Statistical equivalent of the classical TDT for quantitative traits and multivariate phenotypes

Clinical end-point traits are usually governed by quantitative precursors. Hence, there is active research interest in developing statistical methods for association mapping of quantitative traits. Unlike population-based tests for association, family-based tests for transmission disequilibrium are protected against population stratification. In this study, we propose a logistic regression model to test the association for quantitative traits based on a trio design. We show that the method can be viewed as a direct extension of the classical transmission diequilibrium test for binary traits to quantitative traits. We evaluate the performance of our method using extensive simulations and compare it with an existing method, family-based association test. We found that the two methods yield comparable powers if all families are considered. However, unlike FBAT, which yields an inflated rate of false positives when noninformative trios with all three individuals’ heterozygous are removed, our method maintains the correct size without compromising too much on power. We show that our method can be easily modified to incorporate multivariate phenotypes. Here, we applied this method to analyse a quantitative endophenotype associated with alcoholism.     

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.     

Statistical Estimation of Correlated Genome Associations to a Quantitative Trait Network

Many complex disease syndromes, such as asthma, consist of a large number of highly related, rather than independent, clinical or molecular phenotypes. This raises a new technical challenge in identifying genetic variations associated simultaneously with correlated traits. In this study, we propose a new statistical framework called graph-guided fused lasso (GFlasso) to directly and effectively incorporate the correlation structure of multiple quantitative traits such as clinical metrics and gene expressions in association analysis. Our approach represents correlation information explicitly among the quantitative traits as a quantitative trait network (QTN) and then leverages this network to encode structured regularization functions in a multivariate regression model over the genotypes and traits. The result is that the genetic markers that jointly influence subgroups of highly correlated traits can be detected jointly with high sensitivity and specificity. While most of the traditional methods examined each phenotype independently and combined the results afterwards, our approach analyzes all of the traits jointly in a single statistical framework. This allows our method to borrow information across correlated phenotypes to discover the genetic markers that perturb a subset of the correlated traits synergistically. Using simulated datasets based on the HapMap consortium and an asthma dataset, we compared the performance of our method with other methods based on single-marker analysis and regression-based methods that do not use any of the relational information in the traits. We found that our method showed an increased power in detecting causal variants affecting correlated traits. Our results showed that, when correlation patterns among traits in a QTN are considered explicitly and directly during a structured multivariate genome association analysis using our proposed methods, the power of detecting true causal SNPs with possibly pleiotropic effects increased significantly without compromising performance on non-pleiotropic SNPs.     

A Comparison of Multivariate Genome-Wide Association Methods

Joint association analysis of multiple traits in a genome-wide association study (GWAS), i.e. a multivariate GWAS, offers several advantages over analyzing each trait in a separate GWAS. In this study we directly compared a number of multivariate GWAS methods using simulated data. We focused on six methods that are implemented in the software packages PLINK, SNPTEST, MultiPhen, BIMBAM, PCHAT and TATES, and also compared them to standard univariate GWAS, analysis of the first principal component of the traits, and meta-analysis of univariate results. We simulated data (N = 1000) for three quantitative traits and one bi-allelic quantitative trait locus (QTL), and varied the number of traits associated with the QTL (explained variance 0.1%), minor allele frequency of the QTL, residual correlation between the traits, and the sign of the correlation induced by the QTL relative to the residual correlation. We compared the power of the methods using empirically fixed significance thresholds (α = 0.05). Our results showed that the multivariate methods implemented in PLINK, SNPTEST, MultiPhen and BIMBAM performed best for the majority of the tested scenarios, with a notable increase in power for scenarios with an opposite sign of genetic and residual correlation. All multivariate analyses resulted in a higher power than univariate analyses, even when only one of the traits was associated with the QTL. Hence, use of multivariate GWAS methods can be recommended, even when genetic correlations between traits are weak.     

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.     

