Shrinkage Estimation Method for Mapping Multiple Quantitative Trait Loci

In this article, shrinkage estimation method for multiple-marker analysis and for mapping multiple quantitative trait loci (QTL) was reviewed. For multiple-marker analysis, Xu (Genetics, 2003, 163:789-801) developed a Bayesian shrinkage estimation (BSE) method. The key to the success of this method is to allow each marker effect have its own variance parameter, which in turn has its own prior distribution so that the variance can be estimated from the data. Under this hierarchical model, a large number of markers can be handled although most of them may have negligible effects. Under epistatic genetic model, however, the running time is very long. To overcome this problem, a novel method of incorporating the idea described above into maximum likelihood, known as penalized likelihood method, was proposed. A simulated study showed that this method can handle a model with multiple effects, which are ten times larger than the sample size. For multiple QTL analysis, two modified versions for the BSE method were introduced: one is the fixed-interval method and another is the variable-interval method. The former deals with markers with intermediate density, and the latter can handle markers with extremely high density as well as model with epistatic effects. For the detection of epistatic effects, penalized likelihood method and the variable-interval approach of the BSE method are available.     

多QTL定位的压缩估计方法

本文综述了多标记分析和多QTL定位的压缩估计方法。对于前者,Xu（Genetics,2003,163：789—801）首先提出了Bayesian压缩估计方法。其关键在于让每个效应有一个特定的方差参数,而该方差又服从一定的先验分布,以致能从资料中估计之。由此,能够同时估计大量分子标记基因座的遗传效应,即使大多数标记的效应是可忽略的。然而,对于上位性遗传模型,其运算时间还是过长。为此,笔者将上述思想嵌入极大似然法,提出了惩罚最大似然方法。模拟研究显示：该方法能处理变量个数大于样本容量10倍左右的线性遗传模型。对于后者,本文详细介绍了基于固定区间和可变区间的Bayesian压缩估计方法。固定区间方法可处理中等密度的分子标记资料;可变区间方法则可分析高密度分子标记资料,甚至是上位性遗传模型。对于上位性检测,已介绍的惩罚最大似然方法和可变区间Bayesian压缩估计方法可供利用。应当指出,压缩估计方法在今后的eQTL和QTN定位以及基因互作网络分析等研究中也是有应用价值的。     

A penalized maximum likelihood method for estimating epistatic effects of QTL

Although epistasis is an important phenomenon in the genetics and evolution of complex traits, epistatic effects are hard to estimate. The main problem is due to the overparameterized epistatic genetic models. An epistatic genetic model should include potential pair-wise interaction effects of all loci. However, the model is saturated quickly as the number of loci increases. Therefore, a variable selection technique is usually considered to exclude those interactions with negligible effects. With such techniques, we may run a high risk of missing some important interaction effects by not fully exploring the extremely large parameter space of models. We develop a penalized maximum likelihood method. The method developed here adopts a penalty that depends on the values of the parameters. The penalized likelihood method allows spurious QTL effects to be shrunk towards zero, while QTL with large effects are estimated with virtually no shrinkage. A simulation study shows that the new method can handle a model with a number of effects 15 times larger than the sample size. Simulation studies also show that results of the penalized likelihood method are comparable to the Bayesian shrinkage analysis, but the computational speed of the penalized method is orders of magnitude faster.     

Estimation of quantitative trait locus effects with epistasis by variational Bayes algorithms

