Multivariate character process models for the analysis of two or more correlated function-valued traits

Various methods, including random regression, structured antedependence models, and character process models, have been proposed for the genetic analysis of longitudinal data and other function-valued traits. For univariate problems, the character process models have been shown to perform well in comparison to alternative methods. The aim of this article is to present an extension of these models to the simultaneous analysis of two or more correlated function-valued traits. Analytical forms for stationary and nonstationary cross-covariance functions are studied. Comparisons with the other approaches are presented in a simulation study and in an example of a bivariate analysis of genetic covariance in age-specific fecundity and mortality in Drosophila. As in the univariate case, bivariate character process models with an exponential correlation were found to be quite close to first-order structured antedependence models. The simulation study showed that the choice of the most appropriate methodology is highly dependent on the covariance structure of the data. The bivariate character process approach proved to be able to deal with quite complex nonstationary and nonsymmetric cross-correlation structures and was found to be the most appropriate for the real data example of the fruit fly Drosophila melanogaster.     

A flexible estimating equations approach for mapping function-valued traits

In genetic studies, many interesting traits, including growth curves and skeletal shape, have temporal or spatial structure. They are better treated as curves or function-valued traits. Identification of genetic loci contributing to such traits is facilitated by specialized methods that explicitly address the function-valued nature of the data. Current methods for mapping function-valued traits are mostly likelihood-based, requiring specification of the distribution and error structure. However, such specification is difficult or impractical in many scenarios. We propose a general functional regression approach based on estimating equations that is robust to misspecification of the covariance structure. Estimation is based on a two-step least-squares algorithm, which is fast and applicable even when the number of time points exceeds the number of samples. It is also flexible due to a general linear functional model; changing the number of covariates does not necessitate a new set of formulas and programs. In addition, many meaningful extensions are straightforward. For example, we can accommodate incomplete genotype data, and the algorithm can be trivially parallelized. The framework is an attractive alternative to likelihood-based methods when the covariance structure of the data is not known. It provides a good compromise between model simplicity, statistical efficiency, and computational speed. We illustrate our method and its advantages using circadian mouse behavioral data.     

Structured antedependence models for genetic analysis of repeated measures on multiple quantitative traits

Simultaneous analysis of correlated traits that change with time is an important issue in genetic analyses. Several methodologies have already been proposed for the genetic analysis of longitudinal data on single traits, in particular random regression and character process models. Although the latter proved, in most cases, to compare favourably to alternative approaches for analysis of single function-valued traits, they do not allow a straightforward extension to the multivariate case. In this paper, another methodology (structured antedependence models) is proposed, and methods are derived for the genetic analysis of two or more correlated function-valued traits. Multivariate analyses are presented of fertility and mortality in Drosophila and of milk, fat and protein yields in dairy cattle. These models offer a substantial flexibility for the correlation structure, even in the case of complex non-stationary patterns, and perform better than multivariate random regression models, with fewer parameters.     

Variation,selection and evolution of function-valued traits

We describe an emerging framework for understanding variation, selection and evolution of phenotypic traits that are mathematical functions. We use one specific empirical example – thermal performance curves (TPCs) for growth rates of caterpillars – to demonstrate how models for function-valued traits are natural extensions of more familiar, multivariate models for correlated, quantitative traits. We emphasize three main points. First, because function-valued traits are continuous functions, there are important constraints on their patterns of variation that are not captured by multivariate models. Phenotypic and genetic variation in function-valued traits can be quantified in terms of variance-covariance functions and their associated eigenfunctions: we illustrate how these are estimated as well as their biological interpretations for TPCs. Second, selection on a function-valued trait is itself a function, defined in terms of selection gradient functions. For TPCs, the selection gradient describes how the relationship between an organism's performance and its fitness varies as a function of its temperature. We show how the form of the selection gradient function for TPCs relates to the frequency distribution of environmental states (caterpillar temperatures) during selection. Third, we can predict evolutionary responses of function-valued traits in terms of the genetic variance-covariance and the selection gradient functions. We illustrate how non-linear evolutionary responses of TPCs may occur even when the mean phenotype and the selection gradient are themselves linear functions of temperature. Finally, we discuss some of the methodological and empirical challenges for future studies of the evolution of function-valued traits.     