数量遗传学中一种新的求综合性状的方法

本文运用申农(Shannon)提供的最大熵原理,提出一种构成单一综合性状的新方法,并以此与数量遗传学中的多元统计法作了比较。在作多元遗传分析吋,常用多元统计法求出多个数量性状的综合性状,再对这些相互关联的基本性状作主成份分析或用典范相关进行遗传分析。本文提出了不同于多元统计学的另一种新的方法——最大熵法求出多个数量性状的单一综合性状值。它具有数学结构简单,过程明晰,结果简明等优点。     

The genetic analysis of age-dependent traits: modeling the character process

The extension of classical quantitative genetics to deal with function-valued characters (also called infinite-dimensional characters) such as growth curves, mortality curves, and reaction norms, was begun by Kirkpatrick and co-workers. In this theory, the analogs of variance components for single traits are covariance functions for function-valued traits. In the approach presented here, we employ a variety of parametric models for covariance functions that have a number of desirable properties: the functions (1) are positive definite, (2) can be estimated using procedures like those currently used for single traits, (3) have a small number of parameters, and (4) allow simple hypotheses to be easily tested. The methods are illustrated using data from a large experiment that examined the effects of spontaneous mutations on age-specific mortality rates in Drosophila melanogaster. Our methods are shown to work better than a standard multivariate analysis, which assumes the character value at each age is a distinct character. Advantages over existing methods that model covariance functions as a series of orthogonal polynomials are discussed.     

Multivariate stabilizing selection and pleiotropy in the maintenance of quantitative genetic variation

Abstract. We investigate maintenance of quantitative genetic variation at mutation-selection balance for multiple traits. The intrinsic strength of real stabilizing selection on one of these traits denoted the "target trait" and the observed strength of apparent stabilizing selection on the target trait can be quite different: the latter, which is estimable, is much smaller (i.e., implying stronger selection) than the former. Distinguishing them may enable the mutation load to be relaxed when considering multivariate stabilizing selection. It is shown that both correlations among mutational effects and among strengths of real stabilizing selection on the traits are not important unless they are high. The analysis for independent situations thus provides a good approximation to the case where mutant and stabilizing selection effects are correlated. Multivariate stabilizing selection can be regarded as a combination of stabilizing selection on the target trait and the pleiotropic direct selection on fitness that is solely due to the effects of real stabilizing selection on the hidden traits. As the overall fitness approaches a constant value as the number of traits increases, multivariate stabilizing selection can maintain abundant genetic variance only under quite weak selection. The common observations of high polygenic variance and strong stabilizing selection thus imply that if the mutation-selection balance is the true mechanism of maintenance of genetic variation, the apparent stabilizing selection cannot arise solely by real stabilizing selection simultaneously on many metric traits.     

SELECTION AND EVOLUTION OF CAUSALLY COVARYING TRAITS

When traits cause variation in fitness, the distribution of phenotype, weighted by fitness, necessarily changes. The degree to which traits cause fitness variation is therefore of central importance to evolutionary biology. Multivariate selection gradients are the main quantity used to describe components of trait‐fitness covariation, but they quantify the direct effects of traits on (relative) fitness, which are not necessarily the total effects of traits on fitness. Despite considerable use in evolutionary ecology, path analytic characterizations of the total effects of traits on fitness have not been formally incorporated into quantitative genetic theory. By formally defining “extended” selection gradients, which are the total effects of traits on fitness, as opposed to the existing definition of selection gradients, a more intuitive scheme for characterizing selection is obtained. Extended selection gradients are distinct quantities, differing from the standard definition of selection gradients not only in the statistical means by which they may be assessed and the assumptions required for their estimation from observational data, but also in their fundamental biological meaning. Like direct selection gradients, extended selection gradients can be combined with genetic inference of multivariate phenotypic variation to provide quantitative prediction of microevolutionary trajectories.     

