Factors influencing changes in trait correlations across species after using phylogenetic independent contrasts

Comparative interspecific data sets have been analyzed routinely by phylogenetic methods, generally using Felsenstein’s phylogenetic independent contrasts (PIC) method. However, some authors have suggested that it may not be always necessary to incorporate phylogenetic information into statistical analyses of comparative data due to the low influence of shared history on the distribution of␣character states. The main goal of this paper was to undertake a comparison of results from non-phylogenetic Pearson correlation of tip values (TIPs) and phylogenetic independent contrasts analyses (PICs), using 566 correlation coefficients derived from 65 published papers. From each study we collected the following data: taxonomic group, number of species, type of phylogeny, number of polytomies in the phylogenetic tree, if branch length was transformed or not, trait types, the original correlation coefficient between the traits (TIPs) and the correlation coefficient between the traits using the independent contrasts method (PICs). The slope estimated from a regression of PIC correlations on TIP correlations was lower than one, and a paired t-test showed that correlations from PIC are significantly smaller than those obtained by TIP. Thus, PIC analyses tend to decrease the correlation between traits and usually increases the P-value and, thus, favoring the acceptance of the null hypothesis. Multiple factors, including taxonomic group, trait type and use of branch length transformations affected the change in decision regarding the acceptance of the null hypotheses and differences between PIC and TIP results. Due to the variety of factors affecting the differences between results provided by these methods, we suggest that comparative methods should be applied as a conservative approach to cross-species studies. Despite difficulties in quantifying precisely why these factors affect the differences between PIC and TIP, we also suggest that a better evaluation of evolutionary models underlying trait evolution is still necessary in this context and might explain some of the observed patterns.     

Adaptive constraints and the phylogenetic comparative method: a computer simulation test

Recently, the utility of modern phylogenetic comparative methods (PCMs) has been questioned because of the seemingly restrictive assumptions required by these methods. Although most comparative analyses involve traits thought to be undergoing natural or sexual selection, most PCMs require an assumption that the traits be evolving by less directed random processes, such as Brownian motion (BM). In this study, we use computer simulation to generate data under more realistic evolutionary scenarios and consider the statistical abilities of a variety of PCMs to estimate correlation coefficients from these data. We found that correlations estimated without taking phylogeny into account were often quite poor and never substantially better than those produced by the other tested methods. In contrast, most PCMs performed quite well even when their assumptions were violated. Felsenstein's independent contrasts (FIC) method gave the best performance in many cases, even when weak constraints had been acting throughout phenotypic evolution. When strong constraints acted in opposition to variance-generating (i.e., BM) forces, however, FIC correlation coefficients were biased in the direction of those BM forces. In most cases, all other PCMs tested (phylogenetic generalized least squares, phylogenetic mixed model, spatial autoregression, and phylogenetic eigenvector regression) yielded good statistical performance, regardless of the details of the evolutionary model used to generate the data. Actual parameter estimates given by different PCMs for each dataset, however, were occasionally very different from one another, suggesting that the choice among them should depend on the types of traits and evolutionary processes being considered.     

A genetic analysis of the mandible and maxilla in the rat

A quantitative genetic analysis of eight osteometric traits from the oral region together with femur length in the rat are described. The relevant questions being examined are (1) the heritability of each trait; (2) the relative contribution of genetic, maternal environmental, and residual environmental effects to the correlation between the oral traits, and (3) genetic and nongenetic components of the correlation between the oral traits and overall body size. Some quantitative genetic aspects of covarying traits is briefly reviewed with special emphasis on allometric variation. Phenotypic regression coefficients from log-transformed data (= allometry coefficients) of oral traits onto femur length are partitioned into components due to genetic, maternal environmental, residual environmental, and total environmental causes. All phenotypic regression coefficients and all but one based on an environmentally determined covariance component are significantly different from zero, suggesting a substantial body size effect in the oral region resulting from nonheritable causes. However, three genetic coefficients from regressions of oral traits to femur length are not different from zero, indicating a genetic correlation with body size in five traits but not in three others. A principal components analysis was carried out on the phenotypic, genetic, and environmental correlations of the eight oral traits to provide a multivariate depiction of the components of covariation in the maxilla and mandible of the rat. General multivariate effects due to body size together with group or special effects are demonstrated for each component of morphogenesis.     

