QTL analysis in arbitrary pedigrees with incomplete marker information

Mapping quantitative trait loci (QTL) in arbitrary outbred pedigrees is complicated by the combinatorial possibilities of allele flow relationships and of the founder allelic configurations. Exact methods are only available for rather short and simple pedigrees. Stochastic simulation using Markov chain Monte Carlo (MCMC) integration offers more flexibility. MCMC methods are less natural in a frequentist than in a Bayesian context, which we therefore adopt. Among the MCMC algorithms for updating marker locus genotypes, we implement the descent-graph algorithm. It can be used to update marker locus allele flow relationships and can handle arbitrarily complex pedigrees and missing marker information. Compared with updating marker genotypic information, updating QTL parameters, such as position, effects, and the allele flow relationships is relatively easy with MCMC. We treat the effect of each diploid combination of founder alleles as a random variable and only estimate the variance of these effects, ie, we model diploid genotypic effects instead of the usual partition in additive and dominance effects. This is a variant of the random model approach. The number of QTL alleles is generally unknown. In the Bayesian context, the number of QTL present on a linkage group can be treated as variable. Computer simulations suggest that the algorithm can indeed handle complex pedigrees and detect two QTL on a linkage group, but that the number of individuals in a single extended family is limited to about 50 to 100 individuals.     

Accurate partition of individuals into full-sib families from genetic data without parental information.

Two Markov chain Monte Carlo algorithms are proposed that allow the partitioning of individuals into full-sib groups using single-locus genetic marker data when no parental information is available. These algorithms present a method of moving through the sibship configuration space and locating the configuration that maximizes an overall score on the basis of pairwise likelihood ratios of being full-sib or unrelated or maximizes the full joint likelihood of the proposed family structure. Using these methods, up to 757 out of 759 Atlantic salmon were correctly classified into 12 full-sib families of unequal size using four microsatellite markers. Large-scale simulations were performed to assess the sensitivity of the procedures to the number of loci and number of alleles per locus, the allelic distribution type, the distribution of families, and the independent knowledge of population allelic frequencies. The number of loci and the number of alleles per locus had the most impact on accuracy. Very good accuracy can be obtained with as few as four loci when they have at least eight alleles. Accuracy decreases when using allelic frequencies estimated in small target samples with skewed family distributions with the pairwise likelihood approach. We present an iterative approach that partly corrects that problem. The full likelihood approach is less sensitive to the precision of allelic frequencies estimates but did not perform as well with the large data set or when little information was available (e.g., four loci with four alleles).     

Bayesian mapping of quantitative trait loci under complicated mating designs

Quantitative trait loci (QTL) are easily studied in a biallelic system. Such a system requires the cross of two inbred lines presumably fixed for alternative alleles of the QTL. However, development of inbred lines can be time consuming and cost ineffective for species with long generation intervals and severe inbreeding depression. In addition, restriction of the investigation to a biallelic system can sometimes be misleading because many potentially important allelic interactions do not have a chance to express and thus fail to be detected. A complicated mating design involving multiple alleles mimics the actual breeding system. However, it is difficult to develop the statistical model and algorithm using the classical maximum-likelihood method. In this study, we investigate the application of a Bayesian method implemented via the Markov chain Monte Carlo (MCMC) algorithm to QTL mapping under arbitrarily complicated mating designs. We develop the method under a mixed-model framework where the genetic values of founder alleles are treated as random and the nongenetic effects are treated as fixed. With the MCMC algorithm, we first draw the gene flows from the founders to the descendants for each QTL and then draw samples of the genetic parameters. Finally, we are able to simultaneously infer the posterior distribution of the number, the additive and dominance variances, and the chromosomal locations of all identified QTL.     

Maximum-likelihood and markov chain monte carlo approaches to estimate inbreeding and effective size from allele frequency changes

Maximum-likelihood and Bayesian (MCMC algorithm) estimates of the increase of the Wright-Malécot inbreeding coefficient, F(t), between two temporally spaced samples, were developed from the Dirichlet approximation of allelic frequency distribution (model MD) and from the admixture of the Dirichlet approximation and the probabilities of fixation and loss of alleles (model MDL). Their accuracy was tested using computer simulations in which F(t) = 10% or less. The maximum-likelihood method based on the model MDL was found to be the best estimate of F(t) provided that initial frequencies are known exactly. When founder frequencies are estimated from a limited set of founder animals, only the estimates based on the model MD can be used for the moment. In this case no method was found to be the best in all situations investigated. The likelihood and Bayesian approaches give better results than the classical F-statistics when markers exhibiting a low polymorphism (such as the SNP markers) are used. Concerning the estimations of the effective population size all the new estimates presented here were found to be better than the F-statistics classically used.     

