On the differences between maximum likelihood and regression interval mapping in the analysis of quantitative trait loci

The differences between maximum-likelihood (ML) and regression (REG) interval mapping in the analysis of quantitative trait loci (QTL) are investigated analytically and numerically by simulation. The analytical investigation is based on the comparison of the solution sets of the ML and REG methods in the estimation of QTL parameters. Their differences are found to relate to the similarity between the conditional posterior and conditional probabilities of QTL genotypes and depend on several factors, such as the proportion of variance explained by QTL, relative QTL position in an interval, interval size, difference between the sizes of QTL, epistasis, and linkage between QTL. The differences in mean squared error (MSE) of the estimates, likelihood-ratio test (LRT) statistics in testing parameters, and power of QTL detection between the two methods become larger as (1) the proportion of variance explained by QTL becomes higher, (2) the QTL locations are positioned toward the middle of intervals, (3) the QTL are located in wider marker intervals, (4) epistasis between QTL is stronger, (5) the difference between QTL effects becomes larger, and (6) the positions of QTL get closer in QTL mapping. The REG method is biased in the estimation of the proportion of variance explained by QTL, and it may have a serious problem in detecting closely linked QTL when compared to the ML method. In general, the differences between the two methods may be minor, but can be significant when QTL interact or are closely linked. The ML method tends to be more powerful and to give estimates with smaller MSEs and larger LRT statistics. This implies that ML interval mapping can be more accurate, precise, and powerful than REG interval mapping. The REG method is faster in computation, especially when the number of QTL considered in the model is large. Recognizing the factors affecting the differences between REG and ML interval mapping can help an efficient strategy, using both methods in QTL mapping to be outlined.     

Linkage analysis: principles and methods for the analysis of human quantitative traits.

Currently, mapping genes for complex human traits relies on two complementary approaches, linkage and association analyses. Both suffer from several methodological and theoretical limitations, which can considerably increase the type-1 error rate and reduce the power to map human quantitative trait loci (QTL). This review focuses on linkage methods for QTL mapping. It summarizes the most common linkage statistics used, namely Haseman-Elston-based methods, variance components, and statistics that condition on trait values. Methods developed more recently that accommodate the X-chromosome, parental imprinting and allelic association in linkage analysis are also summarized. The type-I error rate and power of these methods are discussed. Finally, rough guidelines are provided to help guide the choice of linkage statistics.     

Power of different sampling strategies to detect quantitative trait loci variance effects

Summary Many studies have shown that segregating quantitative trait loci (QTL) can be detected via linkage to genetic markers. Power to detect a QTL effect on the trait mean as a function of the number of individuals genotyped for the marker is increased by selectively genotyping individuals with extreme values for the quantitative trait. Computer simulations were employed to study the effect of various sampling strategies on the statistical power to detect QTL variance effects. If only individuals with extreme phenotypes for the quantitative trait are selected for genotyping, then power to detect a variance effect is less than by random sampling. If 0.2 of the total number of individuals genotyped are selected from the center of the distribution, then power to detect a variance effect is equal to that obtained with random selection. Power to detect a variance effect was maximum when 0.2 to 0.5 of the individuals selected for genotyping were selected from the tails of the distribution and the remainder from the center.     

A comparison of regression interval mapping and multiple interval mapping for linked QTL

Regression interval mapping and multiple interval mapping are compared with regard to mapping linked quantitative trait loci (QTL) in inbred-line cross experiments. For that purpose, a simulation study was performed using genetic models with two linked QTL. Data were simulated for F(2) populations of different sizes and with all QTL and marker alleles fixed for alternative alleles in the parental lines. The criteria for comparison are power of QTL identification and the accuracy of the QTL position and effect estimates. Further, the estimates of the relative QTL variance are assessed. There are distinct differences in the QTL position estimates between the two methods. Multiple interval mapping tends to be more powerful as compared to regression interval mapping. Multiple interval mapping further leads to more accurate QTL position and QTL effect estimates. The superiority increased with wider marker intervals and larger population sizes. If QTL are in repulsion, the differences between the two methods are very pronounced. For both methods, the reduction of the marker interval size from 10 to 5 cM increases power and greatly improves QTL parameter estimates. This contrasts with findings in the literature for single QTL scenarios, where a marker density of 10 cM is generally considered as sufficient. The use of standard (asymptotic) statistical theory for the computation of the standard errors of the QTL position and effect estimates proves to give much too optimistic standard errors for regression interval mapping as well as for multiple interval mapping.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits, which usually show discontinuous distribution and less information, using conventional statistical methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits, which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence, Bayesian estimates of all interested variables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study, utilities of Bayesian-MCMC are demonstrated using simulated several animal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model, three samplers basing on MCMC, including Gibbs sampling, Metropolis algorithm and reversible jump MCMC, were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases, the accuracy of the parameter estimates will be improved. When the true QTL has a small effect, using outbred population experiment design with large family size is the optimal mapping strategy.     

