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.     

A Score-Statistic Approach for the Mapping of Quantitative-Trait Loci with Sibships of Arbitrary Size

The Haseman-Elston method is widely used for the mapping of quantitative-trait loci. However, this method does not use all the information in the data, because it only considers the sib-pair trait-value difference. In addition, the Haseman-Elston method was developed for independent sib pairs; its generalization to nonindependent sib pairs is not straightforward. Here we introduce a score test statistic derived from a normal likelihood based on multiplex sibship data, conditional on identical-by-descent sharing statuses. This score test is asymptotically equivalent to the corresponding likelihood-ratio test, but it is much easier to implement. Because the proposed test uses all of the trait values, it makes more efficient use of the data than does the Haseman-Elston method. The proposed test is naturally applicable to sibships of arbitrary size. The finite-sample properties of the proposed score statistic are evaluated via simulations.     

A unified Haseman-Elston method for testing linkage with quantitative traits

The Haseman and Elston (H-E) method uses a simple linear regression to model the squared trait difference of sib pairs with the shared allele identical by descent (IBD) at marker locus for linkage testing. Under this setting, the squared mean-corrected trait sum is also linearly related to the IBD sharing. However, the resulting slope estimate for either model is not efficient. In this report, we propose a simple linkage test that optimally uses information from the estimates of both models. We also demonstrate that the new test is more powerful than both the traditional one and the recently revisited H-E methods.     

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.     

Regression-based sib pair linkage analysis for binary traits

The Haseman-Elston (HE) regression method offers a mathematically and computationally simpler alternative to variance-components (VC) models for the linkage analysis of quantitative traits. However, current versions of HE regression and VC models are not optimised for binary traits. Here, we present a modified HE regression and a liability-threshold VC model for binary-traits. The new HE method is based on the regression of a linear combination of the trait squares and the trait cross-product on the proportion of alleles identical by descent (IBD) at the putative locus, for sibling pairs. We have implemented both the new HE regression-based method and have performed analytic and simulation studies to assess its type 1 error rate and power under a range of conditions. These studies showed that the new HE method is well-behaved under the null hypothesis in large samples, is more powerful than both the original and the revisited HE methods, and is approximately equivalent in power to the liability-threshold VC model.     

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.     

Testing the robustness of the new Haseman-Elston quantitative-trait loci-mapping procedure

Variance components (VC) techniques have emerged as among the more powerful methods for detection of quantitative-trait loci (QTL) in linkage analysis. Allison et al. found that, with particularly marked leptokurtosis in the phenotypic distribution and moderate-to-high residual sibling correlation, maximum likelihood (ML) VC methods may produce a severe excess of type I errors. The new Haseman-Elston (NHE) method is a least-squares-based VC method for mapping of QTL in sib pairs (Elston et al.). Using simulation, we investigate the robustness of the NHE to marked nonnormality, by means of the same distributions and worst-case conditions identified by Allison et al. for the ML approach (i.e., 100 pairs; high residual sibling correlation). Results showed that, when marked nonnormality is present, the NHE can be used without severe type I error-rate inflation, even at very small alpha levels.     

Haseman-Elston weighted by marker informativity

In the Haseman-Elston approach the squared phenotypic difference is regressed on the proportion of alleles shared identical by descent (IBD) to map a quantitative trait to a genetic marker. In applications the IBD distribution is estimated and usually cannot be determined uniquely owing to incomplete marker information. At Genetic Analysis Workshop (GAW) 13, Jacobs et al. [BMC Genet 2003, 4(Suppl 1):S82] proposed to improve the power of the Haseman-Elston algorithm by weighting for information available from marker genotypes. The authors did not show, however, the validity of the employed asymptotic distribution. In this paper, we use the simulated data provided for GAW 14 and show that weighting Haseman-Elston by marker information results in increased type I error rates. Specifically, we demonstrate that the number of significant findings throughout the chromosome is significantly increased with weighting schemes. Furthermore, we show that the classical Haseman-Elston method keeps its nominal significance level when applied to the same data. We therefore recommend to use Haseman-Elston with marker informativity weights only in conjunction with empirical p-values. Whether this approach in fact yields an increase in power needs to be investigated further.     

Genomewide scan of hoarding in sib pairs in which both sibs have Gilles de la Tourette syndrome

A genome scan of the hoarding phenotype (a component of obsessive-compulsive disorder) was conducted on 77 sib pairs collected by the Tourette Syndrome Association International Consortium for Genetics (TSAICG). All sib pairs were concordant for a diagnosis of Gilles de la Tourette syndrome (GTS). However, the analyses reported here were conducted for hoarding as both a dichotomous trait and a quantitative trait. Not all sib pairs in the sample were concordant for hoarding. Standard linkage analyses were performed using GENEHUNTER and Haseman-Elston methods. In addition, novel analyses with a recursive-partitioning technique were employed. Significant allele sharing was observed for both the dichotomous and the quantitative hoarding phenotypes for markers at 4q34-35 (P=.0007), by use of GENEHUNTER, and at 5q35.2-35.3 (P=.000002) and 17q25 (P=.00002), by use of the revisited Haseman-Elston method. The 4q site is in proximity to D4S1625, which was identified by the TSAICG as a region linked to the GTS phenotype. The recursive-partitioning technique examined multiple markers simultaneously. Results suggest joint effects of specific loci on 5q and 4q, with an overall P value of.000003. Although P values were not adjusted for multiple comparison, nearly all were much smaller than the customary significance level of.0001 for genomewide scans.     

