Box–Cox transformation for QTL mapping

The maximum likelihood method of QTL mapping assumes that the phenotypic values of a quantitative trait follow a normal distribution. If the assumption is violated, some forms of transformation should be taken to make the assumption approximately true. The Box–Cox transformation is a general transformation method which can be applied to many different types of data. The flexibility of the Box–Cox transformation is due to a variable, called transformation factor, appearing in the Box–Cox formula. We developed a maximum likelihood method that treats the transformation factor as an unknown parameter, which is estimated from the data simultaneously along with the QTL parameters. The method makes an objective choice of data transformation and thus can be applied to QTL analysis for many different types of data. Simulation studies show that (1) Box–Cox transformation can substantially increase the power of QTL detection; (2) Box–Cox transformation can replace some specialized transformation methods that are commonly used in QTL mapping; and (3) applying the Box–Cox transformation to data already normally distributed does not harm the result.     

Functional mapping of dynamic traits with robust t-distribution

Functional mapping has been a powerful tool in mapping quantitative trait loci (QTL) underlying dynamic traits of agricultural or biomedical interest. In functional mapping, multivariate normality is often assumed for the underlying data distribution, partially due to the ease of parameter estimation. The normality assumption however could be easily violated in real applications due to various reasons such as heavy tails or extreme observations. Departure from normality has negative effect on testing power and inference for QTL identification. In this work, we relax the normality assumption and propose a robust multivariate t-distribution mapping framework for QTL identification in functional mapping. Simulation studies show increased mapping power and precision with the t distribution than that of a normal distribution. The utility of the method is demonstrated through a real data analysis.     

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.     

Estimating genetic variance contributed by a quantitative trait locus: A random model approach

Detecting quantitative trait loci (QTL) and estimating QTL variances (represented by the squared QTL effects) are two main goals of QTL mapping and genome-wide association studies (GWAS). However, there are issues associated with estimated QTL variances and such issues have not attracted much attention from the QTL mapping community. Estimated QTL variances are usually biased upwards due to estimation being associated with significance tests. The phenomenon is called the Beavis effect. However, estimated variances of QTL without significance tests can also be biased upwards, which cannot be explained by the Beavis effect; rather, this bias is due to the fact that QTL variances are often estimated as the squares of the estimated QTL effects. The parameters are the QTL effects and the estimated QTL variances are obtained by squaring the estimated QTL effects. This square transformation failed to incorporate the errors of estimated QTL effects into the transformation. The consequence is biases in estimated QTL variances. To correct the biases, we can either reformulate the QTL model by treating the QTL effect as random and directly estimate the QTL variance (as a variance component) or adjust the bias by taking into account the error of the estimated QTL effect. A moment method of estimation has been proposed to correct the bias. The method has been validated via Monte Carlo simulation studies. The method has been applied to QTL mapping for the 10-week-body-weight trait from an F₂ mouse population.     

Mapping quantitative trait loci for traits defined as ratios

Many traits are defined as ratios of two quantitative traits. Methods of QTL mapping for regular quantitative traits are not optimal when applied to ratios due to lack of normality for traits defined as ratios. We develop a new method of QTL mapping for traits defined as ratios. The new method uses a special linear combination of the two component traits, and thus takes advantage of the normal property of the new variable. Simulation study shows that the new method can substantially increase the statistical power of QTL detection relative to the method which treats ratios as regular quantitative traits. The new method also outperforms the method that uses Box-Cox transformed ratio as the phenotype. A real example of QTL mapping for relative growth rate in soybean demonstrates that the new method can detect more QTL than existing methods of QTL mapping for traits defined as ratios.     

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.     

