Polytomies and phylogenetically independent contrasts: examination of the bounded degrees of freedom approach

We examined the effect of soft polytomies on the performance (Type I error rate and bias) of Felsenstein's (1985; Am. Nat. 125:1-15) method of phylogenetically independent contrasts for estimating a bivariate correlation. We specifically tested the adequacy of bounding degrees of freedom, as suggested by Purvis and Garland (1993; Syst. Biol. 42:569-575). We simulated bivariate character evolution under Brownian motion (assumed by independent contrasts) and eight other models on five phylogenetic trees. For non-Brownian motion simulations, the adequacy of branch-length standardization was checked with a simple diagnostic (Garland et al., 1992; Syst. Biol. 41:18-32), and transformations were applied as indicated. Surprisingly, soft polytomies tended to have negligible effects on Type I error rates when models other than Brownian motion were used. Overall, and irrespective of evolutionary model, degrees of freedom were appropriately bounded for hypothesis testing, and unbiased estimates of the correlation coefficient were obtained. Our results, along with those of previous simulation studies, suggest that independent contrasts can reliably be applied to real data, even with phylogenetic uncertainty.     

Assessment of the relative merits of a few methods to detect evolutionary trends

Some of the most basic questions about the history of life concern evolutionary trends. These include determining whether or not metazoans have become more complex over time, whether or not body size tends to increase over time (the Cope-Depéret rule), or whether or not brain size has increased over time in various taxa, such as mammals and birds. Despite the proliferation of studies on such topics, assessment of the reliability of results in this field is hampered by the variability of techniques used and the lack of statistical validation of these methods. To solve this problem, simulations are performed using a variety of evolutionary models (gradual Brownian motion, speciational Brownian motion, and Ornstein-Uhlenbeck), with or without a drift of variable amplitude, with variable variance of tips, and with bounds placed close or far from the starting values and final means of simulated characters. These are used to assess the relative merits (power, Type I error rate, bias, and mean absolute value of error on slope estimate) of several statistical methods that have recently been used to assess the presence of evolutionary trends in comparative data. Results show widely divergent performance of the methods. The simple, nonphylogenetic regression (SR) and variance partitioning using phylogenetic eigenvector regression (PVR) with a broken stick selection procedure have greatly inflated Type I error rate (0.123-0.180 at a 0.05 threshold), which invalidates their use in this context. However, they have the greatest power. Most variants of Felsenstein's independent contrasts (FIC; five of which are presented) have adequate Type I error rate, although two have a slightly inflated Type I error rate with at least one of the two reference trees (0.064-0.090 error rate at a 0.05 threshold). The power of all contrast-based methods is always much lower than that of SR and PVR, except under Brownian motion with a strong trend and distant bounds. Mean absolute value of error on slope of all FIC methods is slightly higher than that of phylogenetic generalized least squares (PGLS), SR, and PVR. PGLS performs well, with low Type I error rate, low error on regression coefficient, and power comparable with some FIC methods. Four variants of skewness analysis are examined, and a new method to assess significance of results is presented. However, all have consistently low power, except in rare combinations of trees, trend strength, and distance between final means and bounds. Globally, the results clearly show that FIC-based methods and PGLS are globally better than nonphylogenetic methods and variance partitioning with PVR. FIC methods and PGLS are sensitive to the model of evolution (and, hence, to branch length errors). Our results suggest that regressing raw character contrasts against raw geological age contrasts yields a good combination of power and Type I error rate. New software to facilitate batch analysis is presented.     

