Euclidean nature of phylogenetic distance matrices

Phylogenies are fundamental to comparative biology as they help to identify independent events on which statistical tests rely. Two groups of phylogenetic comparative methods (PCMs) can be distinguished: those that take phylogenies into account by introducing explicit models of evolution and those that only consider phylogenies as a statistical constraint and aim at partitioning trait values into a phylogenetic component (phylogenetic inertia) and one or multiple specific components related to adaptive evolution. The way phylogenetic information is incorporated into the PCMs depends on the method used. For the first group of methods, phylogenies are converted into variance-covariance matrices of traits following a given model of evolution such as Brownian motion (BM). For the second group of methods, phylogenies are converted into distance matrices that are subsequently transformed into Euclidean distances to perform principal coordinate analyses. Here, we show that simply taking the elementwise square root of a distance matrix extracted from a phylogenetic tree ensures having a Euclidean distance matrix. This is true for any type of distances between species (patristic or nodal) and also for trees harboring multifurcating nodes. Moreover, we illustrate that this simple transformation using the square root imposes less geometric distortion than more complex transformations classically used in the literature such as the Cailliez method. Given the Euclidean nature of the elementwise square root of phylogenetic distance matrices, the positive semidefinitiveness of the phylogenetic variance-covariance matrix of a trait following a BM model, or related models of trait evolution, can be established. In that way, we build a bridge between the two groups of statistical methods widely used in comparative analysis. These results should be of great interest for ecologists and evolutionary biologists performing statistical analyses incorporating phylogenies.     

The degree and pattern of phylogenetic signal in primate long-bone structure

Interspecific scaling is a fundamental tool for comparative studies of primate long-bone structure and adaptation. However, scaling analyses based on conventional statistical methods can lead to false positives regarding adaptive relationships when traits exhibit strong phylogenetic signal. This problem can be addressed through the use of phylogenetic comparative methods (PCMs). To date, PCMs have not been incorporated into comparative studies of primate long-bone structure because it has been assumed that long-bone structure is free of phylogenetic signal once appropriately scaled. To test this assumption, we evaluated the degree of phylogenetic signal in three types of long-bone structural traits (bone length, articular surface areas, and cross-sectional geometric properties) from 17 quadrupedal primate species. We compared the pattern of phylogenetic signal in raw trait values and residual trait values after regression against body mass, bone length, and the product of body mass x bone length. Our results show that significant phylogenetic signal is present in all traits before scaling, due in part to their strong covariance with body mass. After scaling, bone length still exhibits strong phylogenetic signal, but articular surface areas do not, and cross-sectional properties exhibit different levels of signal depending on the variable used to scale the data. These results suggest that PCMs should be incorporated into interspecific studies of bone length and perhaps cross-sectional geometric properties. Our results also demonstrate that tests for phylogenetic signal prior to implementing a PCM should focus on residual variance, not individual traits.     

Nonlinear relationships and phylogenetically independent contrasts

The method of phylogenetically independent contrasts is commonly used for exploring cross-taxon relationships between traits. Here we show that this phylogenetic comparative method (PCM) can fail to detect correlated evolution when the underlying relationship between traits is nonlinear. Simulations indicate that statistical power can be dramatically reduced when independent contrasts analysis is used on nonlinear relationships. We also reanalyze a published data set and demonstrate that ignoring nonlinearity can affect biological inferences. We suggest that researchers consider the shape of the relationship between traits when using independent contrasts analysis. Alternative PCMs may be more appropriate if data cannot be transformed to meet assumptions of linearity.     