Robust Bayesian mapping of quantitative trait loci using Student-t distribution for residual

In most quantitative trait loci (QTL) mapping studies, phenotypes are assumed to follow normal distributions. Deviations from this assumption may affect the accuracy of QTL detection, leading to detection of false positive QTL. To improve the robustness of QTL mapping methods, we replace the normal distribution assumption for residuals in a multiple QTL model with a Student-t distribution that is able to accommodate residual outliers. A Robust Bayesian mapping strategy is proposed on the basis of the Bayesian shrinkage analysis for QTL effects. The simulations show that Robust Bayesian mapping approach can substantially increase the power of QTL detection when the normality assumption does not hold and applying it to data already normally distributed does not influence the result. The proposed QTL mapping method is applied to mapping QTL for the traits associated with physics–chemical characters and quality in rice. Similarly to the simulation study in the real data case the robust approach was able to detect additional QTLs when compared to the traditional approach. The program to implement the method is available on request from the first or the corresponding author. Xin Wang and Zhongze Piao contributed equally to this study.     

A Bayesian Nonparametric Approach for Mapping Dynamic Quantitative Traits

In biology, many quantitative traits are dynamic in nature. They can often be described by some smooth functions or curves. A joint analysis of all the repeated measurements of the dynamic traits by functional quantitative trait loci (QTL) mapping methods has the benefits to (1) understand the genetic control of the whole dynamic process of the quantitative traits and (2) improve the statistical power to detect QTL. One crucial issue in functional QTL mapping is how to correctly describe the smoothness of trajectories of functional valued traits. We develop an efficient Bayesian nonparametric multiple-loci procedure for mapping dynamic traits. The method uses the Bayesian P-splines with (nonparametric) B-spline bases to specify the functional form of a QTL trajectory and a random walk prior to automatically determine its degree of smoothness. An efficient deterministic variational Bayes algorithm is used to implement both (1) the search of an optimal subset of QTL among large marker panels and (2) estimation of the genetic effects of the selected QTL changing over time. Our method can be fast even on some large-scale data sets. The advantages of our method are illustrated on both simulated and real data sets.     

Composite Interval Mapping Based on Lattice Design for Error Control May Increase Power of Quantitative Trait Locus Detection

Experimental error control is very important in quantitative trait locus (QTL) mapping. Although numerous statistical methods have been developed for QTL mapping, a QTL detection model based on an appropriate experimental design that emphasizes error control has not been developed. Lattice design is very suitable for experiments with large sample sizes, which is usually required for accurate mapping of quantitative traits. However, the lack of a QTL mapping method based on lattice design dictates that the arithmetic mean or adjusted mean of each line of observations in the lattice design had to be used as a response variable, resulting in low QTL detection power. As an improvement, we developed a QTL mapping method termed composite interval mapping based on lattice design (CIMLD). In the lattice design, experimental errors are decomposed into random errors and block-within-replication errors. Four levels of block-within-replication errors were simulated to show the power of QTL detection under different error controls. The simulation results showed that the arithmetic mean method, which is equivalent to a method under random complete block design (RCBD), was very sensitive to the size of the block variance and with the increase of block variance, the power of QTL detection decreased from 51.3% to 9.4%. In contrast to the RCBD method, the power of CIMLD and the adjusted mean method did not change for different block variances. The CIMLD method showed 1.2- to 7.6-fold higher power of QTL detection than the arithmetic or adjusted mean methods. Our proposed method was applied to real soybean (Glycine max) data as an example and 10 QTLs for biomass were identified that explained 65.87% of the phenotypic variation, while only three and two QTLs were identified by arithmetic and adjusted mean methods, respectively.     

Multiple QTL mapping in related plant populations via a pedigree-analysis approach

QTL mapping experiments in plant breeding may involve multiple populations or pedigrees that are related through their ancestors. These known relationships have often been ignored for the sake of statistical analysis, despite their potential increase in power of mapping. We describe here a Bayesian method for QTL mapping in complex plant populations and reported the results from its application to a (previously analysed) potato data set. This Bayesian method was originally developed for human genetics data, and we have proved that it is useful for complex plant populations as well, based on a sensitivity analysis that was performed here. The method accommodates robustness to complex structures in pedigree data, full flexibility in the estimation of the number of QTL across multiple chromosomes, thereby accounting for uncertainties in the transmission of QTL and marker alleles due to incomplete marker information, and the simultaneous inclusion of non-genetic factors affecting the quantitative trait.     

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.     

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.     

A simple linear regression method for quantitative trait loci linkage analysis with censored observations

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.     

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.     