PHYLOGENETIC ANALYSES OF THE CORRELATED EVOLUTION OF CONTINUOUS CHARACTERS: A SIMULATION STUDY

We use computer simulation to compare the statistical properties of several methods that have been proposed for estimating the evolutionary correlation between two continuous traits, and define alternative evolutionary correlations that may be of interest. We focus on Felsenstein's (1985) method and some variations of it and on several “minimum evolution” methods (of which the procedure of Huey and Bennett [1987] is a special case), as compared with a nonphylogenetic correlation. The last, a simple correlation of trait values across the tips of a phylogeny, virtually always yields inflated Type I error rates, relatively low power, and relatively poor estimates of evolutionary correlations. We therefore cannot recommend its use. In contrast, Felsenstein's (1985) method yields acceptable significance tests, high power, and good estimates of what we term the input correlation and the standardized realized evolutionary correlation, given complete phylogenetic information and knowledge of the rate and mode of character change (e.g., gradual and proportional to time [“Brownian motion”] or punctuational, with change only at speciation events). Inaccurate branch length information may affect any method adversely, but only rarely does it cause Felsenstein's (1985) method to perform worse than do the others tested. Other proposed methods generally yield inflated Type I error rates and have lower power. However, certain minimum evolution methods (although not the specific procedure used by Huey and Bennett [1987]) often provide more accurate estimates of what we term the unstandardized realized evolutionary correlation, and their use is recommended when estimation of this correlation is desired. We also demonstrate how correct Type I error rates can be obtained for any method by reference to an empirical null distribution derived from computer simulations, and provide practical suggestions on choosing an analytical method, based both on the evolutionary correlation of interest and on the availability of branch lengths and knowledge of the model of evolutionary change appropriate for the characters being analyzed. Computer programs that implement the various methods and that will simulate (correlated) character evolution along a known phylogeny are available from the authors on request. These programs can be used to test the effectiveness of any new methods that might be proposed, and to check the generality of our conclusions with regard to other phylogenies.     

Taxon sampling, correlated evolution, and independent contrasts

Independent contrasts are widely used to incorporate phylogenetic information into studies of continuous traits, particularly analyses of evolutionary trait correlations, but the effects of taxon sampling on these analyses have received little attention. In this paper, simulations were used to investigate the effects of taxon sampling patterns and alternative branch length assignments on the statistical performance of correlation coefficients and sign tests; "full-tree" analyses based on contrasts at all nodes and "paired-comparisons" based only on contrasts of terminal taxon pairs were also compared. The simulations showed that random samples, with respect to the traits under consideration, provide statistically robust estimates of trait correlations. However, exact significance tests are highly dependent on appropriate branch length information; equal branch lengths maintain lower Type I error than alternative topological approaches, and adjusted critical values of the independent contrast correlation coefficient are provided for use with equal branch lengths. Nonrandom samples, with respect to univariate or bivariate trait distributions, introduce discrepancies between interspecific and phylogenetically structured analyses and bias estimates of underlying evolutionary correlations. Examples of nonrandom sampling processes may include community assembly processes, convergent evolution under local adaptive pressures, selection of a nonrandom sample of species from a habitat or life-history group, or investigator bias. Correlation analyses based on species pairs comparisons, while ignoring deeper relationships, entail significant loss of statistical power and as a result provide a conservative test of trait associations. Paired comparisons in which species differ by a large amount in one trait, a method introduced in comparative plant ecology, have appropriate Type I error rates and high statistical power, but do not correctly estimate the magnitude of trait correlations. Sign tests, based on full-tree or paired-comparison approaches, are highly reliable across a wide range of sampling scenarios, in terms of Type I error rates, but have very low power. These results provide guidance for selecting species and applying comparative methods to optimize the performance of statistical tests of trait associations.     

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.     

