The power of interval mapping of quantitative trait loci, using selected sib pairs.

The interval-mapping procedure of Fulker and Cardon for analysis of a quantitative-trait loci (QTL) is extended for application to selected samples of sib pairs. Phenotypic selection of sib pairs, which is known to yield striking increases in power when a single marker is used, provides further increases in power when the interval-mapping approach is used. The greatest benefits of the combined approach are apparent with coarse maps, where QTLs of relatively modest (15%-20%) heritability can be detected with widely spaced markers (40-60 cM apart) in reasonably sized sibling samples. Useful information concerning QTL location is afforded by interval mapping in both selected and unselected samples.     

Quantitative trait loci for yield components in oil palm (Elaeis guineensis Jacq.)

The development of an oil palm RFLP marker map has enabled marker-based QTL mapping studies to be undertaken. Information from 153 RFLP markers was used in combination with phenotypic data from an F₂ population to estimate the position and effects of quantitative trait loci (QTLs) for traits including yield of fruit and its components and measures of vegetative growth. The mapping population consisted of 84 palms segregating for the major gene influencing shell thickness. Marker data were analysed to produce a linkage map consisting of 22 linkage groups. The QTL mapping analysis was carried out by interval mapping and single-marker analysis for the unlinked markers; significance thresholds were generated by permutation. Using both single-marker and interval-mapping analysis significant marker associated QTL effects were found for 11 of the 13 traits analysed. The results of interval-mapping analysis of fruit weight, petiole cross section and rachis length, and ratios of shell:fruit, mesocarp:fruit and kernel:fruit indicated significant (P<0.05) QTLs at the genome-wide threshold. The putative QTLs were associated with between 8.2% and 44.0% of the phenotypic variation, with an average of 27% for the single-marker analysis and 19% for the interval-mapping analysis. The higher percentage of phenotypic variation explained in the single-marker analysis, when compared to the interval-mapping analysis, is likely to be due to the lower stringency associated with the single-marker analysis. Large dominance deviations were associated with a sizeable proportion of the putative QTLs. The ultimate objective of mapping QTLs in commercial populations is to utilise novel breeding strategies such as marker-assisted selection (MAS). The potential impact of MAS in oil palm breeding programmes is discussed. Received: 26 June 2000 / Accepted: 24 October 2000     

Bayesian shrinkage analysis of quantitative trait Loci for dynamic traits

Many quantitative traits are measured repeatedly during the life of an organism. Such traits are called dynamic traits. The pattern of the changes of a dynamic trait is called the growth trajectory. Studying the growth trajectory may enhance our understanding of the genetic architecture of the growth trajectory. Recently, we developed an interval-mapping procedure to map QTL for dynamic traits under the maximum-likelihood framework. We fit the growth trajectory by Legendre polynomials. The method intended to map one QTL at a time and the entire QTL analysis involved scanning the entire genome by fitting multiple single-QTL models. In this study, we propose a Bayesian shrinkage analysis for estimating and mapping multiple QTL in a single model. The method is a combination between the shrinkage mapping for individual quantitative traits and the Legendre polynomial analysis for dynamic traits. The multiple-QTL model is implemented in two ways: (1) a fixed-interval approach where a QTL is placed in each marker interval and (2) a moving-interval approach where the position of a QTL can be searched in a range that covers many marker intervals. Simulation study shows that the Bayesian shrinkage method generates much better signals for QTL than the interval-mapping approach. We propose several alternative methods to present the results of the Bayesian shrinkage analysis. In particular, we found that the Wald test-statistic profile can serve as a mechanism to test the significance of a putative QTL.     

