Mapping quantitative trait loci with dominant markers in four-way crosses

 A common problem in mapping quantitative trait loci (QTLs) is that marker data are often incomplete. This includes missing data, dominant markers, and partially informative markers, arising in outbred populations. Here we briefly present an iteratively re-weighted least square method (IRWLS) to incorporate dominant and missing markers for mapping QTLs in four-way crosses under a heterogeneous variance model. The algorithm uses information from all markers in a linkage group to infer the QTL genotype. Monte Carlo simulations indicate that with half dominant markers, QTL detection is almost as efficient as with all co-dominant markers. However, the precision of the estimated QTL parameters generally decreases as more markers become missing or dominant. Notable differences are observed on the standard deviation of the estimated QTL position for varying levels of marker information content. The method is relatively simple so that more complex models including multiple QTLs or fixed effects can be fitted. Finally, the method can be readily extended to QTL mapping in full-sib families. Received: 16 June 1998 / Accepted: 29 September 1998     

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.     

Mapping QTLs for traits measured as percentages

Many quantitative traits are measured as percentages. As a result, the assumption of a normal distribution for the residual errors of such percentage data is often violated. However, most quantitative trait locus (QTL) mapping procedures assume normality of the residuals. Therefore, proper data transformation is often recommended before statistical analysis is conducted. We propose the probit transformation to convert percentage data into variables with a normal distribution. The advantage of the probit transformation is that it can handle measurement errors with heterogeneous variance and correlation structure in a statistically sound manner. We compared the results of this data transformation with other transformations and found that this method can substantially increase the statistical power of QTL detection. We develop the QTL mapping procedure based on the maximum likelihood methodology implemented via the expectation-maximization algorithm. The efficacy of the new method is demonstrated using Monte Carlo simulation.     

Generalized linear model for interval mapping of quantitative trait loci

We developed a generalized linear model of QTL mapping for discrete traits in line crossing experiments. Parameter estimation was achieved using two different algorithms, a mixture model-based EM (expectation–maximization) algorithm and a GEE (generalized estimating equation) algorithm under a heterogeneous residual variance model. The methods were developed using ordinal data, binary data, binomial data and Poisson data as examples. Applications of the methods to simulated as well as real data are presented. The two different algorithms were compared in the data analyses. In most situations, the two algorithms were indistinguishable, but when large QTL are located in large marker intervals, the mixture model-based EM algorithm can fail to converge to the correct solutions. Both algorithms were coded in C++ and interfaced with SAS as a user-defined SAS procedure called PROC QTL.     

Interval mapping of quantitative trait loci for time-to-event data with the proportional hazards mixture cure model

Interval mapping using normal mixture models has been an important tool for analyzing quantitative traits in experimental organisms. When the primary phenotype is time-to-event, it is natural to use survival models such as Cox's proportional hazards model instead of normal mixtures to model the phenotype distribution. An extra challenge for modeling time-to-event data is that the underlying population may consist of susceptible and nonsusceptible subjects. In this article, we propose a semiparametric proportional hazards mixture cure model which allows missing covariates. We discuss applications to quantitative trait loci (QTL) mapping when the primary trait is time-to-event from a population of mixed susceptibility. This model can be used to characterize QTL effects on both susceptibility and time-to-event distribution, and to estimate QTL location. The model can naturally incorporate covariate effects of other risk factors. Maximum likelihood estimates for the parameters in the model as well as their corresponding variance estimates can be obtained numerically using an EM-type algorithm. The proposed methods are assessed by simulations under practical settings and illustrated using a real data set containing survival times of mice after infection with Listeria monocytogenes. An extension to multiple intervals is also discussed.     

Joint mapping of quantitative trait Loci for multiple binary characters

Joint mapping for multiple quantitative traits has shed new light on genetic mapping by pinpointing pleiotropic effects and close linkage. Joint mapping also can improve statistical power of QTL detection. However, such a joint mapping procedure has not been available for discrete traits. Most disease resistance traits are measured as one or more discrete characters. These discrete characters are often correlated. Joint mapping for multiple binary disease traits may provide an opportunity to explore pleiotropic effects and increase the statistical power of detecting disease loci. We develop a maximum-likelihood method for mapping multiple binary traits. We postulate a set of multivariate normal disease liabilities, each contributing to the phenotypic variance of one disease trait. The underlying liabilities are linked to the binary phenotypes through some underlying thresholds. The new method actually maps loci for the variation of multivariate normal liabilities. As a result, we are able to take advantage of existing methods of joint mapping for quantitative traits. We treat the multivariate liabilities as missing values so that an expectation-maximization (EM) algorithm can be applied here. We also extend the method to joint mapping for both discrete and continuous traits. Efficiency of the method is demonstrated using simulated data. We also apply the new method to a set of real data and detect several loci responsible for blast resistance in rice.     

