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.     

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 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.     

A multivariate method for detecting genetic linkage, with application to a pedigree with an adverse lipoprotein phenotype.

The robust or model-free method for detecting linkage developed by Haseman and Elston for data from sib pairs is extended to incorporate observations of multiple traits on each individual. A method is proposed that estimates the linear function that results in the strongest correlation between the squared pair differences in the trait measurements and identity by descent at a marker locus. The method is illustrated by the study of apolipoprotein and cholesterol levels in individuals from a large family that had many members diagnosed with coronary heart disease.     

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.     

Selection and subsequent analysis of sib pair data for QTL detection.

Haseman and Elston (1972) developed a robust regression method for the detection of linkage between a marker and a quantitative trait locus (QTL) using sib pair data. The principle underlying this method is that the difference in phenotypes between pairs of sibs becomes larger as they share a decreasing number of alleles at a particular QTL identical by descent (IBD) from their parents. In this case, phenotypically very different sibs will also on average share a proportion of alleles IBD at any marker linked to the QTL that is lower than the expected value of 0.5. Thus, the deviation of the proportion of marker alleles IBD from the expected value in pairs of sibs selected to be phenotypically different (i.e. discordant) can provide a test for the presence of a QTL. A simple regression method for QTL detection in sib pairs selected for high phenotypic differences is presented here. The power of the analytical method was found to be greater than the power obtained using the standard analysis when samples of sib pairs with high phenotypic differences were used. However, the use of discordant sib pairs was found to be less powerful for QTL detection than alternative selective genotyping schemes based on the phenotypic values of the sibs except with intense selection, when its advantage was only marginal. The most effective selection scheme overall was the use of sib pairs from entire families selected on the basis of high within-family variance for the trait in question. There is little effect of selection on QTL position estimates, which are in good agreement with the simulated values. However, QTL variance estimates are biased to a greater or lesser degree, depending on the selection method.     

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.     

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.     

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.     

An efficient method to handle the ‘large <Emphasis Type="Italic">p</Emphasis>, small <Emphasis Type="Italic">n</Emphasis>’ problem for genomewide association studies using Haseman–Elston regression

The ‘large p, small n’ problem in genomewide association studies (GWAS) is an important subject in genetic studies. Many approaches have been proposed for this issue, but none of them successfully combine the Haseman–Elston (H–E) regression with sliding-window scan approaches in GWAS. In this article, we extended H–E regression to GWAS, and replaced original data with different measurements of phenotype of sib pairs. Meanwhile, we also applied hidden Markov model to infer identity by state. Using subsequent simulation studies, we found that it had higher statistical power than the corresponding single-marker association studies. The advantage of the H–E regression was also sufficient to capture about 48.01% of the quantitative trait locus (QTL). Meanwhile, the results show that the power decreases with the increase in the number of QTLs, and the power of H–E regression is sensitive to heritability.     

Matrix representation of the Haseman-Elston method.

The Hasemann-Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3x3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.     

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.     

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.     

Dissecting the correlation structure of a bivariate phenotype: Common genes or shared environment?

High correlations between two quantitative traits may be either due to common genetic factors or common environmental factors or a combination of both. In this study, we develop statistical methods to extract the genetic contribution to the total correlation between the components of a bivariate phenotype. Using data on bivariate phenotypes and marker genotypes for sib-pairs, we propose a test for linkage between a common QTL and a marker locus based on the conditional cross-sib trait correlations (trait 1 of sib 1—trait 2 of sib 2 and conversely) given the identity-by-descent (i.b.d.) sharing at the marker locus. We use Monte-Carlo simulations to evaluate the performance of the proposed test under different trait parameters and quantitative trait distributions. An application of the method is illustrated using data on two alcohol-related phenotypes from a project on the collaborative study on the genetics of alcoholism.     

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.     

Detection of human genetic linkage: foundations

Starting with the basics of inheritance, recombination, and genetic map distance, we develop the theory of linkage analysis considering data on an individual or on a sib pair, with and without data on parents. We discuss the effects of ascertainment of individuals with certain traits and the effects of allelic associations between a trait and a linked marker locus.     

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.     

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.     

Selective genotyping for QTL detection using sib pair analysis in outbred populations with hierarchical structures

A simulation study illustrates the effects of the inclusion of half-sib pairs as well as the effects of selective genotyping on the power of detection and the parameter estimates in a sib pair analysis of data from an outbred population. The power of QTL detection obtained from samples of sib pairs selected according to their within family variance or according to the mean within family variance within half sib family was compared and contrasted with the power obtained when only full sib pair analysis was used. There was an increase in power (4–16%) and decrease in the bias of parameter estimates with the use of half-sib information. These improvements in power and parameter estimates depended on the number of the half sib pairs (half sib family size). Almost the same power as that obtained using all the available sib pairs could be achieved by selecting only 50–60% the animals. The most effective method was to select both full and half sib pairs on the basis of high within full sib family variance for the trait in question. The QTL position estimates were in general slightly biased towards the center of the chromosome and the QTL variance estimates were biased upwards, there being quite large differences in bias depending on the selection method.     

Evidence for a new Graves disease susceptibility locus at chromosome 18q21

Graves disease (GD) is a common autoimmune thyroid disorder that is inherited as a complex multigenic trait. By using a single microsatellite marker at each locus, we screened the type 1 diabetes loci IDDM4, IDDM5, IDDM6, IDDM8, and IDDM10 and the fucosyltransferase-2 locus for linkage in sib pairs with GD. This showed a two-point nonparametric linkage (NPL) score of 1.57 (P=.06) at the IDDM6 marker D18S41, but NPL scores were <1.0 at the other five loci. Thus, the investigation of the IDDM6 locus was extended by genotyping 11 microsatellite markers spanning 48 cM across chromosome 18q12-q22 in 81 sib pairs affected with autoimmune thyroid disease (AITD). Multipoint analysis, designating all AITD sib pairs as affected, showed a peak NPL score of 3.46 (P=.0003), at the marker D18S487. Designation of only GD cases as affected (74 sib pairs) showed a peak NPL score of 3.09 (P=.001). Linkage to this region has been demonstrated in type 1 diabetes (IDDM6), rheumatoid arthritis, and systemic lupus erythematosus, which suggests that this locus may have a role in several forms of autoimmunity.