Bayesian mapping of quantitative trait loci under the identity-by-descent-based variance component model

Variance component analysis of quantitative trait loci (QTL) is an important strategy of genetic mapping for complex traits in humans. The method is robust because it can handle an arbitrary number of alleles with arbitrary modes of gene actions. The variance component method is usually implemented using the proportion of alleles with identity-by-descent (IBD) shared by relatives. As a result, information about marker linkage phases in the parents is not required. The method has been studied extensively under either the maximum-likelihood framework or the sib-pair regression paradigm. However, virtually all investigations are limited to normally distributed traits under a single QTL model. In this study, we develop a Bayes method to map multiple QTL. We also extend the Bayesian mapping procedure to identify QTL responsible for the variation of complex binary diseases in humans under a threshold model. The method can also treat the number of QTL as a parameter and infer its posterior distribution. We use the reversible jump Markov chain Monte Carlo method to infer the posterior distributions of parameters of interest. The Bayesian mapping procedure ends with an estimation of the joint posterior distribution of the number of QTL and the locations and variances of the identified QTL. Utilities of the method are demonstrated using a simulated population consisting of multiple full-sib families.     

Using the expectation or the distribution of the identity by descent for mapping quantitative trait loci under the random model.

We examine the ability of four implementations of the random model to map quantitative trait loci (QTLs). The implementations use either the expectation or the distribution of the identity-by-descent value at a putative QTL and either a 2 x 1 vector of sib-pair traits or their scalar difference. When the traits of both sibs are used, there is little difference between the expectation and distribution methods, while the expectation method suffers in both precision and power when the difference between traits is used. This is consistent with the prediction that the difference between the expectation and distribution methods is inversely proportional to the amount of information available for mapping. We find, though, that the amount of information must be very low for this difference to be noticeable. This is exemplified when both marker loci are fixed. In this case, while the expectation method is powerless to detect the QTL, the distribution method can still detect the presence (but not the position) of the QTL 59% of the time (when using trait values) or 14% of the time (when using trait differences). We also note a confounding between estimates of the QTL, polygenic, and error variance. The degree of confounding is small when the vector of trait values is used but can be substantial when the expectation method and trait differences are used. We discuss this in light of the general ability of the random model to partition these components.     

Adding further power to the Haseman and Elston method for detecting linkage in larger sibships: weighting sums and differences

Haseman and Elston (H-E) proposed a robust test to detect linkage between a quantitative trait and a genetic marker. In their method the squared sib-pair trait difference is regressed on the estimated proportion of alleles at a locus shared identical by descent by sib pairs. This method has recently been improved by changing the dependent variable from the squared difference to the mean-corrected product of the sib-pair trait values, a significantly positive regression indicating linkage. Because situations arise in which the original test is more powerful, a further improvement of the H-E method occurs when the dependent variable is changed to a weighted average of the squared sib-pair trait difference and the squared sib-pair mean-corrected trait sum. Here we propose an optimal method of performing this weighting for larger sibships, allowing for the correlation between pairs within a sibship. The optimal weights are inversely proportional to the residual variances obtained from the two different regressions based on the squared sib-pair trait differences and the squared sib-pair mean-corrected trait sums, respectively, allowing for correlations among sib pairs. The proposed method is compared with the existing extension of the H-E approach for larger sibships. Control of the type I error probabilities for sibships of any size can be improved by using a generalized estimating equation approach and the robust sandwich estimate of the variance, or a Monte-Carlo permutation test.     

A novel Markov chain monte carlo approach for constructing accurate meiotic maps

Mapping markers from linkage data continues to be a task performed in many genetic epidemiological studies. Data collected in a study may be used to refine published map estimates and a study may use markers that do not appear in any published map. Furthermore, inaccuracies in meiotic maps can seriously bias linkage findings. To make best use of the available marker information, multilocus linkage analyses are performed. However, two computational issues greatly limit the number of markers currently mapped jointly; the number of candidate marker orders increases exponentially with marker number and computing exact multilocus likelihoods on general pedigrees is computationally demanding. In this article, a new Markov chain Monte Carlo (MCMC) approach that solves both these computational problems is presented. The MCMC approach allows many markers to be mapped jointly, using data observed on general pedigrees with unobserved individuals. The performance of the new mapping procedure is demonstrated through the analysis of simulated and real data. The MCMC procedure performs extremely well, even when there are millions of candidate orders, and gives results superior to those of CRI-MAP.     