Double‐wavelet transform for multisubject task‐induced functional magnetic resonance imaging data

The goal of this article is to model multisubject task‐induced functional magnetic resonance imaging (fMRI) response among predefined regions of interest (ROIs) of the human brain. Conventional approaches to fMRI analysis only take into account temporal correlations, but do not rigorously model the underlying spatial correlation due to the complexity of estimating and inverting the high dimensional spatio‐temporal covariance matrix. Other spatio‐temporal model approaches estimate the covariance matrix with the assumption of stationary time series, which is not always feasible. To address these limitations, we propose a double‐wavelet approach for modeling the spatio‐temporal brain process. Working with wavelet coefficients simplifies temporal and spatial covariance structure because under regularity conditions, wavelet coefficients are approximately uncorrelated. Different wavelet functions were used to capture different correlation structures in the spatio‐temporal model. The main advantages of the wavelet approach are that it is scalable and that it deals with nonstationarity in brain signals. Simulation studies showed that our method could reduce false‐positive and false‐negative rates by taking into account spatial and temporal correlations simultaneously. We also applied our method to fMRI data to study activation in prespecified ROIs in the prefontal cortex. Data analysis showed that the result using the double‐wavelet approach was more consistent than the conventional approach when sample size decreased.     

Grouped generalized estimating equations for longitudinal data analysis

Generalized estimating equation (GEE) is widely adopted for regression modeling for longitudinal data, taking account of potential correlations within the same subjects. Although the standard GEE assumes common regression coefficients among all the subjects, such an assumption may not be realistic when there is potential heterogeneity in regression coefficients among subjects. In this paper, we develop a flexible and interpretable approach, called grouped GEE analysis, to modeling longitudinal data with allowing heterogeneity in regression coefficients. The proposed method assumes that the subjects are divided into a finite number of groups and subjects within the same group share the same regression coefficient. We provide a simple algorithm for grouping subjects and estimating the regression coefficients simultaneously, and show the asymptotic properties of the proposed estimator. The number of groups can be determined by the cross validation with averaging method. We demonstrate the proposed method through simulation studies and an application to a real data set.     

General Regression and Representation Model for Classification

Recently, the regularized coding-based classification methods (e.g. SRC and CRC) show a great potential for pattern classification. However, most existing coding methods assume that the representation residuals are uncorrelated. In real-world applications, this assumption does not hold. In this paper, we take account of the correlations of the representation residuals and develop a general regression and representation model (GRR) for classification. GRR not only has advantages of CRC, but also takes full use of the prior information (e.g. the correlations between representation residuals and representation coefficients) and the specific information (weight matrix of image pixels) to enhance the classification performance. GRR uses the generalized Tikhonov regularization and K Nearest Neighbors to learn the prior information from the training data. Meanwhile, the specific information is obtained by using an iterative algorithm to update the feature (or image pixel) weights of the test sample. With the proposed model as a platform, we design two classifiers: basic general regression and representation classifier (B-GRR) and robust general regression and representation classifier (R-GRR). The experimental results demonstrate the performance advantages of proposed methods over state-of-the-art algorithms.     

Ratios as a size adjustment in morphometrics

Simple ratios in which a measurement variable is divided by a size variable are commonly used but known to be inadequate for eliminating size correlations from morphometric data. Deficiencies in the simple ratio can be alleviated by incorporating regression coefficients describing the bivariate relationship between the measurement and size variables. Recommendations have included: 1) subtracting the regression intercept to force the bivariate relationship through the origin (intercept-adjusted ratios); 2) exponentiating either the measurement or the size variable using an allometry coefficient to achieve linearity (allometrically adjusted ratios); or 3) both subtracting the intercept and exponentiating (fully adjusted ratios). These three strategies for deriving size-adjusted ratios imply different data models for describing the bivariate relationship between the measurement and size variables (i.e., the linear, simple allometric, and full allometric models, respectively). Algebraic rearrangement of the equation associated with each data model leads to a correctly formulated adjusted ratio whose expected value is constant (i.e., size correlation is eliminated). Alternatively, simple algebra can be used to derive an expected value function for assessing whether any proposed ratio formula is effective in eliminating size correlations. Some published ratio adjustments were incorrectly formulated as indicated by expected values that remain a function of size after ratio transformation. Regression coefficients incorporated into adjusted ratios must be estimated using least-squares regression of the measurement variable on the size variable. Use of parameters estimated by any other regression technique (e.g., major axis or reduced major axis) results in residual correlations between size and the adjusted measurement variable. Correctly formulated adjusted ratios, whose parameters are estimated by least-squares methods, do control for size correlations. The size-adjusted results are similar to those based on analysis of least-squares residuals from the regression of the measurement on the size variable. However, adjusted ratios introduce size-related changes in distributional characteristics (variances) that differentially alter relationships among animals in different size classes. © 1993 Wiley-Liss, Inc.     

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.     

Sample size considerations for stepped wedge designs with subclusters

The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.     