Bayesian hierarchical shrinkage methods have been widely used for quantitative trait locus mapping. From the computational perspective, the application of the Markov chain Monte Carlo (MCMC) method is not optimal for high-dimensional problems such as the ones arising in epistatic analysis. Maximum a posteriori (MAP) estimation can be a faster alternative, but it usually produces only point estimates without providing any measures of uncertainty (i.e., interval estimates). The variational Bayes method, stemming from the mean field theory in theoretical physics, is regarded as a compromise between MAP and MCMC estimation, which can be efficiently computed and produces the uncertainty measures of the estimates. Furthermore, variational Bayes methods can be regarded as the extension of traditional expectation-maximization (EM) algorithms and can be applied to a broader class of Bayesian models. Thus, the use of variational Bayes algorithms based on three hierarchical shrinkage models including Bayesian adaptive shrinkage, Bayesian LASSO, and extended Bayesian LASSO is proposed here. These methods performed generally well and were found to be highly competitive with their MCMC counterparts in our example analyses. The use of posterior credible intervals and permutation tests are considered for decision making between quantitative trait loci (QTL) and non-QTL. The performance of the presented models is also compared with R/qtlbim and R/BhGLM packages, using a previously studied simulated public epistatic data set.     

Simultaneous fine mapping of closely linked epistatic quantitative trait loci using combined linkage disequilibrium and linkage with a general pedigree

Causal mutations and their intra- and inter-locus interactions play a critical role in complex trait variation. It is often not easy to detect epistatic quantitative trait loci (QTL) due to complicated population structure requirements for detecting epistatic effects in linkage analysis studies and due to main effects often being hidden by interaction effects. Mapping their positions is even harder when they are closely linked. The data structure requirement may be overcome when information on linkage disequilibrium is used. We present an approach using a mixed linear model nested in an empirical Bayesian approach, which simultaneously takes into account additive, dominance and epistatic effects due to multiple QTL. The covariance structure used in the mixed linear model is based on combined linkage disequilibrium and linkage information. In a simulation study where there are complex epistatic interactions between QTL, it is possible to simultaneously map interacting QTL into a small region using the proposed approach. The estimated variance components are accurate and less biased with the proposed approach compared with traditional models.     

An empirical Bayes method for estimating epistatic effects of quantitative trait loci

Summary .   The genetic variance of a quantitative trait is often controlled by the segregation of multiple interacting loci. Linear model regression analysis is usually applied to estimating and testing effects of these quantitative trait loci (QTL). Including all the main effects and the effects of interaction (epistatic effects), the dimension of the linear model can be extremely high. Variable selection via stepwise regression or stochastic search variable selection (SSVS) is the common procedure for epistatic effect QTL analysis. These methods are computationally intensive, yet they may not be optimal. The LASSO (least absolute shrinkage and selection operator) method is computationally more efficient than the above methods. As a result, it has been widely used in regression analysis for large models. However, LASSO has never been applied to genetic mapping for epistatic QTL, where the number of model effects is typically many times larger than the sample size. In this study, we developed an empirical Bayes method (E-BAYES) to map epistatic QTL under the mixed model framework. We also tested the feasibility of using LASSO to estimate epistatic effects, examined the fully Bayesian SSVS, and reevaluated the penalized likelihood (PENAL) methods in mapping epistatic QTL. Simulation studies showed that all the above methods performed satisfactorily well. However, E-BAYES appears to outperform all other methods in terms of minimizing the mean-squared error (MSE) with relatively short computing time. Application of the new method to real data was demonstrated using a barley dataset.     

Bayesian analyses of multiple epistatic QTL models for body weight and body composition in mice

To comprehensively investigate the genetic architecture of growth and obesity, we performed Bayesian analyses of multiple epistatic quantitative trait locus (QTL) models for body weights at five ages (12 days, 3, 6, 9 and 12 weeks) and body composition traits (weights of two fat pads and five organs) in mice produced from a cross of the F1 between M16i (selected for rapid growth rate) and CAST/Ei (wild-derived strain of small and lean mice) back to M16i. Bayesian model selection revealed a temporally regulated network of multiple QTL for body weight, involving both strong main effects and epistatic effects. No QTL had strong support for both early and late growth, although overlapping combinations of main and epistatic effects were observed at adjacent ages. Most main effects and epistatic interactions had an opposite effect on early and late growth. The contribution of epistasis was more pronounced for body weights at older ages. Body composition traits were also influenced by an interacting network of multiple QTLs. Several main and epistatic effects were shared by the body composition and body weight traits, suggesting that pleiotropy plays an important role in growth and obesity.     

