Bayesian mapping of quantitative trait loci for multiple complex traits with the use of variance components

Complex traits important for humans are often correlated phenotypically and genetically. Joint mapping of quantitative-trait loci (QTLs) for multiple correlated traits plays an important role in unraveling the genetic architecture of complex traits. Compared with single-trait analysis, joint mapping addresses more questions and has advantages for power of QTL detection and precision of parameter estimation. Some statistical methods have been developed to map QTLs underlying multiple traits, most of which are based on maximum-likelihood methods. We develop here a multivariate version of the Bayes methodology for joint mapping of QTLs, using the Markov chain-Monte Carlo (MCMC) algorithm. We adopt a variance-components method to model complex traits in outbred populations (e.g., humans). The method is robust, can deal with an arbitrary number of alleles with arbitrary patterns of gene actions (such as additive and dominant), and allows for multiple phenotype data of various types in the joint analysis (e.g., multiple continuous traits and mixtures of continuous traits and discrete traits). Under a Bayesian framework, parameters--including the number of QTLs--are estimated on the basis of their marginal posterior samples, which are generated through two samplers, the Gibbs sampler and the reversible-jump MCMC. In addition, we calculate the Bayes factor related to each identified QTL, to test coincident linkage versus pleiotropy. The performance of our method is evaluated in simulations with full-sib families. The results show that our proposed Bayesian joint-mapping method performs well for mapping multiple QTLs in situations of either bivariate continuous traits or mixed data types. Compared with the analysis for each trait separately, Bayesian joint mapping improves statistical power, provides stronger evidence of QTL detection, and increases precision in estimation of parameter and QTL position. We also applied the proposed method to a set of real data and detected a coincident linkage responsible for determining bone mineral density and areal bone size of wrist in humans.     

Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.     

Bayesian joint mapping of quantitative trait loci for Gaussian and categorical characters in line crosses

Methodology for joint mapping of quantitative trait loci (QTL) affecting continuous and binary characters in experimental crosses is presented. The procedure consists of a Bayesian Gaussian-threshold model implemented via Markov chain Monte Carlo, which bypasses bottlenecks due to high-dimensional integrals required in maximum likelihood approaches. The method handles multiple binary traits and multiple QTL. Modeling of ordered categorical traits is discussed as well. Features of the method are illustrated using simulated datasets representing a backcross design, and the data are analyzed using mixed-trait and single-trait models. The mixed-trait analysis provides greater detection power of a QTL than a single-trait analysis when the QTL affects two or more traits. The number of QTL inferred in the mixed-trait analysis does not pertain to a specific trait, but the roles of each QTL on specific traits can be assessed from estimates of its effects. The impacts of varying incidence level and sample size on the mixed-trait QTL mapping analysis are investigated as well.     

Mapping Quantitative Trait Loci for Complex Binary Diseases Using Line Crosses

A composite interval gene mapping procedure for complex binary disease traits is proposed in this paper. The binary trait of interest is assumed to be controlled by an underlying liability that is normally distributed. The liability is treated as a typical quantitative character and thus described by the usual quantitative genetics model. Translation from the liability into a binary (disease) phenotype is through the physiological threshold model. Logistic regression analysis is employed to estimate the effects and locations of putative quantitative trait loci (our terminology for a single quantitative trait locus is QTL while multiple loci are referred to as QTLs). Simulation studies show that properties of this mapping procedure mimic those of the composite interval mapping for normally distributed data. Potential utilization of the QTL mapping procedure for resolving alternative genetic models (e.g., single- or two-trait-locus model) is discussed.     

Genetic linkage analysis of a dichotomous trait incorporating a tightly linked quantitative trait in affected sib pairs

Many complex diseases are usually considered as dichotomous traits but are also associated with quantitative biological markers or quantitative risk factors. For such dichotomous traits, although their associated quantitative traits may not directly underly the diagnosis of the disease status, if the associated quantitative trait is also linked to the chromosomal regions linked to the dichotomous trait, then joint analysis of dichotomous and quantitative traits should be more efficient than consideration of them separately. Previous studies have focused on the situation when a dichotomous trait can be modeled by a threshold process acting on a single underlying normal liability distribution. However, for many complex disorders, including most psychiatric disorders, diagnosis is generally based on a set of binary or discrete criteria. These traits cannot be modeled on the basis of a threshold process acting on an underlying continuous trait. We propose a likelihood-based method that efficiently combines such a discrete trait and an associated quantitative trait in the analysis, using affected-sib-pair data. Our simulation studies suggest that joint analysis increases the power to detect linkage of dichotomous traits. We also apply the proposed new method to an asthma genome-scan data set and incorporate the total serum immunoglobulin E level in the analysis.     