Fruit production and floral traits: correlated evolution in <Emphasis Type="Italic">Govenia</Emphasis> (Orchidaceae)

The most common estimate of reproductive success in orchids is usually fruit set. Factors such as resource limitation and certain floral traits may influence reproductive success in animal-pollinated plants. Correlated evolution of reproductive success vs. seven floral traits (inflorescence length, flower number, flower distribution along inflorescence, dorsal sepal length, lateral sepal length, flower color, and column position) was studied in eight species of Govenia. Taxa represented three lineages in the genus. Independent contrasts were calculated on a phylogeny inferred from chloroplast (trnL-F IGS) DNA sequences, and a correlation test and multiple regression were then performed. Two data sets were evaluated, one including all eight species and another excluding G. utriculata, which is autogamous. The historical analyses showed that there is a correlation between reproductive success and dorsal sepal length, column position, and flower number, these correlations suggest that changes in these floral traits usually accompany evolutionary shifts in reproductive success. Multiple regression tests suggest that changes in reproductive success can be explained by shifts in flower number, inflorescence length, column position and by dorsal sepal length. When phylogeny is taken into account, our analyses showed that evolutionary shifts in these floral traits were correlated with changes in reproductive success. Evolutionary correlation between reproductive success and floral traits might be explained by the natural selection of certain floral phenotypes by pollinators.     

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.     

Humeral epiphyseal shape in the felidae: The influence of phylogeny,allometry, and locomotion

Bone morphology of the cats (Mammalia: Felidae) is influenced by many factors, including locomotor mode, body size, hunting methods, prey size and phylogeny. Here, we investigate the shape of the proximal and distal humeral epiphyses in extant species of the felids, based on two‐dimensional landmark configurations. Geometric morphometric techniques were used to describe shape differences in the context of phylogeny, allometry and locomotion. The influence of these factors on epiphyseal shape was assessed using Principal Component Analysis, Linear Discriminant functions and multivariate regression. Phylogenetic Generalised Least Squares was used to examine the association between size or locomotion and humeral epiphyseal shape, after taking a phylogenetic error term into account. Results show marked differences in epiphyseal shape between felid lineages, with a relatively large phylogenetic influence. Additionally, the adaptive influences of size and locomotion are demonstrated, and their influence is independent of phylogeny in most, but not all, cases. Several features of epiphyseal shape are common to the largest terrestrial felids, including a relative reduction in the surface area of the humeral head and increased robusticity of structures that provide attachment for joint‐stabilising muscles, including the medial epicondyle and the greater and lesser tubercles. This increased robusticity is a functional response to the increased loading forces placed on the joints due to large body mass. J. Morphol., 2012. © 2012 Wiley Periodicals, Inc.     

Estimation of doubly labeled water energy expenditure with confidence intervals

Bivariate regression is used to estimate energy expenditure from doubly labeled water data. Two straight lines are fitted to the logarithms of the enrichments of oxygen-18 and deuterium simultaneously as a bivariate regression, so that the correlations between the oxygen and deuterium regression coefficients can be estimated. Maximum likelihood methods are used to extend bivariate regression to unbalanced situations caused by missing observations and to include replicate laboratory determination from the same urine samples, even if one of the replicates is missing. Use of maximum likelihood allows the determination of a confidence interval for the energy expenditure based on the log likelihood surface rather than use of the propagation of variance methods for nonlinear transformations. The model is extended to include the subject's deviations from the two lines as a bivariate continuous-time first-order autoregression to allow for serial correlation in the observations. The analysis of data from two subjects, one without apparent serial correlation and one with serial correlation, is presented.     

Monovalent—divalent cation ratios and the occurrence of phytoplankton, with special reference to the desmids

SUMMARY. We have analysed data from ninety-nine Scottish freshwater lochs, to explore the relationship between water chemistry and phytoplankton assemblages. Our results confirm that there are strong correlations both between the phytoplankton quotient and divalent cation concentrations, particularly Ca⁺⁺, and also between the phytoplankton quotient and the (Na⁺+ K⁺)/(Ca⁺⁺+ Mg⁺⁺) ratio. However, the latter is evidently a spurious relationship, arising from the former association, together with an association between Ca⁺⁺ and the (Na⁺+K⁺)/ (Ca⁺⁺+ Mg⁺⁺) ratio. The observed correlation does not persist if account is taken of concomitant variations in the Ca⁺⁺concentration. This conclusion is suggested both by relatively informal analyses (median polish, partial correlation coefficients) and by more formal modelling and testing (stepwise regression, all-subsets regression).     

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.     