Bayesian model selection for genome-wide epistatic quantitative trait loci analysis

The problem of identifying complex epistatic quantitative trait loci (QTL) across the entire genome continues to be a formidable challenge for geneticists. The complexity of genome-wide epistatic analysis results mainly from the number of QTL being unknown and the number of possible epistatic effects being huge. In this article, we use a composite model space approach to develop a Bayesian model selection framework for identifying epistatic QTL for complex traits in experimental crosses from two inbred lines. By placing a liberal constraint on the upper bound of the number of detectable QTL we restrict attention to models of fixed dimension, greatly simplifying calculations. Indicators specify which main and epistatic effects of putative QTL are included. We detail how to use prior knowledge to bound the number of detectable QTL and to specify prior distributions for indicators of genetic effects. We develop a computationally efficient Markov chain Monte Carlo (MCMC) algorithm using the Gibbs sampler and Metropolis-Hastings algorithm to explore the posterior distribution. We illustrate the proposed method by detecting new epistatic QTL for obesity in a backcross of CAST/Ei mice onto M16i.     

A complete solution for dissecting pure main and epistatic effects of QTL in triple testcross design

Epistasis plays an important role in genetics, evolution and crop breeding. To detect the epistasis, triple test cross (TTC) design had been developed several decades ago. Classical procedures for the TTC design use only linear transformations Z(1), Z(2) and Z(3), calculated from the TTC family means of quantitative trait, to infer the nature of the collective additive, dominance and epistatic effects of all the genes. Although several quantitative trait loci (QTL) mapping approaches in the TTC design have been developed, these approaches do not provide a complete solution for dissecting pure main and epistatic effects. In this study, therefore, we developed a two-step approach to estimate all pure main and epistatic effects in the F(2)-based TTC design under the F(2) and F(∞) metric models. In the first step, with Z(1) and Z(2) the augmented main and epistatic effects in the full genetic model that simultaneously considered all putative QTL on the whole genome were estimated using empirical Bayes approach, and with Z(3) three pure epistatic effects were obtained using two-dimensional genome scans. In the second step, the three pure epistatic effects obtained in the first step were integrated with the augmented epistatic and main effects for the further estimation of all other pure effects. A series of Monte Carlo simulation experiments has been carried out to confirm the proposed method. The results from simulation experiments show that: 1) the newly defined genetic parameters could be rightly identified with satisfactory statistical power and precision; 2) the F(2)-based TTC design was superior to the F(2) and F(2:3) designs; 3) with Z(1) and Z(2) the statistical powers for the detection of augmented epistatic effects were substantively affected by the signs of pure epistatic effects; and 4) with Z(3) the estimation of pure epistatic effects required large sample size and family replication number. The extension of the proposed method in this study to other base populations was further discussed.     

Epistatic Module Detection for Case-Control Studies: A Bayesian Model with a Gibbs Sampling Strategy

The detection of epistatic interactive effects of multiple genetic variants on the susceptibility of human complex diseases is a great challenge in genome-wide association studies (GWAS). Although methods have been proposed to identify such interactions, the lack of an explicit definition of epistatic effects, together with computational difficulties, makes the development of new methods indispensable. In this paper, we introduce epistatic modules to describe epistatic interactive effects of multiple loci on diseases. On the basis of this notion, we put forward a Bayesian marker partition model to explain observed case-control data, and we develop a Gibbs sampling strategy to facilitate the detection of epistatic modules. Comparisons of the proposed approach with three existing methods on seven simulated disease models demonstrate the superior performance of our approach. When applied to a genome-wide case-control data set for Age-related Macular Degeneration (AMD), the proposed approach successfully identifies two known susceptible loci and suggests that a combination of two other loci—one in the gene SGCD and the other in SCAPER—is associated with the disease. Further functional analysis supports the speculation that the interaction of these two genetic variants may be responsible for the susceptibility of AMD. When applied to a genome-wide case-control data set for Parkinson's disease, the proposed method identifies seven suspicious loci that may contribute independently to the disease.     