Statistical Power to Detect Genetic (Co)Variance of Complex Traits Using SNP Data in Unrelated Samples

We have recently developed analysis methods (GREML) to estimate the genetic variance of a complex trait/disease and the genetic correlation between two complex traits/diseases using genome-wide single nucleotide polymorphism (SNP) data in unrelated individuals. Here we use analytical derivations and simulations to quantify the sampling variance of the estimate of the proportion of phenotypic variance captured by all SNPs for quantitative traits and case-control studies. We also derive the approximate sampling variance of the estimate of a genetic correlation in a bivariate analysis, when two complex traits are either measured on the same or different individuals. We show that the sampling variance is inversely proportional to the number of pairwise contrasts in the analysis and to the variance in SNP-derived genetic relationships. For bivariate analysis, the sampling variance of the genetic correlation additionally depends on the harmonic mean of the proportion of variance explained by the SNPs for the two traits and the genetic correlation between the traits, and depends on the phenotypic correlation when the traits are measured on the same individuals. We provide an online tool for calculating the power of detecting genetic (co)variation using genome-wide SNP data. The new theory and online tool will be helpful to plan experimental designs to estimate the missing heritability that has not yet been fully revealed through genome-wide association studies, and to estimate the genetic overlap between complex traits (diseases) in particular when the traits (diseases) are not measured on the same samples.     

Quantitative trait linkage analysis using Gaussian copulas

Mapping and identifying variants that influence quantitative traits is an important problem for genetic studies. Traditional QTL mapping relies on a variance-components (VC) approach with the key assumption that the trait values in a family follow a multivariate normal distribution. Violation of this assumption can lead to inflated type I error, reduced power, and biased parameter estimates. To accommodate nonnormally distributed data, we developed and implemented a modified VC method, which we call the "copula VC method," that directly models the nonnormal distribution using Gaussian copulas. The copula VC method allows the analysis of continuous, discrete, and censored trait data, and the standard VC method is a special case when the data are distributed as multivariate normal. Through the use of link functions, the copula VC method can easily incorporate covariates. We use computer simulations to show that the proposed method yields unbiased parameter estimates, correct type I error rates, and improved power for testing linkage with a variety of nonnormal traits as compared with the standard VC and the regression-based methods.     

An evolutionary quantitative genetics model for phenotypic (co)variances under limited dispersal,with an application to socially synergistic traits

Darwinian evolution consists of the gradual transformation of heritable traits due to natural selection and the input of random variation by mutation. Here, we use a quantitative genetics approach to investigate the coevolution of multiple quantitative traits under selection, mutation, and limited dispersal. We track the dynamics of trait means and of variance–covariances between traits that experience frequency‐dependent selection. Assuming a multivariate‐normal trait distribution, we recover classical dynamics of quantitative genetics, as well as stability and evolutionary branching conditions of invasion analyses, except that due to limited dispersal, selection depends on indirect fitness effects and relatedness. In particular, correlational selection that associates different traits within‐individuals depends on the fitness effects of such associations between‐individuals. We find that these kin selection effects can be as relevant as pleiotropy for the evolution of correlation between traits. We illustrate this with an example of the coevolution of two social traits whose association within‐individuals is costly but synergistically beneficial between‐individuals. As dispersal becomes limited and relatedness increases, associations between‐traits between‐individuals become increasingly targeted by correlational selection. Consequently, the trait distribution goes from being bimodal with a negative correlation under panmixia to unimodal with a positive correlation under limited dispersal.     

When is dispersal for dispersal? Unifying marine and terrestrial perspectives

Recent syntheses on the evolutionary causes of dispersal have focused on dispersal as a direct adaptation, but many traits that influence dispersal have other functions, raising the question: when is dispersal ‘for’ dispersal? We review and critically evaluate the ecological causes of selection on traits that give rise to dispersal in marine and terrestrial organisms. In the sea, passive dispersal is relatively easy and specific morphological, behavioural, and physiological adaptations for dispersal are rare. Instead, there may often be selection to limit dispersal. On land, dispersal is relatively difficult without specific adaptations, which are relatively common. Although selection for dispersal is expected in both systems and traits leading to dispersal are often linked to fitness, systems may differ in the extent to which dispersal in nature arises from direct selection for dispersal or as a by‐product of selection on traits with other functions. Our analysis highlights incompleteness of theories that assume a simple and direct relationship between dispersal and fitness, not just insofar as they ignore a vast array of taxa in the marine realm, but also because they may be missing critically important effects of traits influencing dispersal in all realms.     