Robustness of compound Dirichlet priors for Bayesian inference of branch lengths

We modified the phylogenetic program MrBayes 3.1.2 to incorporate the compound Dirichlet priors for branch lengths proposed recently by Rannala, Zhu, and Yang (2012. Tail paradox, partial identifiability and influential priors in Bayesian branch length inference. Mol. Biol. Evol. 29:325-335.) as a solution to the problem of branch-length overestimation in Bayesian phylogenetic inference. The compound Dirichlet prior specifies a fairly diffuse prior on the tree length (the sum of branch lengths) and uses a Dirichlet distribution to partition the tree length into branch lengths. Six problematic data sets originally analyzed by Brown, Hedtke, Lemmon, and Lemmon (2010. When trees grow too long: investigating the causes of highly inaccurate Bayesian branch-length estimates. Syst. Biol. 59:145-161) are reanalyzed using the modified version of MrBayes to investigate properties of Bayesian branch-length estimation using the new priors. While the default exponential priors for branch lengths produced extremely long trees, the compound Dirichlet priors produced posterior estimates that are much closer to the maximum likelihood estimates. Furthermore, the posterior tree lengths were quite robust to changes in the parameter values in the compound Dirichlet priors, for example, when the prior mean of tree length changed over several orders of magnitude. Our results suggest that the compound Dirichlet priors may be useful for correcting branch-length overestimation in phylogenetic analyses of empirical data sets.     

Linear Regression in Genetic Association Studies

In genomic research phenotype transformations are commonly used as a straightforward way to reach normality of the model outcome. Many researchers still believe it to be necessary for proper inference. Using regression simulations, we show that phenotype transformations are typically not needed and, when used in phenotype with heteroscedasticity, result in inflated Type I error rates. We further explain that important is to address a combination of rare variant genotypes and heteroscedasticity. Incorrectly estimated parameter variability or incorrect choice of the distribution of the underlying test statistic provide spurious detection of associations. We conclude that it is a combination of heteroscedasticity, minor allele frequency, sample size, and to a much lesser extent the error distribution, that matter for proper statistical inference.     

Correlated evolution in fig pollination

This is the first comparative study of correlated evolution between figs (Ficus species, Moraceae) and their pollinators (Hymenoptera: Agaoninae) based on molecular phylogenies of both lineages. Fig relationships based on the internal transcribed spacer region (ITS) of nuclear ribosomal DNA and pollinator relationships inferred from mitochondrial cytochrome oxidase I (COI) sequences enabled the study of correlated evolution based on molecular phylogenies for the largest set of interacting species ever compared. Comparative methods have been applied to tests of adaptation, but the application of these methods in tests of coadaptation, defined as reciprocal evolutionary change in interacting lineages, has received less attention. I have extended tests of correlated evolution between two traits along a phylogeny to the case of interacting lineages, where two traits may or may not share a common phylogenetic history. Independent contrasts and phylogenetic autocorrelation rejected the null hypothesis that trait correlations within lineages are stronger than trait correlations between interacting lineages. Fig style lengths and pollinator ovipositor lengths, for example, were more highly correlated than were pollinator body size and ovipositor length. Mutualistic interactions between figs and their pollinators illustrate the novel ways in which phylogenies and comparative methods can detect patterns of correlated evolution. The most outstanding evidence of correlated evolution between these obligate mutualists is that interacting trait correlations are stronger than within-lineage allometric relationships.     

Neutral theory, phylogenies, and the relationship between phenotypic change and evolutionary rates

The neutral theory of molecular evolution predicts that rates of phenotypic change are largely independent from genotypic change. A recent study by Bromham et al. (2002) confirmed this expectation, finding no evidence for correlated phenotypic and molecular evolutionary rates in animals. We reevaluate this hypothesis, sampling at different taxonomic levels in plants and animals, using Bayesian inference to reconstruct phylogenetic trees and estimate rates of molecular evolution. We use independent contrasts in branch lengths to maximize the information extracted from each of the trees and nodal posterior probabilities to assess the influence of phylogenetic error. Our results indicate that in vascular plants between 2% and 11% of the variation in phenotypic rates of change can be explained by the rate of genotypic change. These results may be explained by the idea that processes that affect general evolutionary rates, such as body size, may also be expected to influence rates of morphological change.     