Bayesian mapping of genomewide interacting quantitative trait loci for ordinal traits

Development of statistical methods and software for mapping interacting QTL has been the focus of much recent research. We previously developed a Bayesian model selection framework, based on the composite model space approach, for mapping multiple epistatic QTL affecting continuous traits. In this study we extend the composite model space approach to complex ordinal traits in experimental crosses. We jointly model main and epistatic effects of QTL and environmental factors on the basis of the ordinal probit model (also called threshold model) that assumes a latent continuous trait underlies the generation of the ordinal phenotypes through a set of unknown thresholds. A data augmentation approach is developed to jointly generate the latent data and the thresholds. The proposed ordinal probit model, combined with the composite model space framework for continuous traits, offers a convenient way for genomewide interacting QTL analysis of ordinal traits. We illustrate the proposed method by detecting new QTL and epistatic effects for an ordinal trait, dead fetuses, in a F(2) intercross of mice. Utility and flexibility of the method are also demonstrated using a simulated data set. Our method has been implemented in the freely available package R/qtlbim, which greatly facilitates the general usage of the Bayesian methodology for genomewide interacting QTL analysis for continuous, binary, and ordinal traits in experimental crosses.     

Bayesian shrinkage analysis of QTLs under shape-adaptive shrinkage priors, and accurate re-estimation of genetic effects

The successful implementation of Bayesian shrinkage analysis of high-dimensional regression models, as often encountered in quantitative trait locus (QTL) mapping, is contingent upon the choice of suitable sparsity-inducing priors. In practice, the shape (that is, the rate of tail decay) of such priors is typically preset, with no regard for the range of plausible alternatives and the fact that the most appropriate shape may depend on the data at hand. This study is presumably the first attempt to tackle this oversight through the shape-adaptive shrinkage prior (SASP) approach, with a focus on the mapping of QTLs in experimental crosses. Simulation results showed that the separation between genuine QTL effects and spurious ones can be made clearer using the SASP-based approach as compared with existing competitors. This feature makes our new method a promising approach to QTL mapping, where good separation is the ultimate goal. We also discuss a re-estimation procedure intended to improve the accuracy of the estimated genetic effects of detected QTLs with regard to shrinkage-induced bias, which may be particularly important in large-scale models with collinear predictors. The re-estimation procedure is relevant to any shrinkage method, and is potentially valuable for many scientific disciplines such as bioinformatics and quantitative genetics, where oversaturated models are booming.     

Locally Epistatic Genomic Relationship Matrices for Genomic Association and Prediction

In plant and animal breeding studies a distinction is made between the genetic value (additive plus epistatic genetic effects) and the breeding value (additive genetic effects) of an individual since it is expected that some of the epistatic genetic effects will be lost due to recombination. In this article, we argue that the breeder can take advantage of the epistatic marker effects in regions of low recombination. The models introduced here aim to estimate local epistatic line heritability by using genetic map information and combining local additive and epistatic effects. To this end, we have used semiparametric mixed models with multiple local genomic relationship matrices with hierarchical designs. Elastic-net postprocessing was used to introduce sparsity. Our models produce good predictive performance along with useful explanatory information.     

An efficient Bayesian model selection approach for interacting quantitative trait loci models with many effects

We extend our Bayesian model selection framework for mapping epistatic QTL in experimental crosses to include environmental effects and gene-environment interactions. We propose a new, fast Markov chain Monte Carlo algorithm to explore the posterior distribution of unknowns. In addition, we take advantage of any prior knowledge about genetic architecture to increase posterior probability on more probable models. These enhancements have significant computational advantages in models with many effects. We illustrate the proposed method by detecting new epistatic and gene-sex interactions for obesity-related traits in two real data sets of mice. Our method has been implemented in the freely available package R/qtlbim (http://www.qtlbim.org) to facilitate the general usage of the Bayesian methodology for genomewide interacting QTL analysis.     