Markov chain Monte Carlo for mapping a quantitative trait locus in outbred populations

A Bayesian approach is presented for mapping a quantitative trait locus (QTL) using the 'Fernando and Grossman' multivariate Normal approximation to QTL inheritance. For this model, a Bayesian implementation that includes QTL position is problematic because standard Markov chain Monte Carlo (MCMC) algorithms do not mix, i.e. the QTL position gets stuck in one marker interval. This is because of the dependence of the covariance structure for the QTL effects on the adjacent markers and may be typical of the 'Fernando and Grossman' model. A relatively new MCMC technique, simulated tempering, allows mixing and so makes possible inferences about QTL position based on marginal posterior probabilities. The model was implemented for estimating variance ratios and QTL position using a continuous grid of allowed positions and was applied to simulated data of a standard granddaughter design. The results showed a smooth mixing of QTL position after implementation of the simulated tempering sampler. In this implementation, map distance between QTL and its flanking markers was artificially stretched to reduce the dependence of markers and covariance. The method generalizes easily to more complicated applications and can ultimately contribute to QTL mapping in complex, heterogeneous, human, animal or plant populations.     

Effects of missing marker and segregation distortion on QTL mapping in F2 populations

Missing marker and segregation distortion are commonly encountered in actual quantitative trait locus (QTL) mapping populations. Our objective in this study was to investigate the impact of the two factors on QTL mapping through computer simulations. Results indicate that detection power decreases with increasing levels of missing markers, and the false discovery rate increases. Missing markers have greater effects on smaller effect QTL and smaller size populations. The effect of missing markers can be quantified by a population with a reduced size similar to the marker missing rate. As for segregation distortion, if the distorted marker is not closely linked with any QTL, it will not have significant impact on QTL mapping; otherwise, the impact of the distortion will depend on the degree of dominance of QTL, frequencies of the three marker types, the linkage distance between the distorted marker and QTL, and the mapping population size. Sometimes, the distortion can result in a higher genetic variance than that of non-distortion, and therefore benefits the detection of linked QTL. A formula of the ratio of genetic variance explained by QTL under distortion and non-distortion was given in this study, so as to easily determine whether the segregation distortion marker (SDM) increases or decreases the QTL detection power. The effect of SDM decreases rapidly as its linkage relationship with QTL becomes looser. In general, distorted markers will not have a great effect on the position and effect estimations of QTL, and their effects can be ignored in large-size mapping populations.     

Genetic mapping of developmental instability: design, model and algorithm

Developmental instability or noise, defined as the phenotypic imprecision of an organism in the face of internal or external stochastic disturbances, has been thought to play an important role in shaping evolutionary processes and patterns. The genetic studies of developmental instability have been based on fluctuating asymmetry (FA) that measures random differences between the left and the right sides of bilateral traits. In this article, we frame an experimental design characterized by a spatial autocorrelation structure for determining the genetic control of developmental instability for those traits that cannot be bilaterally measured. This design allows the residual environmental variance of a quantitative trait to be dissolved into two components due to permanent and random environmental factors. The degree of developmental instability is quantified by the relative proportion of the random residual variance to the total residual variance. We formulate a mixture model to estimate and test the genetic effects of quantitative trait loci (QTL) on the developmental instability of the trait. The genetic parameters including the QTL position, the QTL effects, and spatial autocorrelations are estimated by implementing the EM algorithm within the mixture model framework. Simulation studies were performed to investigate the statistical behavior of the model. A live example for poplar trees was used to map the QTL that control root length growth and its developmental instability from cuttings in water culture.     

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.     

Genome-wide evaluation for quantitative trait loci under the variance component model

The identity-by-descent (IBD) based variance component analysis is an important method for mapping quantitative trait loci (QTL) in outbred populations. The interval-mapping approach and various modified versions of it may have limited use in evaluating the genetic variances of the entire genome because they require evaluation of multiple models and model selection. In this study, we developed a multiple variance component model for genome-wide evaluation using both the maximum likelihood (ML) method and the MCMC implemented Bayesian method. We placed one QTL in every few cM on the entire genome and estimated the QTL variances and positions simultaneously in a single model. Genomic regions that have no QTL usually showed no evidence of QTL while regions with large QTL always showed strong evidence of QTL. While the Bayesian method produced the optimal result, the ML method is computationally more efficient than the Bayesian method. Simulation experiments were conducted to demonstrate the efficacy of the new methods.     