An EM algorithm for mapping quantitative resistance loci

Many disease resistance traits in plants have a polygenic background and the disease phenotypes are modified by environmental factors. As a consequence, the phenotypic values usually show a quantitative variation. The phenotypes of such disease traits, however, are often measured in discrete but ordered categories. These traits are called ordinal traits. In terms of disease resistance, they are called quantitative resistance traits, as opposed to qualitative resistance traits, and are controlled by the quantitative resistance loci (QRL). Classical quantitative trait locus mapping methods are not optimal for ordinal trait analysis because the assumption of normal distribution is violated. Methods for mapping binary trait loci are not suitable either because there are more than two categories in ordinal traits. We developed a maximum likelihood method to map these QRL. The method is implemented via a multicycle expectation-conditional-maximization (ECM) algorithm under the threshold model, where we can estimate both the QRL effects and the thresholds that link the disease liability and the categorical phenotype. The method is verified in simulated data under various combinations of the parameters. An SAS program is available to implement the multicycle ECM algorithm. The program can be downloaded from our website at www.statgen.ucr.edu.     

Mapping quantitative trait loci with epistatic effects

Epistatic variance can be an important source of variation for complex traits. However, detecting epistatic effects is difficult primarily due to insufficient sample sizes and lack of robust statistical methods. In this paper, we develop a Bayesian method to map multiple quantitative trait loci (QTLs) with epistatic effects. The method can map QTLs in complicated mating designs derived from the cross of two inbred lines. In addition to mapping QTLs for quantitative traits, the proposed method can even map genes underlying binary traits such as disease susceptibility using the threshold model. The parameters of interest are various QTL effects, including additive, dominance and epistatic effects of QTLs, the locations of identified QTLs and even the number of QTLs. When the number of QTLs is treated as an unknown parameter, the dimension of the model becomes a variable. This requires the reversible jump Markov chain Monte Carlo algorithm. The utility of the proposed method is demonstrated through analysis of simulation data.     

Multivariate segregation analysis for quantitative traits in line crosses

Segregation analysis is a method of detecting major genes for quantitative traits without using marker information. It serves as an important tool in helping investigators to plan further studies such as quantitative trait loci mapping or more sophisticated genomic analyses. However, current methods of segregation analysis for a single trait typically have low statistical power. We propose a multivariate segregation analysis (MSA) that takes advantage of the correlation structure of multiple quantitative traits to detect major genes. This method not only increases the statistical power, but allows dissection of the genetic architecture underlying the trait complex. In MSA the observed phenotypes of multiple correlated traits are fitted to a multivariate Gaussian mixture model. Model parameters are estimated under the maximum likelihood framework via the expectation-maximization algorithm. The presence of major genes is tested using likelihood ratio test statistics. Pleiotropy is distinguished from close linkage by comparing three possible models using the Bayesian information criterion. Two simulation experiments were performed based on the F(2) mating design. In the first, the statistical properties of MSA under varying heritabilities and sample sizes were investigated and the results compared with those obtained from single-trait analysis. In the second simulation the efficacy of MSA in separating pleiotropy from close linkage was demonstrated. Finally, the new method was applied to real data and detected a major gene responsible for both plant height and tiller number in rice.     

Joint Multipoint Linkage Analysis of Multivariate Qualitative and Quantitative Traits. II. Alcoholism and Event-Related Potentials

The availability of robust quantitative biological markers that are correlated with qualitative psychiatric phenotypes can potentially improve the power of linkage methods to detect quantitative-trait loci influencing psychiatric disorders. We apply a variance-component method for joint multipoint linkage analysis of multivariate discrete and continuous traits to the extended pedigree data from the Collaborative Study on the Genetics of Alcoholism, in a bivariate analysis of qualitative alcoholism phenotypes and quantitative event-related potentials. Joint consideration of the DSM-IV diagnosis of alcoholism and the amplitude of the P300 component of the Cz event-related potential significantly increases the evidence for linkage of these traits to a chromosome 4 region near the class I alcohol dehydrogenase locus ADH3. A likelihood-ratio test for complete pleiotropy is significant, suggesting that the same quantitative-trait locus influences both risk of alcoholism and the amplitude of the P300 component.     

Generalized Linear Model for Mapping Discrete Trait Loci Implemented with LASSO Algorithm

Generalized estimating equation (GEE) algorithm under a heterogeneous residual variance model is an extension of the iteratively reweighted least squares (IRLS) method for continuous traits to discrete traits. In contrast to mixture model-based expectation–maximization (EM) algorithm, the GEE algorithm can well detect quantitative trait locus (QTL), especially large effect QTLs located in large marker intervals in the manner of high computing speed. Based on a single QTL model, however, the GEE algorithm has very limited statistical power to detect multiple QTLs because of ignoring other linked QTLs. In this study, the fast least absolute shrinkage and selection operator (LASSO) is derived for generalized linear model (GLM) with all possible link functions. Under a heterogeneous residual variance model, the LASSO for GLM is used to iteratively estimate the non-zero genetic effects of those loci over entire genome. The iteratively reweighted LASSO is therefore extended to mapping QTL for discrete traits, such as ordinal, binary, and Poisson traits. The simulated and real data analyses are conducted to demonstrate the efficiency of the proposed method to simultaneously identify multiple QTLs for binary and Poisson traits as examples.     