Resistance to Gray Leaf Spot of Maize: Genetic Architecture and Mechanisms Elucidated through Nested Association Mapping and Near-Isogenic Line Analysis

Gray leaf spot (GLS), caused by Cercospora zeae-maydis and Cercospora zeina, is one of the most important diseases of maize worldwide. The pathogen has a necrotrophic lifestyle and no major genes are known for GLS. Quantitative resistance, although poorly understood, is important for GLS management. We used genetic mapping to refine understanding of the genetic architecture of GLS resistance and to develop hypotheses regarding the mechanisms underlying quantitative disease resistance (QDR) loci. Nested association mapping (NAM) was used to identify 16 quantitative trait loci (QTL) for QDR to GLS, including seven novel QTL, each of which demonstrated allelic series with significant effects above and below the magnitude of the B73 reference allele. Alleles at three QTL, qGLS1.04, qGLS2.09, and qGLS4.05, conferred disease reductions of greater than 10%. Interactions between loci were detected for three pairs of loci, including an interaction between iqGLS4.05 and qGLS7.03. Near-isogenic lines (NILs) were developed to confirm and fine-map three of the 16 QTL, and to develop hypotheses regarding mechanisms of resistance. qGLS1.04 was fine-mapped from an interval of 27.0 Mb to two intervals of 6.5 Mb and 5.2 Mb, consistent with the hypothesis that multiple genes underlie highly significant QTL identified by NAM. qGLS2.09, which was also associated with maturity (days to anthesis) and with resistance to southern leaf blight, was narrowed to a 4-Mb interval. The distance between major leaf veins was strongly associated with resistance to GLS at qGLS4.05. NILs for qGLS1.04 were treated with the C. zeae-maydis toxin cercosporin to test the role of host-specific toxin in QDR. Cercosporin exposure increased expression of a putative flavin-monooxygenase (FMO) gene, a candidate detoxification-related gene underlying qGLS1.04. This integrated approach to confirming QTL and characterizing the potential underlying mechanisms advances the understanding of QDR and will facilitate the development of resistant varieties.     

Identification of quantitative trait loci controlling resistance to gray leaf spot disease in maize

Breeding maize for gray leaf spot (GLS) resistance has been hindered by the quantitative nature of the inheritance of GLS resistance and by the limitations of selection under less than optimumal disease pressure. In order to identify the quantitative trait loci (QTLs) controlling GLS resistance, a cross was made between B73 (susceptible) and Va14 (resistant) to generate a large F₂ population. Six GLS disease assessments were made throughout the disease season for over 1000 F₂ plants in 1989, and for 600 F₂-derived F₃ lines replicated in two blocks in 1990. RFLP analysis for78 marker loci representing all ten maize chromosomes was conducted in 239 F₂ individuals including those with the extreme GLS disease phenotypes. The GLS disease scores of the three field evaluations, each averaged over six ratings, were separately used for the interval mapping in order to determine the consistency of the QTL effects. The heavy GLS disease pressure, meticulous disease ratings, and large population size of this study afforded us the sensitivity for detecting QTL effects. QTLs located on three chromosomes (1, 4, and 8) had large effects on GLS resistance, each explaining 35.0–56.0%, 8.8–14.3%, and 7.7–11.0% of the variance, respectively. These three QTL effects were remarkably consistent across three disease evaluations over 2 years and two generations. Smaller QTL effects were also found on chromosomes 2 and 5, but the chromosome-5 effect might be a false positive because it was not repeatable even in the same location. The chromosome-1 QTLs had the largest effect or highest R² reported for any quantitative trait to-date. Except for the chromosome-4 gene, which was from the susceptible parent B73, the resistance alleles at all QTL were derived from Va14. The resistance QTLs on chromosomes 1 and 2 appear to have additive effects, but those on chromosomes 4 and 8 are dominant and recessive, respectively. Significant interaction between the QTLs on chromosomes 1 and 4 was detected in all three evaluations. Cumulatively, the four QTLs identified in this study explained 44, 60, and 68% of the variance in F₂, and in F₃ replications 1 and 2, respectively.     