Spurious linkage between markers in QTL mapping

Selective genotyping of extreme progeny is a powerful method to increase the information content per individual when looking for quantitative trait loci (QTLs) using molecular markers for which a map is known. However, if marker information from the selected individuals is used to construct the map of the markers, this can lead to distorted segregation of the markers that in turn can lead to the estimation of a spurious linkage between independently inherited markers. The mistaken estimation of linkage between independently inherited markers will occur when there are two (or more) independently inherited QTLs linked to two (or more) markers and the same individuals are used to estimate the map of the markers and to do the QTL estimation. The incorrect linkage occurs because in selecting individuals from the tails of the phenotypic distribution we will also be selecting certain combinations of the markers instead of obtaining a random sample of the true distribution of the marker genotypes. Analytical results are outlined and the analyses of a simulated data set illustrate the problems that could arise when data from individuals chosen by selective genotyping are incorrectly employed to construct a marker map. A strategy is proposed to remedy this problem.     

Estimating polygenic effects using markers of the entire genome

Molecular markers have been used to map quantitative trait loci. However, they are rarely used to evaluate effects of chromosome segments of the entire genome. The original interval-mapping approach and various modified versions of it may have limited use in evaluating the genetic effects of the entire genome because they require evaluation of multiple models and model selection. Here we present a Bayesian regression method to simultaneously estimate genetic effects associated with markers of the entire genome. With the Bayesian method, we were able to handle situations in which the number of effects is even larger than the number of observations. The key to the success is that we allow each marker effect to have its own variance parameter, which in turn has its own prior distribution so that the variance can be estimated from the data. Under this hierarchical model, we were able to handle a large number of markers and most of the markers may have negligible effects. As a result, it is possible to evaluate the distribution of the marker effects. Using data from the North American Barley Genome Mapping Project in double-haploid barley, we found that the distribution of gene effects follows closely an L-shaped Gamma distribution, which is in contrast to the bell-shaped Gamma distribution when the gene effects were estimated from interval mapping. In addition, we show that the Bayesian method serves as an alternative or even better QTL mapping method because it produces clearer signals for QTL. Similar results were found from simulated data sets of F(2) and backcross (BC) families.     

Transformation of sib-pair values for the Haseman-Elston method

The squared sib-pair phenotype difference (SQD) has been used as a dependent variable in the Haseman-Elston (H-E) regression quantitative-trait locus (QTL) linkage method, but it has been shown that the SQD does not make full use of linkage information. In this study, we examine the efficiency of SQD in H-E regression compared to other proposed functions of the sib-pair phenotypes. A new function of sib-pair phenotypes, the product of pair values corrected with family mean (PCF), is shown to have desirable properties in many realistic situations. Consistent results were obtained using a combination of large-sample analytic approximations, simulation, and analyses of quantitative-trait data from Genetic Analysis Workshop 10. The advantages of PCF are further improved in the presence of family-specific effects arising from environmental factors or when additional QTLs influence the trait. All of the phenotype functions are incorporated in our new, freely available linkage-mapping program MULTIGENE 1.0 for the PC environment.     

Which strategy is better for linkage analysis: single-nucleotide polymorphisms or microsatellites? Evaluation by identity-by-state – identity-by-descent transformation affected sib-pair method on GAW14 data

The central issue for Genetic Analysis Workshop 14 (GAW14) is the question, which is the better strategy for linkage analysis, the use of single-nucleotide polymorphisms (SNPs) or microsatellite markers? To answer this question we analyzed the simulated data using Duffy's SIB-PAIR program, which can incorporate parental genotypes, and our identity-by-state – identity-by-descent (IBS-IBD) transformation method of affected sib-pair linkage analysis which uses the matrix transformation between IBS and IBD. The advantages of our method are as follows: the assumption of Hardy-Weinberg equilibrium is not necessary; the parental genotype information maybe all unknown; both IBS and its related IBD transformation can be used in the linkage analysis; the determinant of the IBS-IBD transformation matrix provides a quantitative measure of the quality of the marker in linkage analysis. With the originally distributed simulated data, we found that 1) for microsatellite markers there are virtually no differences in types I and II error rates when parental genotypes were or were not used; 2) on average, a microsatellite marker has more power than a SNP marker does in linkage detection; 3) if parental genotype information is used, SNP markers show lower type I error rates than microsatellite markers; and 4) if parental genotypes are not available, SNP markers show considerable variation in type I error rates for different methods.     