Variation in continuous reaction norms: quantifying directions of biological interest

Thermal performance curves are an example of continuous reaction norm curves of common shape. Three modes of variation in these curves--vertical shift, horizontal shift, and generalist-specialist trade-offs--are of special interest to evolutionary biologists. Since two of these modes are nonlinear, traditional methods such as principal components analysis fail to decompose the variation into biological modes and to quantify the variation associated with each mode. Here we present the results of a new method, template mode of variation (TMV), that decomposes the variation into predetermined modes of variation for a particular set of thermal performance curves. We illustrate the method using data on thermal sensitivity of growth rate in Pieris rapae caterpillars. The TMV model explains 67% of the variation in thermal performance curves among families; generalist-specialist trade-offs account for 38% of the total between-family variation. The TMV method implemented here is applicable to both differences in mean and patterns of variation, and it can be used with either phenotypic or quantitative genetic data for thermal performance curves or other continuous reaction norms that have a template shape with a single maximum. The TMV approach may also apply to growth trajectories, age-specific life-history traits, and other function-valued traits.     

Hypothesis testing in comparative and experimental studies of function-valued traits

Many traits of evolutionary interest, when placed in their developmental, physiological, or environmental contexts, are function-valued. For instance, gene expression during development is typically a function of the age of an organism and physiological processes are often a function of environment. In comparative and experimental studies, a fundamental question is whether the function-valued trait of one group is different from another. To address this question, evolutionary biologists have several statistical methods available. These methods can be classified into one of two types: multivariate and functional. Multivariate methods, including univariate repeated-measures analysis of variance (ANOVA), treat each trait as a finite list of data. Functional methods, such as repeated-measures regression, view the data as a sample of points drawn from an underlying function. A key difference between multivariate and functional methods is that functional methods retain information about the ordering and spacing of a set of data values, information that is discarded by multivariate methods. In this study, we evaluated the importance of that discarded information in statistical analyses of function-valued traits. Our results indicate that functional methods tend to have substantially greater statistical power than multivariate approaches to detect differences in a function-valued trait between groups.     

Direct estimation of genetic principal components: simplified analysis of complex phenotypes

Estimating the genetic and environmental variances for multivariate and function-valued phenotypes poses problems for estimation and interpretation. Even when the phenotype of interest has a large number of dimensions, most variation is typically associated with a small number of principal components (eigen-vectors or eigenfunctions). We propose an approach that directly estimates these leading principal components; these then give estimates for the covariance matrices (or functions). Direct estimation of the principal components reduces the number of parameters to be estimated, uses the data efficiently, and provides the basis for new estimation algorithms. We develop these concepts for both multivariate and function-valued phenotypes and illustrate their application in the restricted maximum-likelihood framework.     

Functional mapping of quantitative trait loci underlying the character process: a theoretical framework

Unlike a character measured at a finite set of landmark points, function-valued traits are those that change as a function of some independent and continuous variable. These traits, also called infinite-dimensional characters, can be described as the character process and include a number of biologically, economically, or biomedically important features, such as growth trajectories, allometric scalings, and norms of reaction. Here we present a new statistical infrastructure for mapping quantitative trait loci (QTL) underlying the character process. This strategy, termed functional mapping, integrates mathematical relationships of different traits or variables within the genetic mapping framework. Logistic mapping proposed in this article can be viewed as an example of functional mapping. Logistic mapping is based on a universal biological law that for each and every living organism growth over time follows an exponential growth curve (e.g., logistic or S-shaped). A maximum-likelihood approach based on a logistic-mixture model, implemented with the EM algorithm, is developed to provide the estimates of QTL positions, QTL effects, and other model parameters responsible for growth trajectories. Logistic mapping displays a tremendous potential to increase the power of QTL detection, the precision of parameter estimation, and the resolution of QTL localization due to the small number of parameters to be estimated, the pleiotropic effect of a QTL on growth, and/or residual correlations of growth at different ages. More importantly, logistic mapping allows for testing numerous biologically important hypotheses concerning the genetic basis of quantitative variation, thus gaining an insight into the critical role of development in shaping plant and animal evolution and domestication. The power of logistic mapping is demonstrated by an example of a forest tree, in which one QTL affecting stem growth processes is detected on a linkage group using our method, whereas it cannot be detected using current methods. The advantages of functional mapping are also discussed.     