Bayesian analysis for genetic architecture of dynamic traits

The dissection of the genetic architecture of quantitative traits, including the number and locations of quantitative trait loci (QTL) and their main and epistatic effects, has been an important topic in current QTL mapping. We extend the Bayesian model selection framework for mapping multiple epistatic QTL affecting continuous traits to dynamic traits in experimental crosses. The extension inherits the efficiency of Bayesian model selection and the flexibility of the Legendre polynomial model fitting to the change in genetic and environmental effects with time. We illustrate the proposed method by simultaneously detecting the main and epistatic QTLs for the growth of leaf age in a doubled-haploid population of rice. The behavior and performance of the method are also shown by computer simulation experiments. The results show that our method can more quickly identify interacting QTLs for dynamic traits in the models with many numbers of genetic effects, enhancing our understanding of genetic architecture for dynamic traits. Our proposed method can be treated as a general form of mapping QTL for continuous quantitative traits, being easier to extend to multiple traits and to a single trait with repeat records.     

Predicting Quantitative Traits With Regression Models for Dense Molecular Markers and Pedigree

The availability of genomewide dense markers brings opportunities and challenges to breeding programs. An important question concerns the ways in which dense markers and pedigrees, together with phenotypic records, should be used to arrive at predictions of genetic values for complex traits. If a large number of markers are included in a regression model, marker-specific shrinkage of regression coefficients may be needed. For this reason, the Bayesian least absolute shrinkage and selection operator (LASSO) (BL) appears to be an interesting approach for fitting marker effects in a regression model. This article adapts the BL to arrive at a regression model where markers, pedigrees, and covariates other than markers are considered jointly. Connections between BL and other marker-based regression models are discussed, and the sensitivity of BL with respect to the choice of prior distributions assigned to key parameters is evaluated using simulation. The proposed model was fitted to two data sets from wheat and mouse populations, and evaluated using cross-validation methods. Results indicate that inclusion of markers in the regression further improved the predictive ability of models. An R program that implements the proposed model is freely available.     

Learning genetic epistasis using Bayesian network scoring criteria

Background  

Gene-gene epistatic interactions likely play an important role in the genetic basis of many common diseases. Recently, machine-learning and data mining methods have been developed for learning epistatic relationships from data. A well-known combinatorial method that has been successfully applied for detecting epistasis is Multifactor Dimensionality Reduction (MDR). Jiang et al. created a combinatorial epistasis learning method called BNMBL to learn Bayesian network (BN) epistatic models. They compared BNMBL to MDR using simulated data sets. Each of these data sets was generated from a model that associates two SNPs with a disease and includes 18 unrelated SNPs. For each data set, BNMBL and MDR were used to score all 2-SNP models, and BNMBL learned significantly more correct models. In real data sets, we ordinarily do not know the number of SNPs that influence phenotype. BNMBL may not perform as well if we also scored models containing more than two SNPs. Furthermore, a number of other BN scoring criteria have been developed. They may detect epistatic interactions even better than BNMBL.     

Bayesian shrinkage estimation of quantitative trait loci parameters

Mapping multiple QTL is a typical problem of variable selection in an oversaturated model because the potential number of QTL can be substantially larger than the sample size. Currently, model selection is still the most effective approach to mapping multiple QTL, although further research is needed. An alternative approach to analyzing an oversaturated model is the shrinkage estimation in which all candidate variables are included in the model but their estimated effects are forced to shrink toward zero. In contrast to the usual shrinkage estimation where all model effects are shrunk by the same factor, we develop a Bayesian method that allows the shrinkage factor to vary across different effects. The new shrinkage method forces marker intervals that contain no QTL to have estimated effects close to zero whereas intervals containing notable QTL have estimated effects subject to virtually no shrinkage. We demonstrate the method using both simulated and real data for QTL mapping. A simulation experiment with 500 backcross (BC) individuals showed that the method can localize closely linked QTL and QTL with effects as small as 1% of the phenotypic variance of the trait. The method was also used to map QTL responsible for wound healing in a family of a (MRL/MPJ x SJL/J) cross with 633 F(2) mice derived from two inbred lines.