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.     

Absence of significant linkage between phonological coding dyslexia and chromosome 6p23-21.3, as determined by use of quantitative-trait methods: confirmation of qualitative analyses

We recently reported the absence of significant linkage of phonological coding dyslexia (PCD) to chromosome 6p23-p21.3 in 79 families with at least two affected siblings, even though linkage of dyslexia to this region has been found in four other independent studies. Whereas, in our previous analyses, we used a qualitative (affected, unaffected, or uncertain) PCD phenotype, here we report a reanalysis of linkage to the chromosome 6p region, by use of four quantitative measures of reading disability: phonological awareness, phonological coding, spelling, and rapid-automatized-naming (RAN) speed. The phonological-coding and spelling measures were highly correlated with each other and with the qualitative PCD phenotype, whereas the phonological-awareness and RAN-speed measures were only moderately correlated with the other measures. Using two-point and multipoint quantitative-trait sib-pair linkage analyses and variance-components analyses, we were unable to detect significant evidence for a locus in the 6p23-p21.3 region influencing any of the quantitative reading measures, supporting our previous qualitative linkage results. The most likely explanation for our inability to detect linkage between dyslexia and this region is that families with subtypes of dyslexia linked to this region are underrepresented in our sample, because of either chance or varying ascertainment criteria.     

Joint Multipoint Linkage Analysis of Multivariate Qualitative and Quantitative Traits. II. Alcoholism and Event-Related Potentials

The availability of robust quantitative biological markers that are correlated with qualitative psychiatric phenotypes can potentially improve the power of linkage methods to detect quantitative-trait loci influencing psychiatric disorders. We apply a variance-component method for joint multipoint linkage analysis of multivariate discrete and continuous traits to the extended pedigree data from the Collaborative Study on the Genetics of Alcoholism, in a bivariate analysis of qualitative alcoholism phenotypes and quantitative event-related potentials. Joint consideration of the DSM-IV diagnosis of alcoholism and the amplitude of the P300 component of the Cz event-related potential significantly increases the evidence for linkage of these traits to a chromosome 4 region near the class I alcohol dehydrogenase locus ADH3. A likelihood-ratio test for complete pleiotropy is significant, suggesting that the same quantitative-trait locus influences both risk of alcoholism and the amplitude of the P300 component.     

Transmission-disequilibrium tests for quantitative traits.

The transmission-disequilibrium test (TDT) of Spielman et al. is a family-based linkage-disequilibrium test that offers a powerful way to test for linkage between alleles and phenotypes that is either causal (i.e., the marker locus is the disease/trait allele) or due to linkage disequilibrium. The TDT is equivalent to a randomized experiment and, therefore, is resistant to confounding. When the marker is extremely close to the disease locus or is the disease locus itself, tests such as the TDT can be far more powerful than conventional linkage tests. To date, the TDT and most other family-based association tests have been applied only to dichotomous traits. This paper develops five TDT-type tests for use with quantitative traits. These tests accommodate either unselected sampling or sampling based on selection of phenotypically extreme offspring. Power calculations are provided and show that, when a candidate gene is available (1) these TDT-type tests are at least an order of magnitude more efficient than two common sib-pair tests of linkage; (2) extreme sampling results in substantial increases in power; and (3) if the most extreme 20% of the phenotypic distribution is selectively sampled, across a wide variety of plausible genetic models, quantitative-trait loci explaining as little as 5% of the phenotypic variation can be detected at the .0001 alpha level with <300 observations.     

Regression models for linkage: issues of traits,covariates, heterogeneity,and interaction

Regression methods offer a common framework to analyze linkage for quantitative trait loci as well as linkage for affection status using affected sib-pairs. Although numerous papers on regression methods for linkage have been published, some common themes and important caveats tend to be scattered across the literature. For example, the typical approach is to regress a function of traits on identical-by-descent (IBD) information, but the reversal (regression of IBD on a function of traits) offers important insights. A second example is the use of regression equations to assess linkage heterogeneity or gene-environment interaction, and why these two different etiologies are difficult to distinguish with affected sib-pair data. A third example has to do with the differences, and similarities, between linear regression and non-linear regression methods for affected sib-pair data. The purposes of this paper are to review some recent developments in the linkage regression framework, to emphasize strengths and weaknesses of various proposed methods, and to highlight some important assumptions and caveats.     

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.     