Multiple-interval mapping for ordinal traits

Many statistical methods have been developed to map multiple quantitative trait loci (QTL) in experimental cross populations. Among these methods, multiple-interval mapping (MIM) can map QTL with epistasis simultaneously. However, the previous implementation of MIM is for continuously distributed traits. In this study we extend MIM to ordinal traits on the basis of a threshold model. The method inherits the properties and advantages of MIM and can fit a model of multiple QTL effects and epistasis on the underlying liability score. We study a number of statistical issues associated with the method, such as the efficiency and stability of maximization and model selection. We also use computer simulation to study the performance of the method and compare it to other alternative approaches. The method has been implemented in QTL Cartographer to facilitate its general usage for QTL mapping data analysis on binary and ordinal traits.     

Multiple Trait Analysis of Genetic Mapping for Quantitative Trait Loci

We present in this paper models and statistical methods for performing multiple trait analysis on mapping quantitative trait loci (QTL) based on the composite interval mapping method. By taking into account the correlated structure of multiple traits, this joint analysis has several advantages, compared with separate analyses, for mapping QTL, including the expected improvement on the statistical power of the test for QTL and on the precision of parameter estimation. Also this joint analysis provides formal procedures to test a number of biologically interesting hypotheses concerning the nature of genetic correlations between different traits. Among the testing procedures considered are those for joint mapping, pleiotropy, QTL by environment interaction, and pleiotropy vs. close linkage. The test of pleiotropy (one pleiotropic QTL at a genome position) vs. close linkage (multiple nearby nonpleiotropic QTL) can have important implications for our understanding of the nature of genetic correlations between different traits in certain regions of a genome and also for practical applications in animal and plant breeding because one of the major goals in breeding is to break unfavorable linkage. Results of extensive simulation studies are presented to illustrate various properties of the analyses.     

Joint Multipoint Linkage Analysis of Multivariate Qualitative and Quantitative Traits. I. Likelihood Formulation and Simulation Results

We describe a variance-components method for multipoint linkage analysis that allows joint consideration of a discrete trait and a correlated continuous biological marker (e.g., a disease precursor or associated risk factor) in pedigrees of arbitrary size and complexity. The continuous trait is assumed to be multivariate normally distributed within pedigrees, and the discrete trait is modeled by a threshold process acting on an underlying multivariate normal liability distribution. The liability is allowed to be correlated with the quantitative trait, and the liability and quantitative phenotype may each include covariate effects. Bivariate discrete-continuous observations will be common, but the method easily accommodates qualitative and quantitative phenotypes that are themselves multivariate. Formal likelihood-based tests are described for coincident linkage (i.e., linkage of the traits to distinct quantitative-trait loci [QTLs] that happen to be linked) and pleiotropy (i.e., the same QTL influences both discrete-trait status and the correlated continuous phenotype). The properties of the method are demonstrated by use of simulated data from Genetic Analysis Workshop 10. In a companion paper, the method is applied to data from the Collaborative Study on the Genetics of Alcoholism, in a bivariate linkage analysis of alcoholism diagnoses and P300 amplitude of event-related brain potentials.     

Bayesian quantitative trait loci mapping for multiple traits

Most quantitative trait loci (QTL) mapping experiments typically collect phenotypic data on multiple correlated complex traits. However, there is a lack of a comprehensive genomewide mapping strategy for correlated traits in the literature. We develop Bayesian multiple-QTL mapping methods for correlated continuous traits using two multivariate models: one that assumes the same genetic model for all traits, the traditional multivariate model, and the other known as the seemingly unrelated regression (SUR) model that allows different genetic models for different traits. We develop computationally efficient Markov chain Monte Carlo (MCMC) algorithms for performing joint analysis. We conduct extensive simulation studies to assess the performance of the proposed methods and to compare with the conventional single-trait model. Our methods have been implemented in the freely available package R/qtlbim (http://www.qtlbim.org), which greatly facilitates the general usage of the Bayesian methodology for unraveling the genetic architecture of complex traits.     