On locating multiple interacting quantitative trait loci in intercross designs

A modified version (mBIC) of the Bayesian Information Criterion (BIC) has been previously proposed for backcross designs to locate multiple interacting quantitative trait loci. In this article, we extend the method to intercross designs. We also propose two modifications of the mBIC. First we investigate a two-stage procedure in the spirit of empirical Bayes methods involving an adaptive (i.e., data-based) choice of the penalty. The purpose of the second modification is to increase the power of detecting epistasis effects at loci where main effects have already been detected. We investigate the proposed methods by computer simulations under a wide range of realistic genetic models, with nonequidistant marker spacings and missing data. In the case of large intermarker distances we use imputations according to Haley and Knott regression to reduce the distance between searched positions to not more than 10 cM. Haley and Knott regression is also used to handle missing data. The simulation study as well as real data analyses demonstrates good properties of the proposed method of QTL detection.     

Simple deterministic identity-by-descent coefficients and estimation of QTL allelic effects in full and half sibs

Accurate and rapid methods for the detection of quantitative trait loci (QTLs) and evaluation of consequent allelic effects are required to implement marker-assisted selection in outbred populations. In this study, we present a simple deterministic method for estimating identity-by-descent (IBD) coefficients in full- and half-sib families that can be used for the detection of QTLs via a variance-component approach. In a simulated dataset, IBD coefficients among sibs estimated by the simple deterministic and Markov chain Monte Carlo (MCMC) methods with three or four alleles at each marker locus exhibited a correlation of greater than 0.99. This high correlation was also found in QTL analyses of data from an outbred pig population. Variance component analysis used both the simple deterministic and MCMC methods to estimate IBD coefficients. Both procedures detected a QTL at the same position and gave similar test statistics and heritabilities. The MCMC method, however, required much longer computation than the simple method. The conversion of estimated QTL genotypic effects into allelic effects for use in marker-assisted selection is also demonstrated.     

A haplotypic approach to founder-origin probabilities and outbred QTL analysis

Founder-origin probability methods are used to trace specific chromosomal segments in individual offspring. A haplotypic method was developed for calculating founder-origin probabilities in three-generation outbred pedigrees suited to quantitative trait locus (QTL) analysis. Estimators for expected founder-origin proportions were derived for a linkage group segment, an entire linkage group and a complete haplotype. If the founders are truly outbred, the haplotypic method gives a close approximation when compared with the Haley et al. (1994) method that simultaneously uses all marker information for QTL analysis, and it is less computationally demanding. The chief limitation of the haplotypic method is that some information in two-allele intercross marker-type configurations is ignored. Informativeness of marker arrays is discussed in the framework of founder-origin probabilities and proportions. The haplotypic method can be extended to more complex pedigrees with additional generations.     

Bayesian mapping of QTL in outbred F2 families allowing inference about whether F0 grandparents are homozygous or heterozygous at QTL

In this paper, we propose a new Bayesian method for QTL analysis in outbred F2 families based on Markov chain Monte Carlo (MCMC) estimation allowing inference about whether each of F0 founders (grandparents) is homozygous or heterozygous at QTL. This, in turn, allows us to select a model accurately explaining observations of phenotypes for F2 individuals. The proposed method performs the fitting a statistical model of the two possible QTL states in each F0 grandparent, that is, homozygous and heterozygous at QTL, and gives a posterior distribution for the QTL states in each F0 grandparent. We confine ourselves to the discrimination of two QTL states, homozygous or heterozygous, for each of the F0 grandparents without taking into consideration whether common alleles are shared by F0 grandparents. The statistical model includes allelic effects and dominance effects for each QTL. The number of parameters representing allelic effects and dominance effects is therefore changed depending on the QTL states. A Reversible Jump MCMC technique is used for transition between the models of different dimensions. The effectiveness of the proposed method was investigated using simulation experiments. It was practicable to estimate the QTL states of F0 grandparents as well as the number, the locations and the effects of QTL segregating in an outbred F2 family.     

