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.     

X-linked extension of the revised Haseman-Elston algorithm for linkage analysis in sib pairs

Haseman and Elston (H-E) proposed a regression-based robust test of linkage between a marker and an autosomal quantitative trait locus, using the squared sib pair trait difference as a dependent variable and the proportion of alleles shared identical by descent by the sib pair as an independent variable. Several authors have proposed improvement of the original H-E's seminal work by using an optimal linear combination of squared sum and squared difference as the dependent variable. In this paper, we extend Haseman and Elston's sib pair method to an X-linked locus. We give a general formulation of the complete regression model and details of the regression coefficients in terms of variance components. Simulation results are presented to describe the power of this technique for a theoretical best case scenario.     

两种常用数量性状连锁分析方法的原理和进展

在复杂性状疾病的家系连锁研究中,Haseman-Elston回归分析和方差组成模型是常用的两种数量性状连锁分析方法。前者主要针对同胞对的性状值差或和的平方进行回归分析;后者引用方差组成模型,将数量性状分解为遗传方差和环境方差,可估计二者对表型的影响。两种方法可应用于同胞对、核心家系或扩展家系,定位数量性状基因座。本文对这两种模型的原理、算法及其进展进行了综述,并给出了常用的统计软件包。 Abstract：In this article, we discussed two model-free methods for detecting genetic linkage for quantitative traits, Haseman-Elston regression approach and variance components approach. The former is a regression approach for detecting linkage based on the squared difference or squared sums in quantitative trait values of sib-pairs and their estimated marker IBD scores. The latter can jointly model covariate effects along with variance components, including genetic component and non-genetic sources of variability. We have outlined the model assumption, the algorithm and the extensions for the both methods.     

Nonparametric trend statistic incorporating dispersion differences in sib pair linkage for quantitative traits

The Haseman-Elston (HE) regression method and its extensions are widely used in genetic studies for detecting linkage to quantitative trait loci (QTL) using sib pairs. The principle underlying the simple HE regression method is that the similarity in phenotypes between two siblings increases as they share an increasing number of alleles identical by descent (IBD) from their parents at a particular marker locus. In such a procedure, similarity was identified with the locations, that is, means of groups of sib pairs sharing 0, 1, and 2 alleles IBD. A more powerful, rank-based nonparametric test to detect increasing similarity in sib pairs is presented by combining univariate trend statistics not only of locations, but also of dispersions of the squared phenotypic differences of two siblings for three groups. This trend test does not rely on distributional assumptions, and is applicable to the skewed or leptokurtic phenotypic distributions, in addition to normal or near normal phenotypic distributions. The performances of nonparametric trend statistics, including nonparametric regression slope, are compared with the HE regression methods as genetic linkage strategies.     

Powerful regression-based quantitative-trait linkage analysis of general pedigrees

We present a new method of quantitative-trait linkage analysis that combines the simplicity and robustness of regression-based methods and the generality and greater power of variance-components models. The new method is based on a regression of estimated identity-by-descent (IBD) sharing between relative pairs on the squared sums and squared differences of trait values of the relative pairs. The method is applicable to pedigrees of arbitrary structure and to pedigrees selected on the basis of trait value, provided that population parameters of the trait distribution can be correctly specified. Ambiguous IBD sharing (due to incomplete marker information) can be accommodated in the method by appropriate specification of the variance-covariance matrix of IBD sharing between relative pairs. We have implemented this regression-based method and have performed simulation studies to assess, under a range of conditions, estimation accuracy, type I error rate, and power. For normally distributed traits and in large samples, the method is found to give the correct type I error rate and an unbiased estimate of the proportion of trait variance accounted for by the additive effects of the locus-although, in cases where asymptotic theory is doubtful, significance levels should be checked by simulations. In large sibships, the new method is slightly more powerful than variance-components models. The proposed method provides a practical and powerful tool for the linkage analysis of quantitative traits.     

