of corolla traits

Covariation among traits can modify the evolutionary trajectory of complex structures. This process is thought to operate at a microevolutionary scale, but its long‐term effects remain controversial because trait covariation can itself evolve. Flower morphology, and particularly floral trait (co)variation, has been envisioned as the product of pollinator‐mediated selection. Available evidence suggests that major changes in pollinator assemblages may affect the joint expression of floral traits and their phenotypic integration. We expect species within a monophyletic lineage sharing the same pollinator type will show not only similarity in trait means but also similar phenotypic variance‐covariance structures. Here, we tested this expectation using eighteen Salvia species pollinated either by bees or by hummingbirds. Our findings indicated a nonsignificant multivariate phylogenetic signal and a decoupling between means and variance‐covariance phenotypic matrices of floral traits during the evolution to hummingbird pollination. Mean trait value analyses revealed significant differences between bee‐ and hummingbird‐pollinated Salvia species although fewer differences were detected in the covariance structure between groups. Variance‐covariance matrices were much more similar among bee‐ than hummingbird‐pollinated species. This pattern is consistent with the expectation that, unlike hummingbirds, bees physically manipulate the flower, presumably exerting stronger selection pressures favouring morphological convergence among species. Overall, we conclude that the evolution of hummingbird pollination proceeded through different independent transitions. Thus, although the evolution of hummingbird pollination led to a new phenotypic optimum, the process involved the diversification of the covariance structure.     

Discordance in body size, colour pattern, and advertisement call across genetically distinct populations in a Neotropical anuran (Dendropsophus ebraccatus)

Patterns of intraspecific geographic variation in morphology and behaviour, when examined in a phylogenetic context, can provide insight into the microevolutionary processes driving population divergence and ultimately speciation. In the present study, we quantified behavioural and phenotypic variation among populations from genetically divergent regions in the Central American treefrog, Dendropsophus ebraccatus . Our fine-scale population comparisons demonstrated regional divergence in body size, colour pattern frequencies, and male advertisement call. None of the characters covaried with phylogenetic history or geographic proximity among sampled populations, indicating the importance of highly localized selection pressures and genetic drift in shaping character divergence among isolated regions. The study underscores how multiple phenotypic characters can evolve independently across relatively small spatial scales.  © 2009 The Linnean Society of London, Biological Journal of the Linnean Society , 2009, 97 , 298–313.     

Rapid maximum likelihood ancestral state reconstruction of continuous characters: A rerooting‐free algorithm

Ancestral state reconstruction is a method used to study the evolutionary trajectories of quantitative characters on phylogenies. Although efficient methods for univariate ancestral state reconstruction under a Brownian motion model have been described for at least 25 years, to date no generalization has been described to allow more complex evolutionary models, such as multivariate trait evolution, non‐Brownian models, missing data, and within‐species variation. Furthermore, even for simple univariate Brownian motion models, most phylogenetic comparative R packages compute ancestral states via inefficient tree rerooting and full tree traversals at each tree node, making ancestral state reconstruction extremely time‐consuming for large phylogenies. Here, a computationally efficient method for fast maximum likelihood ancestral state reconstruction of continuous characters is described. The algorithm has linear complexity relative to the number of species and outperforms the fastest existing R implementations by several orders of magnitude. The described algorithm is capable of performing ancestral state reconstruction on a 1,000,000‐species phylogeny in fewer than 2 s using a standard laptop, whereas the next fastest R implementation would take several days to complete. The method is generalizable to more complex evolutionary models, such as phylogenetic regression, within‐species variation, non‐Brownian evolutionary models, and multivariate trait evolution. Because this method enables fast repeated computations on phylogenies of virtually any size, implementation of the described algorithm can drastically alleviate the computational burden of many otherwise prohibitively time‐consuming tasks requiring reconstruction of ancestral states, such as phylogenetic imputation of missing data, bootstrapping procedures, Expectation‐Maximization algorithms, and Bayesian estimation. The described ancestral state reconstruction algorithm is implemented in the Rphylopars functions anc.recon and phylopars.