Generalized character process models: estimating the genetic basis of traits that cannot be observed and that change with age or environmental conditions

The genetic analysis of characters that change as a function of some independent and continuous variable has received increasing attention in the biological and statistical literature. Previous work in this area has focused on the analysis of normally distributed characters that are directly observed. We propose a framework for the development and specification of models for a quantitative genetic analysis of function-valued characters that are not directly observed, such as genetic variation in age-specific mortality rates or complex threshold characters. We employ a hybrid Markov chain Monte Carlo algorithm involving a Monte Carlo EM algorithm coupled with a Markov chain approximation to the likelihood, which is quite robust and provides accurate estimates of the parameters in our models. The methods are investigated using simulated data and are applied to a large data set measuring mortality rates in the fruit fly, Drosophila melanogaster.     

A Simple Regression-Based Method to Map Quantitative Trait Loci Underlying Function-Valued Phenotypes

Most statistical methods for quantitative trait loci (QTL) mapping focus on a single phenotype. However, multiple phenotypes are commonly measured, and recent technological advances have greatly simplified the automated acquisition of numerous phenotypes, including function-valued phenotypes, such as growth measured over time. While methods exist for QTL mapping with function-valued phenotypes, they are generally computationally intensive and focus on single-QTL models. We propose two simple, fast methods that maintain high power and precision and are amenable to extensions with multiple-QTL models using a penalized likelihood approach. After identifying multiple QTL by these approaches, we can view the function-valued QTL effects to provide a deeper understanding of the underlying processes. Our methods have been implemented as a package for R, funqtl.     

Discrete and morphometric traits reveal contrasting patterns and processes in the macroevolutionary history of a clade of scorpions

Many palaeontological studies have investigated the evolution of entire body plans, generally relying on discrete character‐taxon matrices. In contrast, macroevolutionary studies performed by neontologists have mostly focused on morphometric traits. Although these data types are very different, some studies have suggested that they capture common patterns. Nonetheless, the tests employed to support this claim have not explicitly incorporated a phylogenetic framework and may therefore be susceptible to confounding effects due to the presence of common phylogenetic structure. We address this question using the scorpion genus Brachistosternus Pocock 1893 as case study. We make use of a time‐calibrated multilocus molecular phylogeny, and compile discrete and traditional morphometric data sets, both capturing the overall morphology of the organisms. We find that morphospaces derived from these matrices are significantly different, and that the degree of discordance cannot be replicated by simulations of random character evolution. Moreover, we find strong support for contrasting modes of evolution, with discrete characters being congruent with an ‘early burst’ scenario whereas morphometric traits suggest species‐specific adaptations to have driven morphological evolution. The inferred macroevolutionary dynamics are therefore contingent on the choice of character type. Finally, we confirm that metrics of correlation fail to detect these profound differences given common phylogenetic structure in both data sets, and that methods incorporating a phylogenetic framework and accounting for expected covariance should be favoured.     

Up hill, down dale: quantitative genetics of curvaceous traits

'Repeated' measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as 'function-valued' (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances. Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multi-dimensional analyses are discussed.     