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.     

Quantitative trait linkage analysis using Gaussian copulas

Mapping and identifying variants that influence quantitative traits is an important problem for genetic studies. Traditional QTL mapping relies on a variance-components (VC) approach with the key assumption that the trait values in a family follow a multivariate normal distribution. Violation of this assumption can lead to inflated type I error, reduced power, and biased parameter estimates. To accommodate nonnormally distributed data, we developed and implemented a modified VC method, which we call the "copula VC method," that directly models the nonnormal distribution using Gaussian copulas. The copula VC method allows the analysis of continuous, discrete, and censored trait data, and the standard VC method is a special case when the data are distributed as multivariate normal. Through the use of link functions, the copula VC method can easily incorporate covariates. We use computer simulations to show that the proposed method yields unbiased parameter estimates, correct type I error rates, and improved power for testing linkage with a variety of nonnormal traits as compared with the standard VC and the regression-based methods.     

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.     

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.     

Two-locus models of disease: comparison of likelihood and nonparametric linkage methods.

The power to detect linkage for likelihood and nonparametric (Haseman-Elston, affected-sib-pair, and affected-pedigree-member) methods is compared for the case of a common, dichotomous trait resulting from the segregation of two loci. Pedigree data for several two-locus epistatic and heterogeneity models have been simulated, with one of the loci linked to a marker locus. Replicate samples of 20 three-generation pedigrees (16 individuals/pedigree) were simulated and then ascertained for having at least 6 affected individuals. The power of linkage detection calculated under the correct two-locus model is only slightly higher than that under a single locus model with reduced penetrance. As expected, the nonparametric linkage methods have somewhat lower power than does the lod-score method, the difference depending on the mode of transmission of the linked locus. Thus, for many pedigree linkage studies, the lod-score method will have the best power. However, this conclusion depends on how many times the lod score will be calculated for a given marker. The Haseman-Elston method would likely be preferable to calculating lod scores under a large number of genetic models (i.e., varying both the mode of transmission and the penetrances), since such an analysis requires an increase in the critical value of the lod criterion. The power of the affected-pedigree-member method is lower than the other methods, which can be shown to be largely due to the fact that marker genotypes for unaffected individuals are not used.     

A two-stage variable-stringency semiparametric method for mapping quantitative-trait loci with the use of genomewide-scan data on sib pairs

Genomewide scans for mapping loci have proved to be extremely powerful and popular. We present a semiparametric method of mapping a quantitative-trait locus (QTL) or QTLs with the use of sib-pair data generated from a two-stage genomic scan. In a two-stage genomic scan, either the entire genome or a large portion of the genome is saturated with low-density markers at the first stage. At the second stage, the intervals that are identified as probable locations of the trait loci, by means of analysis of data from the first stage, are then saturated with higher-density markers. These data are then analyzed for fine mapping of the loci. Our statistical strategy for analysis of data from the first stage is a low-stringency method based on the rank correlation of squared trait-difference values of the sib pairs and the estimated identity-by-descent scores at the marker loci. We suggest the use of a low-stringency method at the first stage, to save on computational time and to avoid missing any marker interval that may contain the trait loci. For analysis of data from the second stage, we have developed a high-stringency nonparametric-regression approach, using the kernel-smoothing technique. Through extensive simulations, we show that this approach is more powerful than is a currently used method for mapping QTLs by use of sib pairs, particularly in the presence of dominance and epistatic effects at the trait loci.     

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.     

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 tobit variance-component method for linkage analysis of censored trait data

Variance-component (VC) methods are flexible and powerful procedures for the mapping of genes that influence quantitative traits. However, traditional VC methods make the critical assumption that the quantitative-trait data within a family either follow or can be transformed to follow a multivariate normal distribution. Violation of the multivariate normality assumption can occur if trait data are censored at some threshold value. Trait censoring can arise in a variety of ways, including assay limitation or confounding due to medication. Valid linkage analyses of censored data require the development of a modified VC method that directly models the censoring event. Here, we present such a model, which we call the "tobit VC method." Using simulation studies, we compare and contrast the performance of the traditional and tobit VC methods for linkage analysis of censored trait data. For the simulation settings that we considered, our results suggest that (1) analyses of censored data by using the traditional VC method lead to severe bias in parameter estimates and a modest increase in false-positive linkage findings, (2) analyses with the tobit VC method lead to unbiased parameter estimates and type I error rates that reflect nominal levels, and (3) the tobit VC method has a modest increase in linkage power as compared with the traditional VC method. We also apply the tobit VC method to censored data from the Finland-United States Investigation of Non-Insulin-Dependent Diabetes Mellitus Genetics study and provide two examples in which the tobit VC method yields noticeably different results as compared with the traditional method.     

Linkage analysis of a derived glucose phenotype in the Genetic Analysis Workshop 13 simulated data using a variety of Haseman-Elston based regression methods

A variety of Haseman-Elston type regression procedures were used to perform a genome scan across five chromosomes, using replicates 1-5 of the Genetic Analysis Workshop 13 simulated data. The traits of interest were variables corresponding to 'baseline' and 'slope' effects derived from the fasting glucose phenotypes. Performance in terms of detecting the locations of known trait loci was poor for all methods, even when all five replicates were combined to produce a large data set (9230 sib pairs). All methods performed well, however, when applied to new simulated data in which the true genetic effects were allowed to explain a greater proportion of the overall variance.     

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.