Transformations of count data for tests of interaction in factorial and split-plot experiments

In applied entomological experiments, when the response is a count-type variable, certain transformation remedies such as the square root, logarithm (log), or rank transformation are often used to normalize data before analysis of variance. In this study, we examine the usefulness of these transformations by reanalyzing field-collected data from a split-plot experiment and by performing a more comprehensive simulation study of factorial and split-plot experiments. For field-collected data, significant interactions were dependent upon the type of transformation. For the simulation study, Poisson distributed errors were used for a 2 by 2 factorial arrangement, in both randomized complete block and split-plot settings. Various sizes of main effects were induced, and type I error rates and powers of the tests for interaction were examined for the raw response values, log-, square root-, and rank-transformed responses. The aligned rank transformation also was investigated because it has been shown to perform well in testing interactions in factorial arrangements. We found that for testing interactions, the untransformed response and the aligned rank response performed best (preserved nominal type I error rates), whereas the other transformations had inflated error rates when main effects were present. No evaluations of the tests for main effects or simple effects have been conducted. Potentially these transformations will still be necessary when performing these tests.     

Adding time-calibrated branch lengths to the Asteraceae supertree

New inference techniques,such as supertrees,have improved the construction of large phylogenies,helping to reveal the tree of life.In addition,these large phylogenies have enhanced the study of other evolutionary questions,such as whether traits have evolved in a neutral or adaptive way,or what factors have influenced diversification.However,supertrees usually lack branch lengths,which are necessary for all these issues to be investigated.Here,divergence times within the largest family of flowering plants,namely the Asteraceae,are reviewed to estimate time-calibrated branch lengths in the supertree of this lineage.An inconsistency between estimated dates of basal branching events and the earliest asteraceous fossil pollen record was detected.In addition,the impact of different methods of branch length assignment on the total number of transitions between states in the reconstruction of sexual system evolution in Asteraceae was investigated.At least for this dataset,different branch length assignation approaches influenced maximum likelihood(ML)reconstructions only and not Bayesian ones.Therefore,the selection of different branch length information is not arbitrary and should be carefully assessed,at least when ML approaches are being used.The reviewed divergence times and the estimated time-calibrated branch lengths provide a useful tool for future phylogenetic comparative and macroevolutionary studies of Asteraceae.     

Adding time‐calibrated branch lengths to the Asteraceae supertree

Abstract New inference techniques, such as supertrees, have improved the construction of large phylogenies, helping to reveal the tree of life. In addition, these large phylogenies have enhanced the study of other evolutionary questions, such as whether traits have evolved in a neutral or adaptive way, or what factors have influenced diversification. However, supertrees usually lack branch lengths, which are necessary for all these issues to be investigated. Here, divergence times within the largest family of flowering plants, namely the Asteraceae, are reviewed to estimate time‐calibrated branch lengths in the supertree of this lineage. An inconsistency between estimated dates of basal branching events and the earliest asteraceous fossil pollen record was detected. In addition, the impact of different methods of branch length assignment on the total number of transitions between states in the reconstruction of sexual system evolution in Asteraceae was investigated. At least for this dataset, different branch length assignation approaches influenced maximum likelihood (ML) reconstructions only and not Bayesian ones. Therefore, the selection of different branch length information is not arbitrary and should be carefully assessed, at least when ML approaches are being used. The reviewed divergence times and the estimated time‐calibrated branch lengths provide a useful tool for future phylogenetic comparative and macroevolutionary studies of Asteraceae.     

Stochastic mapping of morphological characters

Many questions in evolutionary biology are best addressed by comparing traits in different species. Often such studies involve mapping characters on phylogenetic trees. Mapping characters on trees allows the nature, number, and timing of the transformations to be identified. The parsimony method is the only method available for mapping morphological characters on phylogenies. Although the parsimony method often makes reasonable reconstructions of the history of a character, it has a number of limitations. These limitations include the inability to consider more than a single change along a branch on a tree and the uncoupling of evolutionary time from amount of character change. We extended a method described by Nielsen (2002, Syst. Biol. 51:729-739) to the mapping of morphological characters under continuous-time Markov models and demonstrate here the utility of the method for mapping characters on trees and for identifying character correlation.     

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.     

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.     