Patterns of cranial shape diversification during the phylogenetic branching process of New World monkeys (Primates: Platyrrhini)

One of the central topics in evolutionary biology is understanding the processes responsible for phenotypic diversification related to ecological factors. New World monkeys are an excellent reference system to investigate processes of diversification at macroevolutionary scales. Here, we investigate the cranial shape diversification related to body size and ecology during the phylogenetic branching process of platyrrhines. To investigate this diversification, we used geometric morphometric techniques, a molecular phylogenetic tree, ecological data and phylogenetic comparative methods. Our statistical analyses demonstrated that the phylogenetic branching process is the most important dimension to understand cranial shape variation among extant platyrrhines and suggested that the main shape divergence among the four principal platyrrhine clades probably occurred during the initial branching process. The phylogenetic conservatism, which is the retention of ancestral traits over time within the four principal platyrrhine clades, could be the most important characteristic of platyrrhine cranial shape diversification. Different factors might have driven early shape divergence and posterior relative conservatism, including genetic drift, stabilizing selection, genetic constraints owing to pleiotropy, developmental or functional constraint, lack of genetic variation, among others. Understanding the processes driving the diversification among platyrrhines will probably require further palaeontological, phylogenetic and comparative studies.     

The Independent Evolution Method Is Not a Viable Phylogenetic Comparative Method

Phylogenetic comparative methods (PCMs) use data on species traits and phylogenetic relationships to shed light on evolutionary questions. Recently, Smaers and Vinicius suggested a new PCM, Independent Evolution (IE), which purportedly employs a novel model of evolution based on Felsenstein’s Adaptive Peak Model. The authors found that IE improves upon previous PCMs by producing more accurate estimates of ancestral states, as well as separate estimates of evolutionary rates for each branch of a phylogenetic tree. Here, we document substantial theoretical and computational issues with IE. When data are simulated under a simple Brownian motion model of evolution, IE produces severely biased estimates of ancestral states and changes along individual branches. We show that these branch-specific changes are essentially ancestor-descendant or “directional” contrasts, and draw parallels between IE and previous PCMs such as “minimum evolution”. Additionally, while comparisons of branch-specific changes between variables have been interpreted as reflecting the relative strength of selection on those traits, we demonstrate through simulations that regressing IE estimated branch-specific changes against one another gives a biased estimate of the scaling relationship between these variables, and provides no advantages or insights beyond established PCMs such as phylogenetically independent contrasts. In light of our findings, we discuss the results of previous papers that employed IE. We conclude that Independent Evolution is not a viable PCM, and should not be used in comparative analyses.     

Adaptation and the comparative method

Over the past two decades, it has become widely accepted that phylogenies need to be incorporated into statistical analyses of interspecific data. However, recent debate has focused on whether it is appropriate to apply phylogenetic comparative methods (PCMs) to the study of adaptation. Although some of the criticisms are serious, it is premature to stop applying PCMs altogether. New statistical methods designed explicitly for the comparative study of adaptation overcome these criticisms and offer fresh insights into the evolution of phenotypes.     

Recent developments in the analysis of comparative data

Comparative methods can be used to test ideas about adaptation by identifying cases of either parallel or convergent evolutionary change across taxa. Phylogenetic relationships must be known or inferred if comparative methods are to separate the cross-taxonomic covariation among traits associated with evolutionary change from that attributable to common ancestry. Only the former can be used to test ideas linking convergent or parallel evolutionary change to some aspect of the environment. The comparative methods that are currently available differ in how they manage the effects brought about by phylogenetic relationships. One method is applicable only to discrete data, and uses cladistic techniques to identify evolutionary events that depart from phylogenetic trends. Techniques for continuous variables attempt to control for phylogenetic effects in a variety of ways. One method examines the taxonomic distribution of variance to identify the taxa within which character variation is small. The method assumes that taxa with small amounts of variation are those in which little evolutionary change has occurred, and thus variation is unlikely to be independent of ancestral trends. Analyses are then concentrated among taxa that show more variation, on the assumption that greater evolutionary change in the character has taken place. Several methods estimate directly the extent to which ancestry can predict the observed variation of a character, and subtract the ancestral effect to reveal variation of phylogeny. Yet another can remove phylogenetic effects if the true phylogeny is known. One class of comparative methods controls for phylogenetic effects by searching for comparative trends within rather than across taxa. With current knowledge of phylogenies, there is a trade-off in the choice of a comparative method: those that control phylogenetic effects with greater certainty are either less applicable to real data, or they make restrictive or untestable assumptions. Those that rely on statistical patterns to infer phylogenetic effects may not control phylogeny as efficiently but are more readily applied to existing data sets.     