Mixture generalized linear models for multiple interval mapping of quantitative trait Loci in experimental crosses

Summary .  Quantitative trait loci mapping in experimental organisms is of great scientific and economic importance. There has been a rapid advancement in statistical methods for quantitative trait loci mapping. Various methods for normally distributed traits have been well established. Some of them have also been adapted for other types of traits such as binary, count, and categorical traits. In this article, we consider a unified mixture generalized linear model (GLIM) for multiple interval mapping in experimental crosses. The multiple interval mapping approach was proposed by Kao, Zeng, and Teasdale (1999, Genetics 152, 1203–1216) for normally distributed traits. However, its application to nonnormally distributed traits has been hindered largely by the lack of an efficient computation algorithm and an appropriate mapping procedure. In this article, an effective expectation–maximization algorithm for the computation of the mixture GLIM and an epistasis-effect-adjusted multiple interval mapping procedure is developed. A real data set, Radiata Pine data, is analyzed and the data structure is used in simulation studies to demonstrate the desirable features of the developed method.     

Mapping Quantitative Trait Loci onto a Phylogenetic Tree

Despite advances in genetic mapping of quantitative traits and in phylogenetic comparative approaches, these two perspectives are rarely combined. The joint consideration of multiple crosses among related taxa (whether species or strains) not only allows more precise mapping of the genetic loci (called quantitative trait loci, QTL) that contribute to important quantitative traits, but also offers the opportunity to identify the origin of a QTL allele on the phylogenetic tree that relates the taxa. We describe a formal method for combining multiple crosses to infer the location of a QTL on a tree. We further discuss experimental design issues for such endeavors, such as how many crosses are required and which sets of crosses are best. Finally, we explore the method's performance in computer simulations, and we illustrate its use through application to a set of four mouse intercrosses among five inbred strains, with data on HDL cholesterol.     

QTL mapping of complex binary traits in an advanced intercross line

An advanced intercross line (AIL) is an easier and more cost-effective approach compared to recombinant inbred lines for fine mapping of quantitative trait loci (QTL) identified by F(2) designs. In an AIL, a complex binary trait can be mapped through analysis of either continuously distributed proxy traits for the liability of the binary trait or the liability itself, the latter presenting the greater statistical challenge. In another work, we successfully applied both approaches in an AIL to fine map previously identified QTL underlying anatomical parameters of the cardiac inter-atrial septum including patent foramen ovale. Here, we describe the statistical methods that we used to analyse complex binary traits in our AIL design. This is achieved using a likelihood-based method, with the expectation-maximisation algorithm allowing use of standard logistic regression methods for model fitting.     

Gene mapping and marker clustering using Shannon's mutual information

Finding the causal genetic regions underlying complex traits is one of the main aims in human genetics. In the context of complex diseases, which are believed to be controlled by multiple contributing loci of largely unknown effect and position, it is especially important to develop general yet sensitive methods for gene mapping. We discuss the use of Shannon's information theory for population-based gene mapping of discrete and quantitative traits and for marker clustering. Various measures of mutual information were employed in order to develop a comprehensive framework for gene mapping analyses. An algorithm aimed at finding so-called relevance chains of causal markers is proposed. Moreover, entropy measures are used in conjunction with multidimensional scaling to visualize clusters of genetic markers. The relevance chain algorithm successfully detected the two causal regions in a simulated scenario. The approach has also been applied to a published clinical study on autoimmune (Graves') disease. Results were consistent with those of standard statistical methods, but identified an additional locus of interest in the promoter region of the associated gene CTLA4. The developed software is freely available at http://www.lnt.ei.tum.de/download/InfoGeneMap/.     

Dimension reduction for mapping mRNA abundance as quantitative traits

The advent of sophisticated genomic techniques for gene mapping and microarray analysis has provided opportunities to map mRNA abundance to quantitative trait loci (QTL) throughout the genome. Unfortunately, simple mapping of each individual mRNA trait on the scale of a typical microarray experiment is computationally intensive, subject to high sample variance, and therefore underpowered. However, this problem can be addressed by capitalizing on correlation among the large number of mRNA traits. We present a method to reduce the dimensionality for mapping gene expression data as quantitative traits. We used a blind method, principal components, and a sighted method, hierarchical clustering seeded by disease relevant traits, to define new traits composed of a small collection of promising mRNAs. We validated the principle of our approach by mapping the expression levels of metabolism genes in a population of F(2)-ob/ob mice derived from the BTBR and C57BL/6J strains. We found that lipogenic and gluconeogenic mRNAs, which are known targets of insulin action, were closely associated with the insulin trait. Multiple interval mapping and Bayesian interval mapping of this new trait revealed significant linkages to chromosome regions that were contained in loci associated with type 2 diabetes in this same mouse sample. As a further statistical refinement, we show that principal component analysis also effectively reduced dimensions for mapping phenotypes composed of mRNA abundances.