Study on mapping Quantitative Trait Loci for animal complex binary traits using Bayesian-Markov chain Monte Carlo approach

It is a challenging issue to map Quantitative Trait Loci (QTL) underlying complex discrete traits,which usually show discontinuous distribution and less information,using conventional statisti-cal methods. Bayesian-Markov chain Monte Carlo (Bayesian-MCMC) approach is the key procedure in mapping QTL for complex binary traits,which provides a complete posterior distribution for QTL parameters using all prior information. As a consequence,Bayesian estimates of all interested vari-ables can be obtained straightforwardly basing on their posterior samples simulated by the MCMC algorithm. In our study,utilities of Bayesian-MCMC are demonstrated using simulated several ani-mal outbred full-sib families with different family structures for a complex binary trait underlied by both a QTL and polygene. Under the Identity-by-Descent-Based variance component random model,three samplers basing on MCMC,including Gibbs sampling,Metropolis algorithm and reversible jump MCMC,were implemented to generate the joint posterior distribution of all unknowns so that the QTL parameters were obtained by Bayesian statistical inferring. The results showed that Bayesian-MCMC approach could work well and robust under different family structures and QTL effects. As family size increases and the number of family decreases,the accuracy of the parameter estimates will be im-proved. When the true QTL has a small effect,using outbred population experiment design with large family size is the optimal mapping strategy.     

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.     

CONNECTING QTLS TO THE G-MATRIX OF EVOLUTIONARY QUANTITATIVE GENETICS

Evolutionary quantitative genetics has recently advanced in two distinct streams. Many biologists address evolutionary questions by estimating phenotypic selection and genetic (co)variances ( G matrices). Simultaneously, an increasing number of studies have applied quantitative trait locus (QTL) mapping methods to dissect variation. Both conceptual and practical difficulties have isolated these two foci of quantitative genetics. A conceptual integration follows from the recognition that QTL allele frequencies are the essential variables relating the G -matrix to marker-based mapping experiments. Breeding designs initiated from randomly selected parental genotypes can be used to estimate QTL-specific genetic (co)variances. These statistics appropriately distill allelic variation and provide an explicit population context for QTL mapping estimates. Within this framework, one can parse the G -matrix into a set of mutually exclusive genomic components and ask whether these parts are similar or dissimilar in their respective features, for example the magnitude of phenotypic effects and the extent and nature of pleiotropy. As these features are critical determinants of sustained response to selection, the integration of QTL mapping methods into G -matrix estimation can provide a concrete, genetically based experimental program to investigate the evolutionary potential of natural populations.     

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.     

Linear and generalized linear models for the detection of QTL effects on within-subject variability

Quantitative trait loci (QTLs) may affect not only the mean of a trait but also its variability. A special aspect is the variability between multiple measured traits of genotyped animals, such as the within-litter variance of piglet birth weights. The sample variance of repeated measurements is assigned as an observation for every genotyped individual. It is shown that the conditional distribution of the non-normally distributed trait can be approximated by a gamma distribution. To detect QTL effects in the daughter design, a generalized linear model with the identity link function is applied. Suitable test statistics are constructed to test the null hypothesis H(0): No QTL with effect on the within-litter variance is segregating versus H(A): There is a QTL with effect on the variability of birth weight within litter. Furthermore, estimates of the QTL effect and the QTL position are introduced and discussed. The efficiency of the presented tests is compared with a test based on weighted regression. The error probability of the first type as well as the power of QTL detection are discussed and compared for the different tests.     