A score test for the linkage analysis of qualitative and quantitative traits based on identity by descent data from sib-pairs

We propose a general likelihood-based approach to the linkage analysis of qualitative and quantitative traits using identity by descent (IBD) data from sib-pairs. We consider the likelihood of IBD data conditional on phenotypes and test the null hypothesis of no linkage between a marker locus and a gene influencing the trait using a score test in the recombination fraction theta between the two loci. This method unifies the linkage analysis of qualitative and quantitative traits into a single inferential framework, yielding a simple and intuitive test statistic. Conditioning on phenotypes avoids unrealistic random sampling assumptions and allows sib-pairs from differing ascertainment mechanisms to be incorporated into a single likelihood analysis. In particular, it allows the selection of sib-pairs based on their trait values and the analysis of only those pairs having the most informative phenotypes. The score test is based on the full likelihood, i.e. the likelihood based on all phenotype data rather than just differences of sib-pair phenotypes. Considering only phenotype differences, as in Haseman and Elston (1972) and Kruglyak and Lander (1995), may result in important losses in power. The linkage score test is derived under general genetic models for the trait, which may include multiple unlinked genes. Population genetic assumptions, such as random mating or linkage equilibrium at the trait loci, are not required. This score test is thus particularly promising for the analysis of complex human traits. The score statistic readily extends to accommodate incomplete IBD data at the test locus, by using the hidden Markov model implemented in the programs MAPMAKER/SIBS and GENEHUNTER (Kruglyak and Lander, 1995; Kruglyak et al., 1996). Preliminary simulation studies indicate that the linkage score test generally matches or outperforms the Haseman-Elston test, the largest gains in power being for selected samples of sib-pairs with extreme phenotypes.     

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.     

Effects of misspecification of allele frequencies on the type I error rate of model-free linkage analysis

Linkage analyses of simulated quantitative trait data were performed using the Haseman-Elston (H-E) sib pair regression test to investigate the effects of inaccurate allele frequency estimates on the type I error rates of this test. Computer simulations generating a quantitative trait in nuclear families were performed using GASP [1]. Assuming no linkage, several data sets were simulated; they differed in marker allele numbers and frequencies, number of sib pairs and number of sibships. Each set of simulated data was analyzed using (1) all parental marker data, (2) half of the parental marker data, and (3) no parental marker data, using both correct and incorrect allele frequencies in the latter 2 cases. The H-E sib pair linkage method was found to be robust to misspecification of marker allele frequencies regardless of the number of alleles.     

Equivalence between Haseman-Elston and variance-components linkage analyses for sib pairs

The Haseman-Elston regression method offers a simpler alternative to variance-components (VC) models, for the linkage analysis of quantitative traits. However, even the "revisited" method, which uses the cross-product--rather than the squared difference--in sib trait values, is, in general, less powerful than VC models. In this report, we clarify the relative efficiencies of existing Haseman-Elston methods and show how a new Haseman-Elston method can be constructed to have power equivalent to that of VC models. This method uses as the dependent variable a linear combination of squared sums and squared differences, in which the weights are determined by the overall trait correlation between sibs in a population. We show how this method can be used for both the selection of maximally informative sib pairs for genotyping and the subsequent analysis of such selected samples.     

Association tests in nuclear families

We present a conditional likelihood approach for testing linkage disequilibrium in nuclear families having multiple affected offspring. The likelihood, conditioned on the identity-by-descent (IBD) structure of the sibling genotypes, is unaffected by familial correlation in disease status that arises from linkage between a marker locus and the unobserved trait locus. Two such conditional likelihoods are compared: one that conditions on IBD and phase of the transmitted alleles and a second which conditions only on IBD of the transmitted alleles. Under the log-additive model, the first likelihood is equivalent to the allele-counting methods proposed in the literature. The second likelihood is valid under the added assumption of equal male and female recombination fractions. In a simulation study, we demonstrated that in sibships having two or three affected siblings the score test from each likelihood had the correct test size for testing disequilibrium. They also led to equivalent power to detect linkage disequilibrium at the 5% significance level.     

Consanguinity and the sib-pair method: an approach using identity by descent between and within individuals.