Spontaneous Mutation Rate of Modifiers of Metabolism in Drosophila

A rigorous test of our understanding of evolutionary quantitative genetics would be to predict accurately the equilibrium distribution of a character from empirical estimates of the relevant parameters in a mutation-selection-drift balance model. an aspect of this problem that is amenable to experimental analysis is the distribution of the effects of new mutations. This study quantifies the divergence among 200 lines of Drosophila melanogaster as they accumulated mutations on the second chromosome and estimates the rate of increase of variation and covariation in metabolic characters. Amounts of stored triacylglycerol and glycogen and the activities of a series of 12 metabolic enzymes were assayed in a subset of lines at generations 0, 11, 22, 33 and 44. Analyses of the rate of increase in the among-line variance in each trait allowed estimation of V(m)/V(e), the ratio of among-line variance added per generation to the environmental variance. Values of V(m)/V(e) for the second chromosome ranged from 0.0004 to 0.0289 per generation. Six of the 16 characters showed significant departure from a normal distribution, and several lines exhibited large changes in more than one character. The covariance of pairs of traits also was partitioned into a within-line component (environmental covariance, Cov(e)) and an among-line component (mutational covariance, Cov(m)). Both variances and covariance among lines increased over time, as assessed by linear regression, whereas environmental covariance showed no such trend. Results indicate that the quantitative genetic parameters describing the variation in metabolic traits are similar to those of other continuous characters.     

Character analysis in morphological phylogenetics: problems and solutions

Many aspects of morphological phylogenetics are controversial in the theoretical systematics literature and yet are often poorly explained and justified in empirical studies. In this paper, I argue that most morphological characters describe variation that is fundamentally quantitative, regardless of whether they are coded qualitatively or quantitatively by systematists. Given this view, three fundamental problems in morphological character analysis (definition, delimitation, and ordering of character states) may have a common solution: coding morphological characters as continuous quantitative traits. A new parsimony method (step-matrix gap-weighting, a modification of Thiele's approach) is proposed that allows quantitative traits to be analyzed as continuous variables. The problem of scaling or weighting quantitative characters relative to qualitative characters (and to each other) is reviewed, and three possible solutions are described. The new coding method is applied to data from hoplocercid lizards, and the results show the sensitivity of phylogenetic conclusions to different scaling methods. Although some authors reject the use of continuous, overlapping, quantitative characters in phylogenetic analysis, quantitative data from hoplocercid lizards that are coded using the new approach contain significant phylogenetic structure and exhibit levels of homoplasy similar to those seen in data that are coded qualitatively.     

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.     

Genetic analysis of female preference functions as function-valued traits

The genetic analysis of female preferences has been seen as a particularly challenging empirical endeavor because of difficulties in generating suitable preference metrics in experiments large enough to adequately characterize variation. In this article, we take an alternative approach, treating female preference as a function-valued trait and exploiting random-coefficient models to characterize the genetic basis of female preference without measuring preference functions in each individual. Applying this approach to Drosophila bunnanda, in which females assess males through a multivariate contact pheromone system, we gain three valuable insights into the genetic basis of female preference functions. First, most genetic variation was attributable to one eigenfunction, suggesting shared genetic control of preferences for nine male pheromones. Second, genetic variance in female preference functions was not associated with genetic variance in the pheromones, implying that genetic variation in female preference did not maintain genetic variation in male traits. Finally, breeding values for female preference functions were skewed away from the direction of selection on the male traits, suggesting directional selection on female preferences. The genetic analysis of female preference functions as function-valued traits offers a robust statistical framework for investigations of female preference, in addition to alleviating some experimental difficulties associated with estimating variation in preference functions.     