Is horizontal transmission really a problem for phylogenetic comparative methods? A simulation study using continuous cultural traits

Phylogenetic comparative methods (PCMs) provide a potentially powerful toolkit for testing hypotheses about cultural evolution. Here, we build on previous simulation work to assess the effect horizontal transmission between cultures has on the ability of both phylogenetic and non-phylogenetic methods to make inferences about trait evolution. We found that the mode of horizontal transmission of traits has important consequences for both methods. Where traits were horizontally transmitted separately, PCMs accurately reported when trait evolution was not correlated even at the highest levels of horizontal transmission. By contrast, linear regression analyses often incorrectly concluded that traits were correlated. Where simulated trait evolution was not correlated and traits were horizontally transmitted as a pair, both methods inferred increased levels of positive correlation with increasing horizontal transmission. Where simulated trait evolution was correlated, increasing rates of separate horizontal transmission led to decreasing levels of inferred correlation for both methods, but increasing rates of paired horizontal transmission did not. Furthermore, the PCM was also able to make accurate inferences about the ancestral state of traits. These results suggest that under certain conditions, PCMs can be robust to the effects of horizontal transmission. We discuss ways that future work can investigate the mode and tempo of horizontal transmission of cultural traits.     

TRANSLATING BETWEEN MICROEVOLUTIONARY PROCESS AND MACROEVOLUTIONARY PATTERNS: THE CORRELATION STRUCTURE OF INTERSPECIFIC DATA

As species evolve along a phylogenetic tree, we expect closely related species to retain some phenotypic similarities due to their shared evolutionary histories. The amount of expected similarity depends both on the hierarchical phylogenetic structure, and on the specific magnitude and types of evolutionary changes that accumulate during each generation. In this study, we show how models of microevolutionary change can be translated into the resulting macroevolutionary patterns. We illustrate how the structure of phenotypic covariances expected in interspecific measurements can be derived, and how this structure depends on the microevolutionary forces guiding phenotypic change at each generation. We then explore the covariance structure expected from several simple microevolutionary models of phenotypic evolution, including various combinations of random genetic drift, directional selection, stabilizing selection, and environmental change, as well as models of punctuated or burst-like evolution. We find that stabilizing selection leads to patterns of exponential decrease of between species covariance with phylogenetic distance. This is different from the usual linear patterns of decrease assumed in most comparative and systematic methods. Nevertheless, linear patterns of decrease can result from many processes in addition to random genetic drift, such as directional and fluctuating selection as well as modes of punctuated change. Our framework can be used to develop methods for (1) phylogenetic reconstruction; (2) inference of the evolutionary process from comparative data; and (3) conducting or evaluating statistical analyses of comparative data while taking phylogenetic history into account.     

Assessing current adaptation and phylogenetic inertia as explanations of trait evolution: the need for controlled comparisons

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.     

Phylogenetic comparative analysis of life-history variation among populations of the lizard Sceloporus undulatus: an example and prognosis