Paleotemperature proxies from leaf fossils reinterpreted in light of evolutionary history

Present-day correlations between leaf physiognomic traits (shape and size) and climate are widely used to estimate paleoclimate using fossil floras. For example, leaf-margin analysis estimates paleotemperature using the modern relation of mean annual temperature (MAT) and the site-proportion of untoothed-leaf species (NT). This uniformitarian approach should provide accurate paleoclimate reconstructions under the core assumption that leaf-trait variation principally results from adaptive environmental convergence, and because variation is thus largely independent of phylogeny it should be constant through geologic time. Although much research acknowledges and investigates possible pitfalls in paleoclimate estimation based on leaf physiognomy, the core assumption has never been explicitly tested in a phylogenetic comparative framework. Combining an extant dataset of 21 leaf traits and temperature with a phylogenetic hypothesis for 569 species-site pairs at 17 sites, we found varying amounts of non-random phylogenetic signal in all traits. Phylogenetic vs. standard regressions generally support prevailing ideas that leaf-traits are adaptively responding to temperature, but wider confidence intervals, and shifts in slope and intercept, indicate an overall reduced ability to predict climate precisely due to the non-random phylogenetic signal. Notably, the modern-day relation of proportion of untoothed taxa with mean annual temperature (NT-MAT), central in paleotemperature inference, was greatly modified and reduced, indicating that the modern correlation primarily results from biogeographic history. Importantly, some tooth traits, such as number of teeth, had similar or steeper slopes after taking phylogeny into account, suggesting that leaf teeth display a pattern of exaptive evolution in higher latitudes. This study shows that the assumption of convergence required for precise, quantitative temperature estimates using present-day leaf traits is not supported by empirical evidence, and thus we have very low confidence in previously published, numerical paleotemperature estimates. However, interpreting qualitative changes in paleotemperature remains warranted, given certain conditions such as stratigraphically closely-spaced samples with floristic continuity.     

Phylogenetic ANOVA: Group‐clade aggregation,biological challenges,and a refined permutation procedure

Phylogenetic regression is frequently used in macroevolutionary studies, and its statistical properties have been thoroughly investigated. By contrast, phylogenetic ANOVA has received relatively less attention, and the conditions leading to incorrect statistical and biological inferences when comparing multivariate phenotypes among groups remain underexplored. Here, we propose a refined method of randomizing residuals in a permutation procedure (RRPP) for evaluating phenotypic differences among groups while conditioning the data on the phylogeny. We show that RRPP displays appropriate statistical properties for both phylogenetic ANOVA and regression models, and for univariate and multivariate datasets. For ANOVA, we find that RRPP exhibits higher statistical power than methods utilizing phylogenetic simulation. Additionally, we investigate how group dispersion across the phylogeny affects inferences, and reveal that highly aggregated groups generate strong and significant correlations with the phylogeny, which reduce statistical power and subsequently affect biological interpretations. We discuss the broader implications of this phylogenetic group aggregation, and its relation to challenges encountered with other comparative methods where one or a few transitions in discrete traits are observed on the phylogeny. Finally, we recommend that phylogenetic comparative studies of continuous trait data use RRPP for assessing the significance of indicator variables as sources of trait variation.     

A Bayesian extension of phylogenetic generalized least squares: Incorporating uncertainty in the comparative study of trait relationships and evolutionary rates

Phylogenetic comparative methods use tree topology, branch lengths, and models of phenotypic change to take into account nonindependence in statistical analysis. However, these methods normally assume that trees and models are known without error. Approaches relying on evolutionary regimes also assume specific distributions of character states across a tree, which often result from ancestral state reconstructions that are subject to uncertainty. Several methods have been proposed to deal with some of these sources of uncertainty, but approaches accounting for all of them are less common. Here, we show how Bayesian statistics facilitates this task while relaxing the homogeneous rate assumption of the well-known phylogenetic generalized least squares (PGLS) framework. This Bayesian formulation allows uncertainty about phylogeny, evolutionary regimes, or other statistical parameters to be taken into account for studies as simple as testing for coevolution in two traits or as complex as testing whether bursts of phenotypic change are associated with evolutionary shifts in intertrait correlations. A mixture of validation approaches indicates that the approach has good inferential properties and predictive performance. We provide suggestions for implementation and show its usefulness by exploring the coevolution of ankle posture and forefoot proportions in Carnivora.     

Correlated substitution analysis and the prediction of amino acid structural contacts

It has long been suspected that analysis of correlated amino acid substitutions should uncover pairs or clusters of sites that are spatially proximal in mature protein structures. Accordingly, methods based on different mathematical principles such as information theory, correlation coefficients and maximum likelihood have been developed to identify co-evolving amino acids from multiple sequence alignments. Sets of pairs of sites whose behaviour is identified by these methods as correlated are often significantly enriched in pairs of spatially proximal residues. However, relatively high levels of false-positive predictions typically render such methods, in isolation, of little use in the ab initio prediction of protein structure. Misleading signal (or problems with the estimation of significance levels) can be caused by phylogenetic correlations between homologous sequences and from correlation due to factors other than spatial proximity (for example, correlation of sites which are not spatially close but which are involved in common functional properties of the protein). In recent years, several workers have suggested that information from correlated substitutions should be combined with other sources of information (secondary structure, solvent accessibility, evolutionary rates) in an attempt to reduce the proportion of false-positive predictions. We review methods for the detection of correlated amino acid substitutions, compare their relative performance in contact prediction and predict future directions in the field.