A family-based likelihood ratio test for general pedigree structures that allows for genotyping error and missing data

The purpose of this work is the development of a family-based association test that allows for random genotyping errors and missing data and makes use of information on affected and unaffected pedigree members. We derive the conditional likelihood functions of the general nuclear family for the following scenarios: complete parental genotype data and no genotyping errors; only one genotyped parent and no genotyping errors; no parental genotype data and no genotyping errors; and no parental genotype data with genotyping errors. We find maximum likelihood estimates of the marker locus parameters, including the penetrances and population genotype frequencies under the null hypothesis that all penetrance values are equal and under the alternative hypothesis. We then compute the likelihood ratio test. We perform simulations to assess the adequacy of the central chi-square distribution approximation when the null hypothesis is true. We also perform simulations to compare the power of the TDT and this likelihood-based method. Finally, we apply our method to 23 SNPs genotyped in nuclear families from a recently published study of idiopathic scoliosis (IS). Our simulations suggest that this likelihood ratio test statistic follows a central chi-square distribution with 1 degree of freedom under the null hypothesis, even in the presence of missing data and genotyping errors. The power comparison shows that this likelihood ratio test is more powerful than the original TDT for the simulations considered. For the IS data, the marker rs7843033 shows the most significant evidence for our method (p = 0.0003), which is consistent with a previous report, which found rs7843033 to be the 2nd most significant TDTae p value among a set of 23 SNPs.     

Modeling of identity-by-descent processes along a chromosome between haplotypes and their genotyped ancestors

Identity-by-descent probabilities are important for many applications in genetics. Here we propose a method for modeling the transmission of the haplotypes from the closest genotyped relatives along an entire chromosome. The method relies on a hidden Markov model where hidden states correspond to the set of all possible origins of a haplotype within a given pedigree. Initial state probabilities are estimated from average genetic contribution of each origin to the modeled haplotype while transition probabilities are computed from recombination probabilities and pedigree relationships between the modeled haplotype and the various possible origins. The method was tested on three simulated scenarios based on real data sets from dairy cattle, Arabidopsis thaliana, and maize. The mean identity-by-descent probabilities estimated for the truly inherited parental chromosome ranged from 0.94 to 0.98 according to the design and the marker density. The lowest values were observed in regions close to crossing over or where the method was not able to discriminate between several origins due to their similarity. It is shown that the estimated probabilities were correctly calibrated. For marker imputation (or QTL allele prediction for fine mapping or genomic selection), the method was efficient, with 3.75% allelic imputation error rates on a dairy cattle data set with a low marker density map (1 SNP/Mb). The method should prove useful for situations we are facing now in experimental designs and in plant and animal breeding, where founders are genotyped with relatively high markers densities and last generation(s) genotyped with a lower-density panel.     

Towards linkage analysis with markers in linkage disequilibrium by graphical modelling

We review recent developments of MCMC integration methods for computations on graphical models for two applications in statistical genetics: modelling allelic association and pedigree based linkage analysis. We discuss and illustrate estimation of graphical models from haploid and diploid genotypes, and the importance of MCMC updating schemes beyond what is strictly necessary for irreducibility. We then outline an approach combining these methods to compute linkage statistics when alleles at the marker loci are in linkage disequilibrium. Other extensions suitable for analysis of SNP genotype data in pedigrees are also discussed and programs that implement these methods, and which are available from the author's web site, are described. We conclude with a discussion of how this still experimental approach might be further developed.     

Strain distribution patterns for genetic markers in the LSXSS recombinant-inbred series

We present the strain distribution patterns (SDPs) of 118 SSLP markers and three pigmentation genes that have been characterized in 27 strains from the LSXSS RI series. This coarse map provides a resource for linkage studies of phenotypes that are heritable in the LSXSS RI series. The LSXSS recombinant inbred (RI) strains were derived from the Long-Sleep (LS) and Short-Sleep (SS) selected lines of mice that were selected for differential sensitivity to ethanol but are also differentially sensitive to a variety of other alcohols, barbiturates, sedative hypnotics, and general anesthetics. Since the parents were not inbred, two atypical factors are present in these SDPs. First, more than two alleles are frequently found in these RIs, and second, some alleles can be uniquely associated with one or the other parent while other alleles may be found in both parental lines. To validate the markers found in the parental line, we genotyped all parental mice from one generation of both the LS and SS lines, thus leading to a set of marker SDPs that are useful for further phenotypic association and identification of provisional QTLs. Received: 15 November 1995 / Accepted: 6 February 1996     

A simple and rapid method for calculating identity-by-descent matrices using multiple markers

A fast, partly recursive deterministic method for calculating Identity-by-Descent (IBD) probabilities was developed with the objective of using IBD in Quantitative Trait Locus (QTL) mapping. The method combined a recursive method for a single marker locus with a method to estimate IBD between sibs using multiple markers. Simulated data was used to compare the deterministic method developed in the present paper with a stochastic method (LOKI) for precision in estimating IBD probabilities and performance in the task of QTL detection with the variance component approach. This comparison was made in a variety of situations by varying family size and degree of polymorphism among marker loci. The following were observed for the deterministic method relative to MCMC: (i) it was an order of magnitude faster; (ii) its estimates of IBD probabilities were found to agree closely, even though it does not extract information when haplotypes are not known with certainty; (iii) the shape of the profile for the QTL test statistic as a function of location was similar, although the magnitude of the test statistic was slightly smaller; and (iv) the estimates of QTL variance was similar. It was concluded that the method proposed provided a rapid means of calculating the IBD matrix with only a small loss in precision, making it an attractive alternative to the use of stochastic MCMC methods. Furthermore, developments in marker technology providing denser maps would enhance the relative advantage of this method.     