Complex genetic effects in quantitative trait locus identification: a computationally tractable random model for use in F(2) populations

Methodology for mapping quantitative trait loci (QTL) has focused primarily on treating the QTL as a fixed effect. These methods differ from the usual models of genetic variation that treat genetic effects as random. Computationally expensive methods that allow QTL to be treated as random have been explicitly developed for additive genetic and dominance effects. By extending these methods with a variance component method (VCM), multiple QTL can be mapped. We focused on an F(2) crossbred population derived from inbred lines and estimated effects for each individual and their corresponding marker-derived genetic covariances. We present extensions to pairwise epistatic effects, which are computationally intensive because a great many individual effects must be estimated. But by replacing individual genetic effects with average genetic effects for each marker class, genetic covariances are approximated. This substantially reduces the computational burden by reducing the dimensions of covariance matrices of genetic effects, resulting in a remarkable gain in the speed of estimating the variance components and evaluating the residual log-likelihood. Preliminary results from simulations indicate competitiveness of the reduced model with multiple-interval mapping, regression interval mapping, and VCM with individual genetic effects in its estimated QTL positions and experimental power.     

Mapping quantitative trait loci in F2 incorporating phenotypes of F3 progeny

In plants and laboratory animals, QTL mapping is commonly performed using F(2) or BC individuals derived from the cross of two inbred lines. Typical QTL mapping statistics assume that each F(2) individual is genotyped for the markers and phenotyped for the trait. For plant traits with low heritability, it has been suggested to use the average phenotypic values of F(3) progeny derived from selfing F(2) plants in place of the F(2) phenotype itself. All F(3) progeny derived from the same F(2) plant belong to the same F(2:3) family, denoted by F(2:3). If the size of each F(2:3) family (the number of F(3) progeny) is sufficiently large, the average value of the family will represent the genotypic value of the F(2) plant, and thus the power of QTL mapping may be significantly increased. The strategy of using F(2) marker genotypes and F(3) average phenotypes for QTL mapping in plants is quite similar to the daughter design of QTL mapping in dairy cattle. We study the fundamental principle of the plant version of the daughter design and develop a new statistical method to map QTL under this F(2:3) strategy. We also propose to combine both the F(2) phenotypes and the F(2:3) average phenotypes to further increase the power of QTL mapping. The statistical method developed in this study differs from published ones in that the new method fully takes advantage of the mixture distribution for F(2:3) families of heterozygous F(2) plants. Incorporation of this new information has significantly increased the statistical power of QTL detection relative to the classical F(2) design, even if only a single F(3) progeny is collected from each F(2:3) family. The mixture model is developed on the basis of a single-QTL model and implemented via the EM algorithm. Substantial computer simulation was conducted to demonstrate the improved efficiency of the mixture model. Extension of the mixture model to multiple QTL analysis is developed using a Bayesian approach. The computer program performing the Bayesian analysis of the simulated data is available to users for real data analysis.     

Simultaneous detection and fine mapping of quantitative trait loci in mice using heterogeneous stocks

We describe a method to simultaneously detect and fine map quantitative trait loci (QTL) that is especially suited to the mapping of modifier loci in mouse mutant models. The method exploits the high level of historical recombination present in a heterogeneous stock (HS), an outbred population of mice derived from known founder strains. The experimental design is an F(2) cross between the HS and a genetically distinct line, such as one carrying a knockout or transgene. QTL detection is performed by a standard genome scan with approximately 100 markers and fine mapping by typing the same animals using densely spaced markers over those candidate regions detected by the scan. The analysis uses an extension of the dynamic-programming technique employed previously to fine map QTL in HS mice. We show by simulation that a QTL accounting for 5% of the total variance can be detected and fine mapped with >50% probability to within 3 cM by genotyping approximately 1500 animals.     

Mapping quantitative trait loci by an extension of the Haley-Knott regression method using estimating equations

The Haley-Knott (HK) regression method continues to be a popular approximation to standard interval mapping (IM) of quantitative trait loci (QTL) in experimental crosses. The HK method is favored for its dramatic reduction in computation time compared to the IM method, something that is particularly important in simultaneous searches for multiple interacting QTL. While the HK method often approximates the IM method well in estimating QTL effects and in power to detect QTL, it may perform poorly if, for example, there is strong epistasis between QTL or if QTL are linked. Also, it is well known that the estimation of the residual variance by the HK method is biased. Here, we present an extension of the HK method that uses estimating equations based on both means and variances. For normally distributed phenotypes this estimating equation (EE) method is more efficient than the HK method. Furthermore, computer simulations show that the EE method performs well for very different genetic models and data set structures, including nonnormal phenotype distributions, nonrandom missing data patterns, varying degrees of epistasis, and varying degrees of linkage between QTL. The EE method retains key qualities of the HK method such as computational speed and robustness against nonnormal phenotype distributions, while approximating the IM method better in terms of accuracy and precision of parameter estimates and power to detect QTL.     