A simple regression method for mapping quantitative trait loci in line crosses using flanking markers

The use of flanking marker methods has proved to be a powerful tool for the mapping of quantitative trait loci (QTL) in the segregating generations derived from crosses between inbred lines. Methods to analyse these data, based on maximum-likelihood, have been developed and provide good estimates of QTL effects in some situations. Maximum-likelihood methods are, however, relatively complex and can be computationally slow. In this paper we develop methods for mapping QTL based on multiple regression which can be applied using any general statistical package. We use the example of mapping in an F(2) population and show that these regression methods produce very similar results to those obtained using maximum likelihood. The relative simplicity of the regression methods means that models with more than a single QTL can be explored and we give examples of two lined loci and of two interacting loci. Other models, for example with more than two QTL, with environmental fixed effects, with between family variance or for threshold traits, could be fitted in a similar way. The ease, speed of application and generality of regression methods for flanking marker analysis, and the good estimates they obtain, suggest that they should provide the method of choice for the analysis of QTL mapping data from inbred line crosses.     

Genetics of species differences in the wild annual sunflowers, Helianthus annuus and H. petiolaris

Much of our knowledge of speciation genetics stems from quantitative trait locus (QTL) studies. However, interpretations of the size and distribution of QTL underlying species differences are complicated by differences in the way QTL magnitudes are estimated. Also, many studies fail to exploit information about QTL directions or to compare inter- and intraspecific QTL variation. Here, we comprehensively analyze an extensive QTL data set for an interspecific backcross between two wild annual sunflowers, Helianthus annuus and H. petiolaris, interpret different estimates of QTL magnitudes, identify trait groups that have diverged through selection, and compare inter- and intraspecific QTL magnitudes. Our results indicate that even minor QTL (in terms of backcross variance) may be surprisingly large compared to levels of standing variation in the parental species or phenotypic differences between them. Morphological traits, particularly flower morphology, were more strongly or consistently selected than life history or physiological traits. Also, intraspecific QTL were generally smaller than interspecific ones, consistent with the prediction that larger QTL are more likely to spread to fixation across a subdivided population. Our results inform the genetics of species differences in Helianthus and suggest an approach for the simultaneous mapping of inter- and intraspecific QTL.     

Approximating identity-by-descent matrices using multiple haplotype configurations on pedigrees

Identity-by-descent (IBD) matrix calculation is an important step in quantitative trait loci (QTL) analysis using variance component models. To calculate IBD matrices efficiently for large pedigrees with large numbers of loci, an approximation method based on the reconstruction of haplotype configurations for the pedigrees is proposed. The method uses a subset of haplotype configurations with high likelihoods identified by a haplotyping method. The new method is compared with a Markov chain Monte Carlo (MCMC) method (Loki) in terms of QTL mapping performance on simulated pedigrees. Both methods yield almost identical results for the estimation of QTL positions and variance parameters, while the new method is much more computationally efficient than the MCMC approach for large pedigrees and large numbers of loci. The proposed method is also compared with an exact method (Merlin) in small simulated pedigrees, where both methods produce nearly identical estimates of position-specific kinship coefficients. The new method can be used for fine mapping with joint linkage disequilibrium and linkage analysis, which improves the power and accuracy of QTL mapping.     

Linkage between quantitative trait loci and marker loci: resolution power of three statistical approaches in single marker analysis

This paper presents a comparison of three methods of parameter estimation in analysis of linkage between a quantitative trait locus (QTL) and a marker locus: maximum likelihood, mean square for trait cumulative distribution function, and method of moments, employing simulated backcross data. The sensitivity of estimates to violation of assumptions of normality and equal variances were also studied. Some measures of discrepancy between the trait distributions in the QTL groups are considered to evaluate the potential dependence of the resolution capacity of the QTL substitution effect with respect to trait mean value and variance.     

Improving the power of sib pair quantitative trait loci detection by phenotype winsorization

OBJECTIVES: In sib pair studies, quantitative trait loci (QTL) identification may be adversely affected by non-normality in the phenotypic distribution, particularly when subjects falling in the tails of the distribution bias the trait mean or variance. We evaluated the robustness and power of reducing the influence of subjects with extreme phenotypic values by Winsorizing non-normal distributions in three versions of Haseman-Elston regression-based methods of QTL linkage analysis. METHODS: Data were simulated for normal and non-normal distributions. Phenotypic values that correspond to cutoff points at the omega and 1 - omega percentiles of the distribution were identified, and phenotypic values falling outside the boundaries of the omega and 1 - omega cutoff points were replaced by the omega and 1 - omega values, respectively. One million replications were performed for the three tests of linkage for Winsorized and non-Winsorized data. RESULTS: Winsorization reduced conservatism in the tails of the empirical type I error rate for the vast majority of the tests of linkage, increased the power of QTL detection in non-normal data and created a slight negative bias in symmetrical phenotypic distributions. CONCLUSIONS: Winsorizing can improve the power of QTL detection with certain non-normal distributions but can also introduce bias into the estimate of the QTL effect.     