Over the past 15 years, phylogenetic comparative methods (PCMs) have become standard in the study of life-history evolution. To date, most studies have focused on variation among species or higher taxonomic levels, generally revealing the presence of significant phylogenetic effects as well as residual variation potentially attributable to adaptive evolution. Recently, population-level phylogenetic hypotheses have become available for many species, making it possible to apply PCMs directly to the level at which experiments are typically used to test adaptive hypotheses. In this study, we present the results of PCMs applied to life-history variation among populations of the widespread and well-studied lizard Sceloporus undulatus. Using S. undulatus (which may represent four closely related species) as an example, we explore the benefits of using PCMs at the population level, as well as consider the importance of several thorny methodological problems including but not limited to nonindependence of populations, lack of sufficient variation in traits, and the typically small sample sizes dictated by the difficulty of collecting detailed demographic data. We show that phylogenetic effects on life-history variation among populations of S. undulatus appear to be unimportant, and that several classic trade-offs expected by theory and revealed by many interspecific comparisons are absent. Our results suggest that PCMs applied to variation in life-history traits below the species level may be of limited value, but more studies like ours are needed to draw a general conclusion. Finally, we discuss several outstanding problems that face studies seeking to apply PCMs below the species level.     

Assessing the Goodness of Fit of Phylogenetic Comparative Methods: A Meta-Analysis and Simulation Study

Background

Phylogenetic comparative methods (PCMs) have been applied widely in analyzing data from related species but their fit to data is rarely assessed.

Question

Can one determine whether any particular comparative method is typically more appropriate than others by examining comparative data sets?

Data

I conducted a meta-analysis of 122 phylogenetic data sets found by searching all papers in JEB, Blackwell Synergy and JSTOR published in 2002–2005 for the purpose of assessing the fit of PCMs. The number of species in these data sets ranged from 9 to 117.

Analysis Method

I used the Akaike information criterion to compare PCMs, and then fit PCMs to bivariate data sets through REML analysis. Correlation estimates between two traits and bootstrapped confidence intervals of correlations from each model were also compared.

Conclusions

For phylogenies of less than one hundred taxa, the Independent Contrast method and the independent, non-phylogenetic models provide the best fit.For bivariate analysis, correlations from different PCMs are qualitatively similar so that actual correlations from real data seem to be robust to the PCM chosen for the analysis. Therefore, researchers might apply the PCM they believe best describes the evolutionary mechanisms underlying their data.     

Phylogenetic comparative methods and the geographic range size – body size relationship in new world terrestrial carnivora

Most recent papers avoid describing macroecological relationships and interpreting then without a previous control of non-independence in data caused by phylogenetic patterns in data. In this paper, we analyzed the geographic range size – body size relationship for 70 species of New World terrestrial Carnivora (fissipeds) using various phylogenetic comparative methods and simulation procedures to assess their statistical performance. Autocorrelation analyses suggested a strong phylogenetic pattern for body size, but not for geographic range size. The correlation between the two traits was estimated using standard Pearson correlation across species (TIPS) and four different comparative methods: Felsenstein's independent contrasts (PIC), autoregressive method (ARM), phylogenetic eigenvector regression (PVR) and phylogenetic generalized least-squares (PGLS). The correlation between the two variables was significant for all methods, except PIC, in such a way that ecological mechanisms (i.e., minimum viable population or environmental heterogeneity- physiological homeostasis), could be valid explanations for the relationship. Simulations using different O-U processes for each trait were run in order to estimate true Type I errors of each method. Type I errors at 5% were similar for all phylogenetic methods (always lower than 8%), but equal to 13.1% for TIPS. PIC usually performs better than all other methods under Brownian motion evolution, but not in this case using a more complex combination of evolutionary models. So, recent claims that using independent contrasts in ecological research can be too conservative are correct but, on the other hand, using simple across-species correlation is too liberal even under the more complex evolutionary models exhibited by the traits analyzed here.     