Complete multipoint sib-pair analysis of qualitative and quantitative traits.

Sib-pair analysis is an increasingly important tool for genetic dissection of complex traits. Current methods for sib-pair analysis are primarily based on studying individual genetic markers one at a time and thus fail to use the full inheritance information provided by multipoint linkage analysis. In this paper, we describe how to extract the complete multipoint inheritance information for each sib pair. We then describe methods that use this information to map loci affecting traits, thereby providing a unified approach to both qualitative and quantitative traits. Specifically, complete multipoint approaches are presented for (1) exclusion mapping of qualitative traits; (2) maximum-likelihood mapping of qualitative traits; (3) information-content mapping, showing the extent to which all inheritance information has been extracted at each location in the genome; and (4) quantitative-trait mapping, by two parametric methods and one nonparametric method. In addition, we explore the effects of marker density, marker polymorphism, and availability of parents on the information content of a study. We have implemented the analysis methods in a new computer package, MAPMAKER/SIBS. With this computer package, complete multipoint analysis with dozens of markers in hundreds of sib pairs can be carried out in minutes.     

A Bayesian approach for constructing genetic maps when markers are miscoded

The advent of molecular markers has created opportunities for a better understanding of quantitative inheritance and for developing novel strategies for genetic improvement of agricultural species, using information on quantitative trait loci (QTL). A QTL analysis relies on accurate genetic marker maps. At present, most statistical methods used for map construction ignore the fact that molecular data may be read with error. Often, however, there is ambiguity about some marker genotypes. A Bayesian MCMC approach for inferences about a genetic marker map when random miscoding of genotypes occurs is presented, and simulated and real data sets are analyzed. The results suggest that unless there is strong reason to believe that genotypes are ascertained without error, the proposed approach provides more reliable inference on the genetic map.     

A model selection-based interval-mapping method for autopolyploids

While extensive progress has been made in quantitative trait locus (QTL) mapping for diploid species, similar progress in QTL mapping for polyploids has been limited due to the complex genetic architecture of polyploids. To date, QTL mapping in polyploids has focused mainly on tetraploids with dominant and/or codominant markers. Here, we extend this view to include any even ploidy level under a dominant marker system. Our approach first selects the most likely chromosomal marker configurations using a Bayesian selection criterion and then fits an interval-mapping model to each candidate. Profiles of the likelihood-ratio test statistic and the maximum-likelihood estimates (MLEs) of parameters including QTL effects are obtained via the EM algorithm. Putative QTL are then detected using a resampling-based significance threshold, and the corresponding parental configuration is identified to be the underlying parental configuration from which the data are observed. Although presented via pseudo-doubled backcross experiments, this approach can be readily extended to other breeding systems. Our method is applied to single-dose restriction fragment autotetraploid alfalfa data, and the performance is investigated through simulation studies.     

Population-demographic and clinico-genetic characteristics of Huntington chorea in one region of Azerbaijan

The data are presented on prevalence and clinical patterns of Huntington disease in Shamkhor region of Azerbaijan, where about 126.8 thousand inhabitants live. Population, demographic and genealogical data show that high prevalence of Huntington disease in that region is determined by the founder effects, reinforced later by extended reproduction of the population. Linkage analysis using the affected sib-pair method failed to reveal a linkage between Huntington's chorea locus and HLA, AB0, MN systems. Significant probability of linkage to Huntington's chorea locus was calculated for Gc marker.     