Quantitative trait loci (QTL) detection in multicross inbred designs: recovering QTL identical-by-descent status information from marker data

Mapping quantitative trait loci in plants is usually conducted using a population derived from a cross between two inbred lines. The power of such QTL detection and the parameter estimates depend largely on the choice of the two parental lines. Thus, the QTL detected in such populations represent only a small part of the genetic architecture of the trait. In addition, the effects of only two alleles are characterized, which is of limited interest to the breeder, while common pedigree breeding material remains unexploited for QTL mapping. In this study, we extend QTL mapping methodology to a generalized framework, based on a two-step IBD variance component approach, applicable to any type of breeding population obtained from inbred parents. We then investigate with simulated data mimicking conventional breeding programs the influence of different estimates of the IBD values on the power of QTL detection. The proposed method would provide an alternative to the development of specifically designed recombinant populations, by utilizing the genetic variation actually managed by plant breeders. The use of these detected QTL in assisting breeding would thus be facilitated.     

Mapping quantitative trait loci using multiple families of line crosses.

To avoid a loss in statistical power as a result of homozygous individuals being selected as parents of a mapping population, one can use multiple families of line crosses for quantitative trait genetic linkage analysis. Two strategies of combining data are investigated: the fixed-model and the random-model strategies. The fixed-model approach estimates and tests the average effect of gene substitution for each parent, while the random-model approach treats each effect of gene substitution as a random variable and directly estimates and tests the variance of gene substitution. Extensive Monte Carlo simulations verify that the two strategies perform equally well, although the random model is preferable in combining data from a large number of families. Simulations also show that there may be an optimal sampling strategy (number of families vs. number of individuals per family) in which QTL mapping reaches its maximum power and minimum estimation error. Deviation from the optimal strategy reduces the efficiency of the method.     

Interval mapping of quantitative trait loci with selective DNA pooling data

Selective DNA pooling is an efficient method to identify chromosomal regions that harbor quantitative trait loci (QTL) by comparing marker allele frequencies in pooled DNA from phenotypically extreme individuals. Currently used single marker analysis methods can detect linkage of markers to a QTL but do not provide separate estimates of QTL position and effect, nor do they utilize the joint information from multiple markers. In this study, two interval mapping methods for analysis of selective DNA pooling data were developed and evaluated. One was based on least squares regression (LS-pool) and the other on approximate maximum likelihood (ML-pool). Both methods simultaneously utilize information from multiple markers and multiple families and can be applied to different family structures (half-sib, F2 cross and backcross). The results from these two interval mapping methods were compared with results from single marker analysis by simulation. The results indicate that both LS-pool and ML-pool provided greater power to detect the QTL than single marker analysis. They also provide separate estimates of QTL location and effect. With large family sizes, both LS-pool and ML-pool provided similar power and estimates of QTL location and effect as selective genotyping. With small family sizes, however, the LS-pool method resulted in severely biased estimates of QTL location for distal QTL but this bias was reduced with the ML-pool.     

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.     

Mapping quantitative trait loci for binary traits using a heterogeneous residual variance model: an application to Marek's disease susceptibility in chickens

A typical problem in mapping quantitative trait loci (QTLs) comes from missing QTL genotype. A routine method for parameter estimation involving missing data is the mixture model maximum likelihood method. We developed an alternative QTL mapping method that describes a mixture of several distributions by a single model with a heterogeneous residual variance. The two methods produce similar results, but the heterogeneous residual variance method is computationally much faster than the mixture model approach. In addition, the new method can automatically generate sampling variances of the estimated parameters. We derive the new method in the context of QTL mapping for binary traits in a F2 population. Using the heterogeneous residual variance model, we identified a QTL on chromosome IV that controls Marek's disease susceptibility in chickens. The QTL alone explains 7.2% of the total disease variation. This revised version was published online in July 2006 with corrections to the Cover Date.