No substitute for real data: A cautionary note on the use of phylogenies from birth–death polytomy resolvers for downstream comparative analyses

The statistical estimation of phylogenies is always associated with uncertainty, and accommodating this uncertainty is an important component of modern phylogenetic comparative analysis. The birth–death polytomy resolver is a method of accounting for phylogenetic uncertainty that places missing (unsampled) taxa onto phylogenetic trees, using taxonomic information alone. Recent studies of birds and mammals have used this approach to generate pseudoposterior distributions of phylogenetic trees that are complete at the species level, even in the absence of genetic data for many species. Many researchers have used these distributions of phylogenies for downstream evolutionary analyses that involve inferences on phenotypic evolution, geography, and community assembly. I demonstrate that the use of phylogenies constructed in this fashion is inappropriate for many questions involving traits. Because species are placed on trees at random with respect to trait values, the birth–death polytomy resolver breaks down natural patterns of trait phylogenetic structure. Inferences based on these trees are predictably and often drastically biased in a direction that depends on the underlying (true) pattern of phylogenetic structure in traits. I illustrate the severity of the phenomenon for both continuous and discrete traits using examples from a global bird phylogeny.     

STABILIZING SELECTION AND THE COMPARATIVE ANALYSIS OF ADAPTATION

Comparative studies tend to differ from optimality and functionality studies in how they treat adaptation. While the comparative approach focuses on the origin and change of traits, optimality studies assume that adaptations are maintained at an optimum by stabilizing selection. This paper presents a model of adaptive evolution on a macroevolutionary time scale that includes the maintenance of traits at adaptive optima by stabilizing selection as the dominant evolutionary force. Interspecific variation is treated as variation in the position of adaptive optima. The model illustrates how phylogenetic constraints not only lead to correlations between phylogenetically related species, but also to imperfect adaptations. From this model, a statistical comparative method is derived that can be used to estimate the effect of a selective factor on adaptive optima in a way that would be consistent with an optimality study of adaptation to this factor. The method is illustrated with an analysis of dental evolution in fossil horses. The use of comparative methods to study evolutionary trends is also discussed.     

A phylogenetic comparative method for evaluating trait coevolution across two phylogenies for sets of interacting species

Evaluating trait correlations across species within a lineage via phylogenetic regression is fundamental to comparative evolutionary biology, but when traits of interest are derived from two sets of lineages that coevolve with one another, methods for evaluating such patterns in a dual‐phylogenetic context remain underdeveloped. Here, we extend multivariate permutation‐based phylogenetic regression to evaluate trait correlations in two sets of interacting species while accounting for their respective phylogenies. This extension is appropriate for both univariate and multivariate response data, and may use one or more independent variables, including environmental covariates. Imperfect correspondence between species in the interacting lineages can also be accommodated, such as when species in one lineage associate with multiple species in the other, or when there are unmatched taxa in one or both lineages. For both univariate and multivariate data, the method displays appropriate type I error, and statistical power increases with the strength of the trait covariation and the number of species in the phylogeny. These properties are retained even when there is not a 1:1 correspondence between lineages. Finally, we demonstrate the approach by evaluating the evolutionary correlation between traits in fig species and traits in their agaonid wasp pollinators. R computer code is provided.     

Phylogenetic analysis and comparative data: a test and review of evidence

The question is often raised whether it is statistically necessary to control for phylogenetic associations in comparative studies. To investigate this question, we explore the use of a measure of phylogenetic correlation, lambda, introduced by Pagel (1999), that normally varies between 0 (phylogenetic independence) and 1 (species' traits covary in direct proportion to their shared evolutionary history). Simulations show lambda to be a statistically powerful index for measuring whether data exhibit phylogenetic dependence or not and whether it has low rates of Type I error. Moreover, lambda is robust to incomplete phylogenetic information, which demonstrates that even partial information on phylogeny will improve the accuracy of phylogenetic analyses. To assess whether traits generally show phylogenetic associations, we present a quantitative review of 26 published phylogenetic comparative data sets. The data sets include 103 traits and were chosen from the ecological literature in which debate about the need for phylogenetic correction has been most acute. Eighty-eight percent of data sets contained at least one character that displayed significant phylogenetic dependence, and 60% of characters overall (pooled across studies) showed significant evidence of phylogenetic association. In 16% of tests, phylogenetic correlation could be neither supported nor rejected. However, most of these equivocal results were found in small phylogenies and probably reflect a lack of power. We suggest that the parameter lambda be routinely estimated when analyzing comparative data, since it can also be used simultaneously to adjust the phylogenetic correction in a manner that is optimal for the data set, and we present an example of how this may be done.     