To test for linkage between a trait and a marker, one can consider identical marker alleles in related individuals, for instance, sibs. For recessive diseases, it has been shown that some information may be gained from the identity by descent (IBD) of the two alleles of an affected inbred individual at the marker locus. The aim of this paper is to extend the sib-pair method of linkage analysis to the situation of sib pairs sampled from consanguineous populations. This extension takes maximum advantage of the information provided by both the IBD pattern between sibs and allelic identity within each sib of the pair. This is possible through the use of the condensed identity coefficients. Here, we propose a new test of linkage based on a chi2. We compare the performance of this test with that of the classical chi2 test based on the distribution of sib pairs sharing 0, 1, or 2 alleles IBD. For sib pairs from first-cousin matings, the proposed test can better detect the role of a disease-susceptibility (DS) locus. Its power is shown to be greater than that of the classical test, especially for models where the DS allele may be common and incompletely penetrant; that is to say for situations that may be encountered in multifactorial diseases. A study of the impact of inbreeding on the expected proportions of sib pairs sharing 0, 1, or 2 alleles IBD is also performed here. Ignoring inbreeding, when in fact inbreeding exists, increases the rate of type I errors in tests of linkage.     

Imprinting detection by extending a regression-based QTL analysis method

We present an extension of a regression-based quantitative-trait linkage analysis method to incorporate parent-of-origin effects. We separately regressed total, paternal, and maternal IBD sharing on traits’ squared sums and differences. We also developed a test for imprinting that indicates whether there is any difference between the paternal and maternal regression coefficients. Since this method treats the identity-by-descent information as the dependent variable that is conditioned on the trait, it can be readily applied to data from complex ascertainment processes. We performed a simulation study to examine the performance of the method. We found that when using empirical critical values, the method shows identical or higher power compared to existing methods for evaluation of parent-of-origin effect in linkage analysis of quantitative traits. Missing parental genotypes increase the type I error rate of the linkage test and decrease the power of the imprinting test. When the major gene has a low heritability, the power of the method decreases considerably, but the statistical tests still perform well. We also applied a permutation algorithm, which ensures the appropriate type I error rate for the test for imprinting. The method was applied to a data from a study of 6 body size related measures and 23 loci on chromosome 7 for 255 nuclear families. Multipoint identities-by-descent (IBD) were obtained using a modification of the SIMWALK 2 program. A parent-of-origin effect consistent with maternal imprinting was suggested at 99.67–111.26 Mb for body mass index, bioelectrical impedance analysis, waist circumference, and leptin concentration. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. An erratum to this article can be found at     

Combined linkage and association mapping of quantitative trait loci by multiple markers

Using multiple diallelic markers, variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL) based on nuclear families. The objective is to build a model that may fully use marker information for fine association mapping of QTL in the presence of prior linkage. The measures of linkage disequilibrium and the genetic effects are incorporated in the mean coefficients and are decomposed into orthogonal additive and dominance effects. The linkage information is modeled in variance-covariance matrices. Hence, the proposed methods model both association and linkage in a unified model. On the basis of marker information, a multipoint interval mapping method is provided to estimate the proportion of allele sharing identical by descent (IBD) and the probability of sharing two alleles IBD at a putative QTL for a sib-pair. To test the association between the trait locus and the markers, both likelihood-ratio tests and F-tests can be constructed on the basis of the proposed models. In addition, analytical formulas of noncentrality parameter approximations of the F-test statistics are provided. Type I error rates of the proposed test statistics are calculated to show their robustness. After comparing with the association between-family and association within-family (AbAw) approach by Abecasis and Fulker et al., it is found that the method proposed in this article is more powerful and advantageous based on simulation study and power calculation. By power and sample size comparison, it is shown that models that use more markers may have higher power than models that use fewer markers. The multiple-marker analysis can be more advantageous and has higher power in fine mapping QTL. As an application, the Genetic Analysis Workshop 12 German asthma data are analyzed using the proposed methods.     

Weighting improves the "new Haseman-Elston" method

Elston et al. [Genet Epidemiol, in press] apply the results of Wright [Am J Hum Genet 1997;60:740-742] and Drigalenko [Am J Hum Genet 1998;63:1242-1245] to extend the traditional Haseman-Elston regression scheme [Haseman and Elston, Behav Genet 1972;2:3-19] to include not only linkage information contained in the sib pair's squared difference, but also information in their mean-corrected squared sum. The new algorithm detects linkage to a quantitative trait locus by modelling sib pair trait covariance as a function of identity-by-descent status. We demonstrate why this new estimator is suboptimal and can in some cases be inferior to the original Haseman-Elston method. We also describe a simple approach to estimation which improves on this new Haseman-Elston method by incorporating variance-based weights into the test statistic while staying within the linear modelling framework. In support of our theoretical claim, we conduct both a sib pair simulation and an application to GAW 10 sib pair data showing that our new estimator is superior to both the old and new Haseman-Elston schemes currently implemented in the analysis package S.A.G.E. 4.0.     