Combined linkage and linkage disequilibrium QTL mapping in multiple families of maize (Zea mays L.) line crosses highlights complementarities between models based on parental haplotype and single locus polymorphism

Advancements in genotyping are rapidly decreasing marker costs and increasing marker density. This opens new possibilities for mapping quantitative trait loci (QTL), in particular by combining linkage disequilibrium information and linkage analysis (LDLA). In this study, we compared different approaches to detect QTL for four traits of agronomical importance in two large multi-parental datasets of maize (Zea mays L.) of 895 and 928 testcross progenies composed of 7 and 21 biparental families, respectively, and genotyped with 491 markers. We compared to traditional linkage-based methods two LDLA models relying on the dense genotyping of parental lines with 17,728 SNP: one based on a clustering approach of parental line segments into ancestral alleles and one based on single marker information. The two LDLA models generally identified more QTL (60 and 52 QTL in total) than classical linkage models (49 and 44 QTL in total). However, they performed inconsistently over datasets and traits suggesting that a compromise must be found between the reduction of allele number for increasing statistical power and the adequacy of the model to potentially complex allelic variation. For some QTL, the model exclusively based on linkage analysis, which assumed that each parental line carried a different QTL allele, was able to capture remaining variation not explained by LDLA models. These complementarities between models clearly suggest that the different QTL mapping approaches must be considered to capture the different levels of allelic variation at QTL involved in complex traits.     

On the detection of imprinted quantitative trait loci in line crosses: effect of linkage disequilibrium

Imprinted quantitative trait loci (QTL) are commonly reported in studies using line-cross designs, especially in livestock species. It was previously shown that such parent-of-origin effects might result from the nonfixation of QTL alleles in one or both parental lines, rather than from genuine molecular parental imprinting. We herein demonstrate that if linkage disequilibrium exists between marker loci and nonfixed QTL, spurious detection of pseudo-imprinting is increased by an additional 40–80% in scenarios mimicking typical livestock situations. This is due to the fact that imprinting can be tested only in F₂ offspring whose sire and dam have distinct marker genotypes. In the case of linkage disequilibrium between markers and QTL, such parents have a higher chance to have distinct QTL genotypes as well, thus resulting in distinct padumnal and madumnal allele substitution effects, i.e., QTL pseudo-imprinting.     

Development of a STS marker linked to a major locus controlling flour colour in wheat (Triticum aestivum L.)

Flour colour is an important quality trait in the production of bread, noodles and other related end products. Current screening for flour colour in breeding programs requires several grams of flour to be milled. In order to screen large numbers of plants, a rapid PCR-based assay is required. We report here the conversion of a codominant AFLP marker linked to a major locus controlling flour colour in hexaploid wheat, to a sequence tagged site (STS) marker for use in marker-assisted selection (MAS). The two-allelic AFLP bands were cloned and sequenced to allow specific primers to be designed. The primers amplified bands of the expected size in the parental varieties and co-segregated with the original AFLP marker in the mapping population. The primers also amplified alleles of the expected size from the DNA of parental lines of two other related mapping populations. Cultivars that contributed to the pedigree of the original parent `Schomburgk' used to generate the mapping population were also screened to determine the origin of the `yellow' allele.     

Comparison of properties of tests for assessing tumor clonality

SUMMARY: In a recent article Begg et al. (2007, Biometrics 63, 522-530) proposed a statistical test to determine whether or not a diagnosed second primary tumor is biologically independent of the original primary tumor, by comparing patterns of allelic losses at candidate genetic loci. The proposed concordant mutations test is a conditional test, an adaptation of Fisher's exact test, that requires no knowledge of the marginal mutation probabilities. The test was shown to have generally good properties, but is susceptible to anticonservative bias if there is wide variation in mutation probabilities between loci, or if the individual mutation probabilities of the parental alleles for individual patients differ substantially from each other. In this article, a likelihood ratio test is derived in an effort to address these validity issues. This test requires prespecification of the marginal mutation probabilities at each locus, parameters for which some information will typically be available in the literature. In simulations this test is shown to be valid, but to be considerably less efficient than the concordant mutations test for sample sizes (numbers of informative loci) typical of this problem. Much of the efficiency deficit can be recovered, however, by restricting the allelic imbalance parameter estimate to a prespecified range, assuming that this parameter is in the prespecified range.