A fast expectation-maximum algorithm for fine-scale QTL mapping

The recent technology of the single-nucleotide-polymorphism (SNP) array makes it possible to genotype millions of SNP markers on genome, which in turn requires to develop fast and efficient method for fine-scale quantitative trait loci (QTL) mapping. The single-marker association (SMA) is the simplest method for fine-scale QTL mapping, but it usually shows many false-positive signals and has low QTL-detection power. Compared with SMA, the haplotype-based method of Meuwissen and Goddard who assume QTL effect to be random and estimate variance components (VC) with identity-by-descent (IBD) matrices that inferred from unknown historic population is more powerful for fine-scale QTL mapping; furthermore, their method also tends to show continuous QTL-detection profile to diminish many false-positive signals. However, as we know, the variance component estimation is usually very time consuming and difficult to converge. Thus, an extremely fast EMF (Expectation-Maximization algorithm under Fixed effect model) is proposed in this research, which assumes a biallelic QTL and uses an expectation-maximization (EM) algorithm to solve model effects. The results of simulation experiments showed that (1) EMF was computationally much faster than VC method; (2) EMF and VC performed similarly in QTL detection power and parameter estimations, and both outperformed the paired-marker analysis and SMA. However, the power of EMF would be lower than that of VC if the QTL was multiallelic.     

A note on QTL detecting for censored traits

Most existing statistical methods for mapping quantitative trait loci (QTL) assume that the phenotype follows a normal distribution and that it is fully observed. However, some phenotypes have skewed distributions and may be censored. This note proposes a simple and efficient approach to QTL detecting for censored traits with the Cox PH model without estimating the baseline hazard function which is "nuisance".     

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.     

Comparison of methods for analysis of selective genotyping survival data

Survival traits and selective genotyping datasets are typically not normally distributed, thus common models used to identify QTL may not be statistically appropriate for their analysis. The objective of the present study was to compare models for identification of QTL associated with survival traits, in particular when combined with selective genotyping. Data were simulated to model the survival distribution of a population of chickens challenged with Marek disease virus. Cox proportional hazards (CPH), linear regression (LR), and Weibull models were compared for their appropriateness to analyze the data, ability to identify associations of marker alleles with survival, and estimation of effects when all individuals were genotyped (full genotyping) and when selective genotyping was used. Little difference in power was found between the CPH and the LR model for low censoring cases for both full and selective genotyping. The simulated data were not transformed to follow a Weibull distribution and, as a result, the Weibull model generally resulted in less power than the other two models and overestimated effects. Effect estimates from LR and CPH were unbiased when all individuals were genotyped, but overestimated when selective genotyping was used. Thus, LR is preferred for analyzing survival data when the amount of censoring is low because of ease of implementation and interpretation. Including phenotypic data of non-genotyped individuals in selective genotyping analysis increased power, but resulted in LR having an inflated false positive rate, and therefore the CPH model is preferred for this scenario, although transformation of the data may also make the Weibull model appropriate for this case. The results from the research presented herein are directly applicable to interval mapping analyses.     

Interval mapping methods for detecting QTL affecting survival and time-to-event phenotypes

Quantitative trait loci (QTL) are usually searched for using classical interval mapping methods which assume that the trait of interest follows a normal distribution. However, these methods cannot take into account features of most survival data such as a non-normal distribution and the presence of censored data. We propose two new QTL detection approaches which allow the consideration of censored data. One interval mapping method uses a Weibull model (W), which is popular in parametrical modelling of survival traits, and the other uses a Cox model (C), which avoids making any assumption on the trait distribution. Data were simulated following the structure of a published experiment. Using simulated data, we compare W, C and a classical interval mapping method using a Gaussian model on uncensored data (G) or on all data (G'=censored data analysed as though records were uncensored). An adequate mathematical transformation was used for all parametric methods (G, G' and W). When data were not censored, the four methods gave similar results. However, when some data were censored, the power of QTL detection and accuracy of QTL location and of estimation of QTL effects for G decreased considerably with censoring, particularly when censoring was at a fixed date. This decrease with censoring was observed also with G', but it was less severe. Censoring had a negligible effect on results obtained with the W and C methods.     