Quantitative genetics of continuous reaction norms: thermal sensitivity of caterpillar growth rates

A continuous reaction norm or performance curve represents a phenotypic trait of an individual or genotype in which the trait value may vary with some continuous environmental variable. We explore patterns of genetic variation in thermal performance curves of short-term caterpillar growth rate in a population of Pieris rapae. We compare multivariate methods, which treat performance at each test temperature as a distinct trait, with function-valued methods that treat a performance curve as a continuous function. Mean growth rate increased with increasing temperatures from 8 to 35 degrees C, was highest at 35 degrees C, and declined at 40 degrees C. There was substantial and significant variation among full-sib families in their thermal performance curves. Estimates of broad-sense genetic variances and covariances showed that genetic variance in growth rate increased more than 30-fold from low (8-11 degrees C) to high (35-40 degrees C) temperatures, even after differences in mean growth rate across temperatures were removed. Growth rate at 35 and 40 degrees C was negatively correlated genetically, suggesting a genetic trade-off in growth rate at these temperatures; this trade-off may represent either a generalist-specialist trade-off and/or variation in the optimal temperature for growth. The estimated genetic variance-covariance function (G function), the function-valued analog of the variance-covariance matrix (G matrix), was quite bumpy compared with the estimated G matrix; and results of principal component analyses of the G function were difficult to interpret. The use of orthogonal polynomials as the basis functions in current function-valued estimation methods may generate artifacts when the true G function has prominent local features, such as strong negative covariances at nearby temperatures (e.g. at 35 and 40 degrees C); this may be a particular issue for thermal performance curves and other highly nonlinear reaction norms.     

Computerized Analysis of Crustacean Relationships

The strengths and weaknesses of phylogenetic analysis using computers are reviewed from the viewpoint of understanding crustacean evolution. Computerized methods require the explicit presentation of characters and character state homologies. New techniques allow investigators to design evolutionary models into a character data matrix, or to use evolutionary models that make minimal a priori assumptions. The computer analysis relieves the investigator from the highly repetitious testing of trees, allows the concentration on the character state data, and provides objective methods for comparing trees, primarily their length. These are regarded as the strengths of computerized methods. The weaknesses of these methods include the relatively inscrutable nature of the character data matrix compared with the overall ‘gestalt’ of resulting trees, the difficulties of defining discrete homologies within the Crustacea, especially for counts of segmentation, the lack of clear intermediate character states in some multistate segmental characters, and the inability to define evolutionary polarity. These difficulties may be overcome by analysing the data using the minimal assumption models of character evolution, and by a recognition that the trees are a result of the input data, and therefore the data should be criticized, rather than the trees themselves. A ‘consensus’ character data set, including most extant major groups of the Crustacea as well as several key fossils, was assembled and revised by the participants in the workshop. An artificial taxon, ‘ur-crustacean characters’, was introduced to root the tree. Three observations may be made from parsimony analyses using several weighting and tree rooting methods. (1) The currently accepted large scale phylogeny and classification of the Crustacea is not corroborated. (2) The number of supposed plesiomorphic traits possessed by a taxon is not a good index for early derivation in crustacean evolution. (3) The taxon Maxillopoda is not supported by the arrangement of any of the trees.     

Genetic covariance structure of growth in the salamander Ambystoma macrodactylum

The size of an organism at any point during ontogeny often has fitness consequences through either direct selection on size or through selection on size-related morphological, performance, or life history traits. However, the evolutionary response to selection on size across ontogeny (a growth trajectory) may be limited by genetic correlations across ages. Here we characterize the phenotypic and genetic covariance structure of length and mass growth trajectories in a natural population of larval Ambystoma macrodactylum using function-valued quantitative genetic analyses and principal component decomposition. Most of the phenotypic and genetic variation in both growth trajectories appears to be confined to a single principal component describing a pattern of positive covariation among sizes across all ages. Higher order principal components with no significant associated genetic variation were identified for both trajectories, suggesting that evolution towards certain patterns of negative covariation between sizes across ages is constrained. The well-characterized positive relationship between size at metamorphosis and fitness in pond-breeding amphibians predicts that the across-age covariance structure will strongly limit evolution only if there is negative selection on size prior to metamorphosis. The pattern of genetic covariation observed in this study is similar to that observed in other vertebrate taxa, indicating that size may often be highly genetically and phenotypically integrated across ontogeny. Additionally, we find that phenotypic and genetic analyses of growth trajectories can yield qualitatively similar patterns of covariance structure.