Assessing among‐lineage variability in phylogenetic imputation of functional trait datasets

Phylogenetic imputation has recently emerged as a potentially powerful tool for predicting missing data in functional traits datasets. As such, understanding the limitations of phylogenetic modelling in predicting trait values is critical if we are to use them in subsequent analyses. Previous studies have focused on the relationship between phylogenetic signal and clade‐level prediction accuracy, yet variability in prediction accuracy among individual tips of phylogenies remains largely unexplored. Here, we used simulations of trait evolution along the branches of phylogenetic trees to show how the accuracy of phylogenetic imputations is influenced by the combined effects of 1) the amount of phylogenetic signal in the traits and 2) the branch length of the tips to be imputed. Specifically, we conducted cross‐validation trials to estimate the variability in prediction accuracy among individual tips on the phylogenies (hereafter ‘tip‐level accuracy’). We found that under a Brownian motion model of evolution (BM, Pagel't λ = 1), tip‐level accuracy rapidly decreased with increasing tip branch‐lengths, and only tips of approximately 10% or less of the total height of the trees showed consistently accurate predictions (i.e. cross‐validation R‐squared >0.75). When phylogenetic signal was weak, the effect of tip branch‐length was reduced, becoming negligible for traits simulated with λ < 0.7, where accuracy was in any case low. Our study shows that variability in prediction accuracy among individual tips of the phylogeny should be considered when evaluating the reliability of phylogenetically imputed trait values. To address this challenge, we describe a Monte Carlo‐based method that allows one to estimate the expected tip‐level accuracy of phylogenetic predictions for continuous traits. Our approach identifies gaps in functional trait datasets for which phylogenetic imputation performs poorly, and will help ecologists to design more efficient trait collection campaigns by focusing resources on lineages whose trait values are more uncertain.     

The ancestral distance test: what relatedness can reveal about correlated evolution in large lineages with missing character data and incomplete phylogenies

The ancestral distance test is introduced to detect correlated evolution between two binary traits in large phylogenies that may lack resolved subclades, branch lengths, and/or comparative data. We define the ancestral distance as the time separating a randomly sampled taxon from its most recent ancestor (MRA) with extant descendants that have an independent trait. The sampled taxon either has (target sample) or lacks (nontarget sample) a dependent trait. Modeled as a Markov process, we show that the distribution of ancestral distances for the target sample is identical to that of the nontarget sample when characters are uncorrelated, whereas ancestral distances are smaller on average for the target sample when characters are correlated. Simulations suggest that the ancestral distance can be estimated using the time, total branch length, taxonomic rank, or number of speciation events between a sampled taxon and the MRA. These results are shown to be robust to deviations from Markov assumptions. A Monte Carlo technique estimates P-values when fully resolved phylogenies with branch lengths are available, and we evaluate the Monte Carlo approach using a data set with known correlation. Measures of relatedness were found to provide a robust means to test hypotheses of correlated character evolution.     

Evolutionary Diversification of Continuous Traits: Phylogenetic Tests and Application to Seed Size in the California Flora

Evolutionary diversification of a phenotypic trait reflects the tempo and mode of trait evolution, as well as the phylogenetic topology and branch lengths. Comparisons of trait variance between sister groups provide a powerful approach to test for differences in rates of diversification, controlling for differences in clade age. We used simulation analyses under constant rate Brownian motion to develop phylogenetically based F-tests of the ratio of trait variances between sister groups. Random phylogenies were used for a generalized evolutionary null model, so that detailed internal phylogenies are not required, and both gradual and speciational models of evolution were considered. In general, phylogenetically structured tests were more conservative than corresponding parametric statistics (i.e., larger variance ratios are required to achieve significance). The only exception was for comparisons under a speciational evolutionary model when the group with higher variance has very low sample size (number of species). The methods were applied to a large data set on seed size for 1976 species of California flowering plants. Seven of 37 sister-group comparisons were significant for the phylogenetically structured tests (compared to 12 of 37 for the parametric F-test). Groups with higher diversification of seed size generally had a greater diversity of fruit types, life form, or life history as well. The F-test for trait variances provides a simple, phylogenetically structured approach to test for differences in rates of phenotypic diversification and could also provide a valuable tool in the study of adaptive radiations.