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 multiple QTL using linkage disequilibrium and linkage analysis information and multitrait data

A multi-locus QTL mapping method is presented, which combines linkage and linkage disequilibrium (LD) information and uses multitrait data. The method assumed a putative QTL at the midpoint of each marker bracket. Whether the putative QTL had an effect or not was sampled using Markov chain Monte Carlo (MCMC) methods. The method was tested in dairy cattle data on chromosome 14 where the DGAT1 gene was known to be segregating. The DGAT1 gene was mapped to a region of 0.04 cM, and the effects of the gene were accurately estimated. The fitting of multiple QTL gave a much sharper indication of the QTL position than a single QTL model using multitrait data, probably because the multi-locus QTL mapping reduced the carry over effect of the large DGAT1 gene to adjacent putative QTL positions. This suggests that the method could detect secondary QTL that would, in single point analyses, remain hidden under the broad peak of the dominant QTL. However, no indications for a second QTL affecting dairy traits were found on chromosome 14.     

Functional mapping of quantitative trait loci underlying growth trajectories using a transform-both-sides logistic model

The incorporation of developmental control mechanisms of growth has proven to be a powerful tool in mapping quantitative trait loci (QTL) underlying growth trajectories. A theoretical framework for implementing a QTL mapping strategy with growth laws has been established. This framework can be generalized to an arbitrary number of time points, where growth is measured, and becomes computationally more tractable, when the assumption of variance stationarity is made. In practice, however, this assumption is likely to be violated for age-specific growth traits due to a scale effect. In this article, we present a new statistical model for mapping growth QTL, which also addresses the problem of variance stationarity, by using a transform-both-sides (TBS) model advocated by Carroll and Ruppert (1984, Journal of the American Statistical Association 79, 321-328). The TBS-based model for mapping growth QTL cannot only maintain the original biological properties of a growth model, but also can increase the accuracy and precision of parameter estimation and the power to detect a QTL responsible for growth differentiation. Using the TBS-based model, we successfully map a QTL governing growth trajectories to a linkage group in an example of forest trees. The statistical and biological properties of the estimates of this growth QTL position and effect are investigated using Monte Carlo simulation studies. The implications of our model for understanding the genetic architecture of growth are discussed.     

A semiparametric approach for composite functional mapping of dynamic quantitative traits

Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.     

The Cox proportional hazards model with change point: an epidemiologic application

In this paper, we develop the Cox proportional hazards model with special structured time-dependent covariates in the context of prospective epidemiologic studies. Our model possesses the following two features: (i) different relative risk parameters are allowed for early versus late onset of the disease of interest; (ii) an additional parameter is introduced so that specification is not required for the time (age) at which a change of the magnitude of the relative risks takes place, the so-called change point. Some difficulties with statistical inference for the proposed model are briefly discussed, and the large-sample distribution of a test for no change point is derived. As an illustration, we apply the model to a set of data gathered on a group of white male medical students of The Johns Hopkins Medical School enrolled between 1948 and 1964. We examine the hypothesis that the effect of reactivity to the cold pressor test may vary with early versus late onset of hypertension.     

Gene-environment interactions in complex diseases: genetic models and methods for QTL mapping in multiple half-sib populations