Statistical power of QTL mapping methods applied to bacteria counts

Most QTL mapping methods assume that phenotypes follow a normal distribution, but many phenotypes of interest are not normally distributed, e.g. bacteria counts (or colony-forming units, CFU). Such data are extremely skewed to the right and can present a high amount of zero values, which are ties from a statistical point of view. Our objective is therefore to assess the efficiency of four QTL mapping methods applied to bacteria counts: (1) least-squares (LS) analysis, (2) maximum-likelihood (ML) analysis, (3) non-parametric (NP) mapping and (4) nested ANOVA (AN). A transformation based on quantiles is used to mimic observed distributions of bacteria counts. Single positions (1 marker, 1 QTL) as well as chromosome scans (11 markers, 1 QTL) are simulated. When compared with the analysis of a normally distributed phenotype, the analysis of raw bacteria counts leads to a strong decrease in power for parametric methods, but no decrease is observed for NP. However, when a mathematical transformation (MT) is applied to bacteria counts prior to analysis, parametric methods have the same power as NP. Furthermore, parametric methods, when coupled with MT, outperform NP when bacteria counts have a very high proportion of zeros (70.8%). Our results show that the loss of power is mainly explained by the asymmetry of the phenotypic distribution, for parametric methods, and by the existence of ties, for the non-parametric method. Therefore, mapping of QTL for bacterial diseases, as well as for other diseases assessed by a counting process, should focus on the occurrence of ties in phenotypes before choosing the appropriate QTL mapping method.     

Locating multiple interacting quantitative trait Loci using rank-based model selection

In previous work, a modified version of the Bayesian information criterion (mBIC) was proposed to locate multiple interacting quantitative trait loci (QTL). Simulation studies and real data analysis demonstrate good properties of the mBIC in situations where the error distribution is approximately normal. However, as with other standard techniques of QTL mapping, the performance of the mBIC strongly deteriorates when the trait distribution is heavy tailed or when the data contain a significant proportion of outliers. In the present article, we propose a suitable robust version of the mBIC that is based on ranks. We investigate the properties of the resulting method on the basis of theoretical calculations, computer simulations, and a real data analysis. Our simulation results show that for the sample sizes typically used in QTL mapping, the methods based on ranks are almost as efficient as standard techniques when the data are normal and are much better when the data come from some heavy-tailed distribution or include a proportion of outliers.     

Varying coefficient models for mapping quantitative trait loci using recombinant inbred intercrosses

There has been a great deal of interest in the development of methodologies to map quantitative trait loci (QTL) using experimental crosses in the last 2 decades. Experimental crosses in animal and plant sciences provide important data sources for mapping QTL through linkage analysis. The Collaborative Cross (CC) is a renewable mouse resource that is generated from eight genetically diverse founder strains to mimic the genetic diversity in humans. The recombinant inbred intercrosses (RIX) generated from CC recombinant inbred (RI) lines share similar genetic structures of F(2) individuals but with up to eight alleles segregating at any one locus. In contrast to F(2) mice, genotypes of RIX can be inferred from the genotypes of their RI parents and can be produced repeatedly. Also, RIX mice typically do not share the same degree of relatedness. This unbalanced genetic relatedness requires careful statistical modeling to avoid false-positive findings. Many quantitative traits are inherently complex with genetic effects varying with other covariates, such as age. For such complex traits, if phenotype data can be collected over a wide range of ages across study subjects, their dynamic genetic patterns can be investigated. Parametric functions, such as sigmoidal or logistic functions, have been used for such purpose. In this article, we propose a flexible nonparametric time-varying coefficient QTL mapping method for RIX data. Our method allows the QTL effects to evolve with time and naturally extends classical parametric QTL mapping methods. We model the varying genetic effects nonparametrically with the B-spline bases. Our model investigates gene-by-time interactions for RIX data in a very flexible nonparametric fashion. Simulation results indicate that the varying coefficient QTL mapping has higher power and mapping precision compared to parametric models when the assumption of constant genetic effects fails. We also apply a modified permutation procedure to control overall significance level.     