Multiple-Trait Genomic Selection Methods Increase Genetic Value Prediction Accuracy

Genetic correlations between quantitative traits measured in many breeding programs are pervasive. These correlations indicate that measurements of one trait carry information on other traits. Current single-trait (univariate) genomic selection does not take advantage of this information. Multivariate genomic selection on multiple traits could accomplish this but has been little explored and tested in practical breeding programs. In this study, three multivariate linear models (i.e., GBLUP, BayesA, and BayesCπ) were presented and compared to univariate models using simulated and real quantitative traits controlled by different genetic architectures. We also extended BayesA with fixed hyperparameters to a full hierarchical model that estimated hyperparameters and BayesCπ to impute missing phenotypes. We found that optimal marker-effect variance priors depended on the genetic architecture of the trait so that estimating them was beneficial. We showed that the prediction accuracy for a low-heritability trait could be significantly increased by multivariate genomic selection when a correlated high-heritability trait was available. Further, multiple-trait genomic selection had higher prediction accuracy than single-trait genomic selection when phenotypes are not available on all individuals and traits. Additional factors affecting the performance of multiple-trait genomic selection were explored.     

Enhanced efficiency of quantitative trait loci mapping analysis based on multivariate complexes of quantitative traits

An approach to increase the efficiency of mapping quantitative trait loci (QTL) was proposed earlier by the authors on the basis of bivariate analysis of correlated traits. The power of QTL detection using the log-likelihood ratio (LOD scores) grows proportionally to the broad sense heritability. We found that this relationship holds also for correlated traits, so that an increased bivariate heritability implicates a higher LOD score, higher detection power, and better mapping resolution. However, the increased number of parameters to be estimated complicates the application of this approach when a large number of traits are considered simultaneously. Here we present a multivariate generalization of our previous two-trait QTL analysis. The proposed multivariate analogue of QTL contribution to the broad-sense heritability based on interval-specific calculation of eigenvalues and eigenvectors of the residual covariance matrix allows prediction of the expected QTL detection power and mapping resolution for any subset of the initial multivariate trait complex. Permutation technique allows chromosome-wise testing of significance for the whole trait complex and the significance of the contribution of individual traits owing to: (a) their correlation with other traits, (b) dependence on the chromosome in question, and (c) both a and b. An example of application of the proposed method on a real data set of 11 traits from an experiment performed on an F(2)/F(3) mapping population of tetraploid wheat (Triticum durum x T. dicoccoides) is provided.     

Testing for biases in selection on avian reproductive traits and partitioning direct and indirect selection using quantitative genetic models

Key life history traits such as breeding time and clutch size are frequently both heritable and under directional selection, yet many studies fail to document microevolutionary responses. One general explanation is that selection estimates are biased by the omission of correlated traits that have causal effects on fitness, but few valid tests of this exist. Here, we show, using a quantitative genetic framework and six decades of life‐history data on two free‐living populations of great tits Parus major, that selection estimates for egg‐laying date and clutch size are relatively unbiased. Predicted responses to selection based on the Robertson–Price Identity were similar to those based on the multivariate breeder's equation (MVBE), indicating that unmeasured covarying traits were not missing from the analysis. Changing patterns of phenotypic selection on these traits (for laying date, linked to climate change) therefore reflect changing selection on breeding values, and genetic constraints appear not to limit their independent evolution. Quantitative genetic analysis of correlational data from pedigreed populations can be a valuable complement to experimental approaches to help identify whether apparent associations between traits and fitness are biased by missing traits, and to parse the roles of direct versus indirect selection across a range of environments.     