An interval quantitative trait locus (QTL) mapping method for complex polygenic diseases (as binary traits) showing QTL by environment interactions (QEI) was developed for outbred populations on a within-family basis. The main objectives, within the above context, were to investigate selection of genetic models and to compare liability or generalized interval mapping (GIM) and linear regression interval mapping (RIM) methods. Two different genetic models were used: one with main QTL and QEI effects (QEI model) and the other with only a main QTL effect (QTL model). Over 30 types of binary disease data as well as six types of continuous data were simulated and analysed by RIM and GIM. Using table values for significance testing, results show that RIM had an increased false detection rate (FDR) for testing interactions which was attributable to scale effects on the binary scale. GIM did not suffer from a high FDR for testing interactions. The use of empirical thresholds, which effectively means higher thresholds for RIM for testing interactions, could repair this increased FDR for RIM, but such empirical thresholds would have to be derived for each case because the amount of FDR depends on the incidence on the binary scale. RIM still suffered from higher biases (15-100% over- or under-estimation of true values) and high standard errors in QTL variance and location estimates than GIM for QEI models. Hence GIM is recommended for disease QTL mapping with QEI. In the presence of QEI, the model including QEI has more power (20-80% increase) to detect the QTL when the average QTL effect is small (in a situation where the model with a main QTL only is not too powerful). Top-down model selection is proposed in which a full test for QEI is conducted first and then the model is subsequently simplified. Methods and results will be applicable to human, plant and animal QTL mapping experiments.     

Genome-wide association mapping of agronomic traits in sugar beet

Recent results indicate that association mapping in populations from applied plant breeding is a powerful tool to detect QTL which are of direct relevance for breeding. The focus of this study was to unravel the genetic architecture of six agronomic traits in sugar beet. To this end, we employed an association mapping approach, based on a very large population of 924 elite sugar beet lines from applied plant breeding, fingerprinted with 677 single nucleotide polymorphism (SNP) markers covering the entire genome. We show that in this population linkage disequilibrium decays within a short genetic distance and is sufficient for the detection of QTL with a large effect size. To increase the QTL detection power and the mapping resolution a much higher number of SNPs is required. We found that for QTL detection, the mixed model including only the kinship matrix performed best, even in the presence of a considerable population structure. In genome-wide scans, main effect QTL and epistatic QTL were detected for all six traits. Our full two-dimensional epistasis scan revealed that for complex traits there appear to be epistatic master regulators, loci which are involved in a large number of epistatic interactions throughout the genome.     

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.     

Defining the assumptions underlying modeling of epistatic QTL using variance component methods

Variance component models are commonly used to detect quantitative trait loci (QTL) in general pedigrees. The variance-covariance structure of the random QTL effect is given by the identity by descent (IBD) between genotypes. Epistatic effects have previously been modeled, both for unlinked and linked loci, as a random effect with a variance-covariance structure given by the Hadamard product between the IBD matrices of the direct QTL effects. In the original papers, the model was given but not derived. Here, we identify the underlying assumptions of this previously proposed model. It assumes that either an unlinked QTL or a fully informative marker (i.e., all marker alleles are unique in the base generation) is located between the loci. We discuss the need of developing a general algorithm to estimate the variance-covariance structure of the random epistatic effect for linked loci.     

A nonlinear mixed-effect mixture model for functional mapping of dynamic traits

Functional mapping has emerged as a next-generation statistical tool for mapping quantitative trait loci (QTL) that affect complex dynamic traits. In this article, we incorporated the idea of nonlinear mixed-effect (NLME) models into the mixture-based framework of functional mapping, aimed to generalize the spectrum of applications for functional mapping. NLME-based functional mapping, implemented with the linearization algorithm based on the first-order Taylor expansion, can provide reasonable estimates of QTL genotypic-specific curve parameters (fixed effect) and the between-individual variation of these parameters (random effect). Results from simulation studies suggest that the NLME-based model is more general than traditional functional mapping. The new model can be useful for the identification of the ontogenetic patterns of QTL genetic effects during time course.     