Extension of the Haseman-Elston regression model to longitudinal data

We propose an extension to longitudinal data of the Haseman and Elston regression method for linkage analysis. The proposed model is a mixed model having several random effects. As response variable, we investigate the sibship sample mean corrected cross-product (smHE) and the BLUP-mean corrected cross product (pmHE), comparing them with the original squared difference (oHE), the overall mean corrected cross-product (rHE), and the weighted average of the squared difference and the squared mean-corrected sum (wHE). The proposed model allows for the correlation structure of longitudinal data. Also, the model can test for gene x time interaction to discover genetic variation over time. The model was applied in an analysis of the Genetic Analysis Workshop 13 (GAW13) simulated dataset for a quantitative trait simulating systolic blood pressure. Independence models did not preserve the test sizes, while the mixed models with both family and sibpair random effects tended to preserve size well.     

Multipoint linkage analysis. A cautionary note.

Multipoint linkage analysis is commonly used to evaluate linkage of a disease to multiple markers in a small region. Multipoint analysis is particularly powerful when the IBD relations of family members at the trait locus are ambiguous. The increased power arises because, unlike single-marker analyses, multipoint analysis uses haplotype information from several markers to infer the IBD relations. We wish to temper this advantage with a cautionary note: multipoint analysis is sensitive to power loss due to misspecification of intermarker distances. Such misspecification is especially problematic when dealing with closely spaced markers. We present computer simulations comparing the power of single-point and multipoint analyses, both when IBD relations are ambiguous, and when the intermarker distances are misspecified. We conclude that when evaluating markers in a small region to confirm or refute previous findings, a situation in which p values of modest statistical significance are important, single marker analyses may provide more reliable measures of the strength of support for linkage than multipoint statistics.     

Efficiency robust tests for mapping quantitative trait loci using extremely discordant sib pairs

In 1972, Haseman and Elston proposed a pioneering regression method for mapping quantitative trait loci using randomly selected sib pairs. Recently, the statistical power of their method was shown to be increased when extremely discordant sib pairs are ascertained. While the precise genetic model may not be known, prior information that constrains IBD probabilities is often available. We investigate properties of tests that are robust against model uncertainty and show that the power gain from further constraining IBD probabilities is marginal. The additional linkage information contained in the trait values can be incorporated by combining the Haseman-Elston regression method and a robust allele sharing test.     

Composite statistics for QTL mapping with moderately discordant sibling pairs

Extreme discordant sibling-pair (EDSP) designs have been shown in theory to be very powerful for mapping quantitative-trait loci (QTLs) in humans. However, their practical applicability has been somewhat limited by the need to phenotype very large populations to find enough pairs that are extremely discordant. In this paper, we demonstrate that there is also substantial power in pairs that are only moderately discordant, and that designs using moderately discordant pairs can yield a more practical balance between phenotyping and genotyping efforts. The power we demonstrate for moderately discordant pairs stems from a new statistical result. Statistical analysis in discordant-pair studies is generally done by testing for reduced identity by descent (IBD) sharing in the pairs. By contrast, the most commonly-used statistical methods for more standard QTL mapping are Haseman-Elston regression and variance-components analysis. Both of these use statistics that are functions of the trait values given IBD information for the pedigree. We show that IBD sharing statistics and "trait value given IBD" statistics contribute complementary rather than redundant information, and thus that statistics of the two types can be combined to form more powerful tests of linkage. We propose a simple composite statistic, and test it with simulation studies. The simulation results show that our composite statistic increases power only minimally for extremely discordant pairs. However, it boosts the power of moderately discordant pairs substantially and makes them a very practical alternative. Our composite statistic is straightforward to calculate with existing software; we give a practical example of its use by applying it to a Genetic Analysis Workshop (GAW) data set.     

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.     

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.