The analysis of QTL by simultaneous use of the full linkage map

An extension of interval mapping is presented that incorporates all intervals on the linkage map simultaneously. The approach uses a working model in which the sizes of putative QTL for all intervals across the genome are random effects. An outlier detection method is used to screen for possible QTL. Selected QTL are subsequently fitted as fixed effects. This screening and selection approach is repeated until the variance component for QTL sizes is not statistically significant. A comprehensive simulation study is conducted in which map uncertainty is included. The proposed method is shown to be superior to composite interval mapping in terms of power of detection of QTL. There is an increase in the rate of false positive QTL detected when using the new approach, but this rate decreases as the population size increases. The new approach is much simpler computationally. The analysis of flour milling yield in a doubled haploid population illustrates the improved power of detection of QTL using the approach, and also shows how vital it is to allow for sources of non-genetic variation in the analysis.     

Strategies for genetic mapping of categorical traits

The search for efficient and powerful statistical methods and optimal mapping strategies for categorical traits under various experimental designs continues to be one of the main tasks in genetic mapping studies. Methodologies for genetic mapping of categorical traits can generally be classified into two groups, linear and non-linear models. We develop a method based on a threshold model, termed mixture threshold model to handle ordinal (or binary) data from multiple families. Monte Carlo simulations are done to compare its statistical efficiencies and properties of the proposed non-linear model with a linear model for genetic mapping of categorical traits using multiple families. The mixture threshold model has notably higher statistical power than linear models. There may be an optimal sampling strategy (family size vs number of families) in which genetic mapping reaches its maximal power and minimal estimation errors. A single large-sibship family does not necessarily produce the maximal power for detection of quantitative trait loci (QTL) due to genetic sampling of QTL alleles. The QTL allelic model has a marked impact on efficiency of genetic mapping of categorical traits in terms of statistical power and QTL parameter estimation. Compared with a fixed number of QTL alleles (two or four), the model with an infinite number of QTL alleles and normally distributed allelic effects results in loss of statistical power. The results imply that inbred designs (e.g. F₂ or four-way crosses) with a few QTL alleles segregating or reducing number of QTL alleles (e.g. by selection) in outbred populations are desirable in genetic mapping of categorical traits using data from multiple families. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalized linear mixed models for mapping multiple quantitative trait loci

Many biological traits are discretely distributed in phenotype but continuously distributed in genetics because they are controlled by multiple genes and environmental variants. Due to the quantitative nature of the genetic background, these multiple genes are called quantitative trait loci (QTL). When the QTL effects are treated as random, they can be estimated in a single generalized linear mixed model (GLMM), even if the number of QTL may be larger than the sample size. The GLMM in its original form cannot be applied to QTL mapping for discrete traits if there are missing genotypes. We examined two alternative missing genotype-handling methods: the expectation method and the overdispersion method. Simulation studies show that the two methods are efficient for multiple QTL mapping (MQM) under the GLMM framework. The overdispersion method showed slight advantages over the expectation method in terms of smaller mean-squared errors of the estimated QTL effects. The two methods of GLMM were applied to MQM for the female fertility trait of wheat. Multiple QTL were detected to control the variation of the number of seeded spikelets.     

Bayesian functional mapping of dynamic quantitative traits

Without consideration of other linked QTLs responsible for dynamic trait, original functional mapping based on a single QTL model is not optimal for analyzing multiple dynamic trait loci. Despite that composite functional mapping incorporates the effects of genetic background outside the tested QTL in mapping model, the arbitrary choice of background markers also impact on the power of QTL detection. In this study, we proposed Bayesian functional mapping strategy that can simultaneously identify multiple QTL controlling developmental patterns of dynamic traits over the genome. Our proposed method fits the change of each QTL effect with the time by Legendre polynomial and takes the residual covariance structure into account using the first autoregressive equation. Also, Bayesian shrinkage estimation was employed to estimate the model parameters. Especially, we specify the gamma distribution as the prior for the first-order auto-regressive coefficient, which will guarantee the convergence of Bayesian sampling. Simulations showed that the proposed method could accurately estimate the QTL parameters and had a greater statistical power of QTL detection than the composite functional mapping. A real data analysis of leaf age growth in rice is used for the demonstration of our method. It shows that our Bayesian functional mapping can detect more QTLs as compared to composite functional mapping.     

QTL mapping in outbred half-sib families using Bayesian model selection

In this article, we propose a model selection method, the Bayesian composite model space approach, to map quantitative trait loci (QTL) in a half-sib population for continuous and binary traits. In our method, the identity-by-descent-based variance component model is used. To demonstrate the performance of this model, the method was applied to map QTL underlying production traits on BTA6 in a Chinese half-sib dairy cattle population. A total of four QTLs were detected, whereas only one QTL was identified using the traditional least square (LS) method. We also conducted two simulation experiments to validate the efficiency of our method. The results suggest that the proposed method based on a multiple-QTL model is efficient in mapping multiple QTL for an outbred half-sib population and is more powerful than the LS method based on a single-QTL model.