Statistical inference of selection and divergence from a time-dependent Poisson random field model

We apply a recently developed time-dependent Poisson random field model to aligned DNA sequences from two related biological species to estimate selection coefficients and divergence time. We use Markov chain Monte Carlo methods to estimate species divergence time and selection coefficients for each locus. The model assumes that the selective effects of non-synonymous mutations are normally distributed across genetic loci but constant within loci, and synonymous mutations are selectively neutral. In contrast with previous models, we do not assume that the individual species are at population equilibrium after divergence. Using a data set of 91 genes in two Drosophila species, D. melanogaster and D. simulans, we estimate the species divergence time t(div) = 2.16 N(e) (or 1.68 million years, assuming the haploid effective population size N(e) = 6.45 x 10(5) years) and a mean selection coefficient per generation μ(γ) = 1.98/N(e). Although the average selection coefficient is positive, the magnitude of the selection is quite small. Results from numerical simulations are also presented as an accuracy check for the time-dependent model.     

Power of QTL mapping experiments in commercial Atlantic salmon populations, exploiting linkage and linkage disequilibrium and effect of limited recombination in males

Whereas detection and positioning of genes that affect quantitative traits (quantitative trait loci (QTL)) using linkage mapping uses only information from recombinants in the genotyped generations, linkage disequilibrium (LD) mapping uses historical recombinants. Thus, whereas linkage mapping requires large family sizes to detect and accurately position QTL, LD mapping is more dependent on the number of families sampled from the population. In commercial Atlantic salmon breeding programmes, only a small number of individuals per family are routinely phenotyped for traits such as disease resistance and meat colour. In this paper, we assess the power and accuracy of combined linkage disequilibrium linkage analysis (LDLA) to detect QTL in the commercial population using simulation. When 15 half-sib sire families (each sire mated to 30 dams, each dam with 10 progeny) were sampled from the population for genotyping, we were able to detect a QTL explaining 10% of the phenotypic variance in 85% of replicates and position this QTL within 3 cM of the true position in 70% of replicates. When recombination was absent in males, a feature of the salmon genome, power to detect QTL increased; however, the accuracy of positioning the QTL was decreased. By increasing the number of sire families sampled from the population to be genotyped to 30, we were able to increase both the proportion of QTL detected and correctly positioned (even with no recombination in males). QTL with much smaller effect could also be detected. The results suggest that even with the existing recording structure in commercial salmon breeding programmes, there is considerable power to detect and accurately position QTL using LDLA.     

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.     

Power analysis for quantitative trait locus mapping in populations derived by multiple backcrosses

 Populations derived by multiple backcrosses are potentially useful for quantitative trait locus (QTL) mapping studies. Comparisons of relative power to detect QTL using populations derived by multiple back-crosses are needed to make decisions when mapping projects are initiated. The objective of this study was to theoretically compare the power to detect QTL in populations derived by multiple backcrosses relative to mapping in a recombinant inbred population of equal size. Backcrossing results in a reduction in genetic variance with each generation and also results in an increasing frequency of the recurrent parent marker genotype. The relevant outcome for QTL mapping is a reduction in genetic variance to partition between marker genotype classes and increasing unbalance of the number of individuals contributing to the mean of the marker genotypes. Both of these factors lead to a decrease in the power to detect a QTL as the number of backcross generations increases. Experimental error was held constant with the populations compared. From a theoretical standpoint, backcross-derived populations offer few advantages for QTL detection. If, however, a backcrossing approach is the most efficient method to achieve a desired breeding objective and if QTL detection is an objective of equal or less importance, backcross-derived populations are a reasonable approach to QTL detection. Received: 4 August 1996 / Accepted: 4 April 1997     

远交群体动态性状基因定位的似然分析Ⅰ.理论方法

受动物遗传育种中用来估计动态性状育种值的随机回归测定日模型思想的启发 ,将关于时间 (测定日期 )的Legendre多项式镶嵌在遗传模型的每个遗传效应中 ,以刻画QTL对动态性状变化过程的作用 ,从而建立起动态性状基因定位的数学模型。利用远交设计群体 ,阐述了动态性状基因定位的似然分析原理 ,推导了定位参数似然估计的EM法两步求解过程。结合动态性状遗传分析的特点和普通数量性状基因定位研究进展 ,还提出了有关动态性状基因定位进一步研究的设想