An empirical Bayes method for updating inferences in analysis of quantitative trait loci using information from related genome scans

Individual genome scans for quantitative trait loci (QTL) mapping often suffer from low statistical power and imprecise estimates of QTL location and effect. This lack of precision yields large confidence intervals for QTL location, which are problematic for subsequent fine mapping and positional cloning. In prioritizing areas for follow-up after an initial genome scan and in evaluating the credibility of apparent linkage signals, investigators typically examine the results of other genome scans of the same phenotype and informally update their beliefs about which linkage signals in their scan most merit confidence and follow-up via a subjective-intuitive integration approach. A method that acknowledges the wisdom of this general paradigm but formally borrows information from other scans to increase confidence in objectivity would be a benefit. We developed an empirical Bayes analytic method to integrate information from multiple genome scans. The linkage statistic obtained from a single genome scan study is updated by incorporating statistics from other genome scans as prior information. This technique does not require that all studies have an identical marker map or a common estimated QTL effect. The updated linkage statistic can then be used for the estimation of QTL location and effect. We evaluate the performance of our method by using extensive simulations based on actual marker spacing and allele frequencies from available data. Results indicate that the empirical Bayes method can account for between-study heterogeneity, estimate the QTL location and effect more precisely, and provide narrower confidence intervals than results from any single individual study. We also compared the empirical Bayes method with a method originally developed for meta-analysis (a closely related but distinct purpose). In the face of marked heterogeneity among studies, the empirical Bayes method outperforms the comparator.     

Genomewide association analysis in diverse inbred mice: power and population structure

The discovery of quantitative trait loci (QTL) in model organisms has relied heavily on the ability to perform controlled breeding to generate genotypic and phenotypic diversity. Recently, we and others have demonstrated the use of an existing set of diverse inbred mice (referred to here as the mouse diversity panel, MDP) as a QTL mapping population. The use of the MDP population has many advantages relative to traditional F(2) mapping populations, including increased phenotypic diversity, a higher recombination frequency, and the ability to collect genotype and phenotype data in community databases. However, these methods are complicated by population structure inherent in the MDP and the lack of an analytical framework to assess statistical power. To address these issues, we measured gene expression levels in hypothalamus across the MDP. We then mapped these phenotypes as quantitative traits with our association algorithm, resulting in a large set of expression QTL (eQTL). We utilized these eQTL, and specifically cis-eQTL, to develop a novel nonparametric method for association analysis in structured populations like the MDP. These eQTL data confirmed that the MDP is a suitable mapping population for QTL discovery and that eQTL results can serve as a gold standard for relative measures of statistical power.     

QTL fine mapping by measuring and testing for Hardy-Weinberg and linkage disequilibrium at a series of linked marker loci in extreme samples of populations

It has recently been demonstrated that fine-scale mapping of a susceptibility locus for a complex disease can be accomplished on the basis of deviations from Hardy-Weinberg (HW) equilibrium at closely linked marker loci among affected individuals. We extend this theory to fine-scale localization of a quantitative-trait locus (QTL) from extreme individuals in populations, by means of HW and linkage-disequilibrium (LD) analyses. QTL mapping and/or linkage analyses can establish a large genomic region ( approximately 30 cM) that contains a QTL. The QTL can be fine mapped by examination of the degree of deviation from HW and LD at a series of closely linked marker loci. The tests can be performed for samples of individuals belonging to either high or low percentiles of the phenotype distribution or for combined samples of these extreme individuals. The statistical properties (the power and the size) of the tests of this fine-mapping approach are investigated and are compared extensively, under various genetic models and parameters for the QTL and marker loci. On the basis of the results, a two-stage procedure that uses extreme samples and different tests (for HW and LD) is suggested for QTL fine mapping. This two-step procedure is economic and powerful and can accurately narrow a genomic region containing a QTL from approximately 30-1 cM, a range that renders physical mapping feasible for identification of the QTL. In addition, the relationship between parameterizations of complex diseases, by means of penetrance, and those of complex quantitative traits, by means of genotypic values, is outlined. This means that many statistical genetic methods developed for searching for susceptibility loci of complex diseases can be directly adopted and/or extended to QTL mapping for quantitative traits.