Intraspecific variation in sperm length is negatively related to sperm competition in passerine birds

Spermatozoa are among the most diversified cells in the animal kingdom, but the underlying evolutionary forces affecting intraspecific variation in sperm morphology are poorly understood. It has been hypothesized that sperm competition is a potent selection pressure on sperm variation within species. Here, we examine intraspecific variation in total sperm length of 22 wild passerine bird species (21 genera, 11 families) in relation to the risk of sperm competition, as expressed by the frequency of extrapair paternity and relative testis size. We demonstrate, by using phylogenetic comparative methods, that between-male variation in sperm length within species is closely and negatively linked to the risk of sperm competition. This relationship was even stronger when only considering species in which data on sperm length and extrapair paternity originated from the same populations. Intramale variation in sperm length within species was also negatively, although nonsignificantly, related to sperm competition risk. Our findings suggest that postcopulatory sexual selection is a powerful evolutionary force reducing the intraspecific phenotypic variation in sperm-size traits, potentially driving the diversification of sperm morphology across populations and species.     

PHYLOGENIES,SPATIAL AUTOREGRESSION,AND THE COMPARATIVE METHOD: A COMPUTER SIMULATION TEST

Brownian motion computer simulation was used to test the statistical properties of a spatial autoregressive method in estimating evolutionary correlations between two traits using interspecific comparative data. When applied with a phylogeny of 42 species, the method exhibited reasonable Type I and II error rates. Estimation abilities were comparable to those of independent contrasts and minimum evolution (parsimony) methods, and generally superior to a traditional nonphylogenetic approach (not taking phylogenies into account at all). However, the autoregressive method performed extremely poorly with a smaller phylogeny (15 species) and with nearly independent (“star”) phylogenies. In both of these situations, any phylogenetic autocorrelation present in the data was not detected by the method. Results show how diagnostic techniques (e.g., Moran's I) can be useful in detecting and avoiding such situations, but that such techniques should not be used as definitive evidence that phylogenetic correlation is not present in a set of comparative data. The correction factor (α) proposed by Gittleman and Kot (1990) for use in weighting phylogenetic information had little effect in most analyses of 15 or 42 species with incorrect phylogenetic information, and may require much larger sample sizes before significant improvement is shown. With the sample sizes tested in this study, however, the autoregressive method implemented with this correction factor and correct phylogenetic information led to downwardly biased estimates of the absolute magnitude of the evolutionary correlation between two traits. Cautions and recommendations for implemention of the spatial autoregressive method are given; computer programs to conduct the analyses are available on request.     

Evolution of morphological integration in the skull of Carnivora (Mammalia): Changes in Canidae lead to increased evolutionary potential of facial traits

Morphological integration refers to the fact that different phenotypic traits of organisms are not fully independent from each other, and tend to covary to different degrees. The covariation among traits is thought to reflect properties of the species' genetic architecture and thus can have an impact on evolutionary responses. Furthermore, if morphological integration changes along the history of a group, inferences of past selection regimes might be problematic. Here, we evaluated the stability and evolution of the morphological integration of skull traits in Carnivora by using evolutionary simulations and phylogenetic comparative methods. Our results show that carnivoran species are able to respond to natural selection in a very similar way. Our comparative analyses show that the phylogenetic signal for pattern of integration is lower than that observed for morphology (trait averages), and that integration was stable throughout the evolution of the group. That notwithstanding, Canidae differed from other families by having higher integration, evolvability, flexibility, and allometric coefficients on the facial region. These changes might have allowed canids to rapidly adapt to different food sources, helping to explain not only the phenotypic diversification of the family, but also why humans were able to generate such a great diversity of dog breeds through artificial selection.