Quantitative trait locus analysis of longitudinal quantitative trait data in complex pedigrees

There is currently considerable interest in genetic analysis of quantitative traits such as blood pressure and body mass index. Despite the fact that these traits change throughout life they are commonly analyzed only at a single time point. The genetic basis of such traits can be better understood by collecting and effectively analyzing longitudinal data. Analyses of these data are complicated by the need to incorporate information from complex pedigree structures and genetic markers. We propose conducting longitudinal quantitative trait locus (QTL) analyses on such data sets by using a flexible random regression estimation technique. The relationship between genetic effects at different ages is efficiently modeled using covariance functions (CFs). Using simulated data we show that the change in genetic effects over time can be well characterized using CFs and that including parameters to model the change in effect with age can provide substantial increases in power to detect QTL compared with repeated measure or univariate techniques. The asymptotic distributions of the methods used are investigated and methods for overcoming the practical difficulties in fitting CFs are discussed. The CF-based techniques should allow efficient multivariate analyses of many data sets in human and natural population genetics.     

Selecting Informative Traits for Multivariate Quantitative Trait Locus Mapping Helps to Gain Optimal Power

A major consideration in multitrait analysis is which traits should be jointly analyzed. As a common strategy, multitrait analysis is performed either on pairs of traits or on all of traits. To fully exploit the power of multitrait analysis, we propose variable selection to choose a subset of informative traits for multitrait quantitative trait locus (QTL) mapping. The proposed method is very useful for achieving optimal statistical power for QTL identification and for disclosing the most relevant traits. It is also a practical strategy to effectively take advantage of multitrait analysis when the number of traits under consideration is too large, making the usual multivariate analysis of all traits challenging. We study the impact of selection bias and the usage of permutation tests in the context of variable selection and develop a powerful implementation procedure of variable selection for genome scanning. We demonstrate the proposed method and selection procedure in a backcross population, using both simulated and real data. The extension to other experimental mapping populations is straightforward.     

Smoothing of the bivariate LOD score for non-normal quantitative traits

Variance component analysis provides an efficient method for performing linkage analysis for quantitative traits. However, type I error of variance components-based likelihood ratio testing may be affected when phenotypic data are non-normally distributed (especially with high values of kurtosis). This results in inflated LOD scores when the normality assumption does not hold. Even though different solutions have been proposed to deal with this problem with univariate phenotypes, little work has been done in the multivariate case. We present an empirical approach to adjust the inflated LOD scores obtained from a bivariate phenotype that violates the assumption of normality. Using the Collaborative Study on the Genetics of Alcoholism data available for the Genetic Analysis Workshop 14, we show how bivariate linkage analysis with leptokurtotic traits gives an inflated type I error. We perform a novel correction that achieves acceptable levels of type I error.     

Multivariate heredity of melanin-based coloration,body mass and immunity

The genetic covariation among different traits may cause the appearance of correlated response to selection on multivariate phenotypes. Genes responsible for the expression of melanin-based color traits are also involved in other important physiological functions such as immunity and metabolism by pleiotropy, suggesting the possibility of multivariate evolution. However, little is known about the relationship between melanin coloration and these functions at the additive genetic level in wild vertebrates. From a multivariate perspective, we simultaneously explored inheritance and selection of melanin coloration, body mass and phytohemagglutinin (PHA)-mediated immune response by using long-term data over an 18-year period collected in a wild population of the common kestrel Falco tinnunculus. Pedigree-based quantitative genetic analyses showed negative genetic covariance between melanin-based coloration and body mass in male adults and positive genetic covariance between body mass and PHA-mediated immune response in fledglings as predicted by pleiotropic effects of melanocortin receptor activity. Multiple selection analyses showed an increased fitness in male adults with intermediate phenotypic values for melanin color and body mass. In male fledglings, there was evidence for a disruptive selection on rump gray color, but a stabilizing selection on PHA-mediated immune response. Our results provide an insight into the evolution of multivariate traits genetically related with melanin-based coloration. The differences in multivariate inheritance and selection between male and female kestrels might have resulted in sexual dimorphism in size and color. When pleiotropic effects are present, coloration can evolve through a complex pathway involving correlated response to selection on multivariate traits.