A logistic regression mixture model for interval mapping of genetic trait loci affecting binary phenotypes

Often in genetic research, presence or absence of a disease is affected by not only the trait locus genotypes but also some covariates. The finite logistic regression mixture models and the methods under the models are developed for detection of a binary trait locus (BTL) through an interval-mapping procedure. The maximum-likelihood estimates (MLEs) of the logistic regression parameters are asymptotically unbiased. The null asymptotic distributions of the likelihood-ratio test (LRT) statistics for detection of a BTL are found to be given by the supremum of a chi2-process. The limiting null distributions are free of the null model parameters and are determined explicitly through only four (backcross case) or nine (intercross case) independent standard normal random variables. Therefore a threshold for detecting a BTL in a flanking marker interval can be approximated easily by using a Monte Carlo method. It is pointed out that use of a threshold incorrectly determined by reading off a chi2-probability table can result in an excessive false BTL detection rate much more severely than many researchers might anticipate. Simulation results show that the BTL detection procedures based on the thresholds determined by the limiting distributions perform quite well when the sample sizes are moderately large.     

A sib-pair approach to interval mapping of quantitative trait loci.

An interval mapping procedure based on the sib-pair method of Haseman and Elston is developed, and simulation studies are carried out to explore its properties. The procedure is analogous to other interval mapping procedures used with experimental material, such as plants and animals, and yields very similar results in terms of the location and effect size of a quantitative trait locus (QTL). The procedure offers an advantage over the conventional Haseman and Elston approach, in terms of power, and provides useful information concerning the location of a QTL. Because of its simplicity, the method readily lends itself to the analysis of selected samples for increased power and the evaluation of multilocus models of complex phenotypes.     

Relationship estimation by Markov-process models in a sib-pair linkage study

The results of sib-pair linkage studies may be compromised if a substantial number of putative sib pairs are not actually sib pairs. For classification of pairs in a sib-pair genome scan, I propose multipoint methods that are based on a Markov-process model of allele sharing along the chromosome. These methods can be implemented by standard algorithms that compute multipoint marker allele-sharing probabilities for sib pairs. When marker data from at least half the genome are used, misclassification rates are small. The methods will be implemented in an upcoming version of the computer software package S.A.G.E.     

Mapping multiple Quantitative Trait Loci by Bayesian classification

We developed a classification approach to multiple quantitative trait loci (QTL) mapping built upon a Bayesian framework that incorporates the important prior information that most genotypic markers are not cotransmitted with a QTL or their QTL effects are negligible. The genetic effect of each marker is modeled using a three-component mixture prior with a class for markers having negligible effects and separate classes for markers having positive or negative effects on the trait. The posterior probability of a marker's classification provides a natural statistic for evaluating credibility of identified QTL. This approach performs well, especially with a large number of markers but a relatively small sample size. A heat map to visualize the results is proposed so as to allow investigators to be more or less conservative when identifying QTL. We validated the method using a well-characterized data set for barley heading values from the North American Barley Genome Mapping Project. Application of the method to a new data set revealed sex-specific QTL underlying differences in glucose-6-phosphate dehydrogenase enzyme activity between two Drosophila species. A simulation study demonstrated the power of this approach across levels of trait heritability and when marker data were sparse.     

Comparing the distribution of RAPD and RFLP markers in a high density linkage map of sugar beet.

The distribution of RAPD markers was compared with that of RFLP markers in a high density linkage map of sugar beet. The same mapping population of 161 F2 individuals was used to generate all the marker data. The total map comprises 160 RAPD and 248 RFLP markers covering 508 cM. Both the RAPD and the RFLP markers show a high degree of clustering over the nine linkage groups. The pattern is compatible with a strong distal localization of recombination in the sugar beet. It leads generally to one major cluster of markers in the centre of each linkage group. In regions of high marker density, dominant RAPD markers present in either linkage phase and codominant RFLP markers are subclustered relative to each other. This phenomenon is shown to be attributable to: (i) effects of the mapping procedure when dominant and codominant data are combined, (ii) effects of the mapping procedure when dominant data in both linkage phases are combined, and (iii) genuine differences in the way RAPD and RFLP markers are recruited.