Joint Multipoint Linkage Analysis of Multivariate Qualitative and Quantitative Traits. I. Likelihood Formulation and Simulation Results

We describe a variance-components method for multipoint linkage analysis that allows joint consideration of a discrete trait and a correlated continuous biological marker (e.g., a disease precursor or associated risk factor) in pedigrees of arbitrary size and complexity. The continuous trait is assumed to be multivariate normally distributed within pedigrees, and the discrete trait is modeled by a threshold process acting on an underlying multivariate normal liability distribution. The liability is allowed to be correlated with the quantitative trait, and the liability and quantitative phenotype may each include covariate effects. Bivariate discrete-continuous observations will be common, but the method easily accommodates qualitative and quantitative phenotypes that are themselves multivariate. Formal likelihood-based tests are described for coincident linkage (i.e., linkage of the traits to distinct quantitative-trait loci [QTLs] that happen to be linked) and pleiotropy (i.e., the same QTL influences both discrete-trait status and the correlated continuous phenotype). The properties of the method are demonstrated by use of simulated data from Genetic Analysis Workshop 10. In a companion paper, the method is applied to data from the Collaborative Study on the Genetics of Alcoholism, in a bivariate linkage analysis of alcoholism diagnoses and P300 amplitude of event-related brain potentials.     

Linkage of a Gene Causing High Bone Mass to Human Chromosome 11 (11q12-13)

The purpose of this paper is to report the linkage of a genetic locus (designated "HBM") in the human genome to a phenotype of very high spinal bone density, using a single extended pedigree. We measured spinal bone-mineral density, spinal Z(BMD), and collected blood from 22 members of this kindred. DNA was genotyped on an Applied Biosystems model 377 (ABI PRISM Linkage Mapping Sets; Perkin Elmer Applied Biosystems), by use of fluorescence-based marker sets that included 345 markers. Both two-point and multipoint linkage analyses were performed, by use of affected/unaffected and quantitative-trait models. Spinal Z(BMD) for affected individuals (N = 12) of the kindred was 5.54 +/- 1.40; and for unaffected individuals (N = 16) it was 0.41 +/- 0.81. The trait was present in affected individuals 18-86 years of age, suggesting that HBM influences peak bone mass. The only region of linkage was to a series of markers on chromosome 11 (11q12-13). The highest LOD score (5.21) obtained in two-point analysis, when a quantitative-trait model was used, was at D11S987. Multipoint analysis using a quantitative-trait model confirmed the linkage, with a LOD score of 5.74 near marker D11S987. HBM demonstrates the utility of spinal Z(BMD) as a quantitative bone phenotype that can be used for linkage analysis. Osteoporosis pseudoglioma syndrome also has been mapped to this region of chromosome 11. Identification of the causal gene for both traits will be required for determination of whether a single gene with different alleles that determine a wide range of peak bone densities exists in this region.     

Combined linkage and association sib-pair analysis for quantitative traits.

An extension to current maximum-likelihood variance-components procedures for mapping quantitative-trait loci in sib pairs that allows a simultaneous test of allelic association is proposed. The method involves modeling of the allelic means for a test of association, with simultaneous modeling of the sib-pair covariance structure for a test of linkage. By partitioning of the mean effect of a locus into between- and within-sibship components, the method controls for spurious associations due to population stratification and admixture. The power and efficacy of the method are illustrated through simulation of various models of both real and spurious association.     

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.     

Parameter Estimation and Quantitative Parametric Linkage Analysis with GENEHUNTER-QMOD

Objective: We present a parametric method for linkage analysis of quantitative phenotypes. The method provides a test for linkage as well as an estimate of different phenotype parameters. We have implemented our new method in the program GENEHUNTER-QMOD and evaluated its properties by performing simulations. Methods: The phenotype is modeled as a normally distributed variable, with a separate distribution for each genotype. Parameter estimates are obtained by maximizing the LOD score over the normal distribution parameters with a gradient-based optimization called PGRAD method. Results: The PGRAD method has lower power to detect linkage than the variance components analysis (VCA) in case of a normal distribution and small pedigrees. However, it outperforms the VCA and Haseman-Elston regression for extended pedigrees, nonrandomly ascertained data and non-normally distributed phenotypes. Here, the higher power even goes along with conservativeness, while the VCA has an inflated type I error. Parameter estimation tends to underestimate residual variances but performs better for expectation values of the phenotype distributions. Conclusion: With GENEHUNTER-QMOD, a powerful new tool is provided to explicitly model quantitative phenotypes in the context of linkage analysis. It is freely available at http://www.helmholtz-muenchen.de/genepi/downloads.     

Multipoint quantitative-trait linkage analysis in general pedigrees.

Multipoint linkage analysis of quantitative-trait loci (QTLs) has previously been restricted to sibships and small pedigrees. In this article, we show how variance-component linkage methods can be used in pedigrees of arbitrary size and complexity, and we develop a general framework for multipoint identity-by-descent (IBD) probability calculations. We extend the sib-pair multipoint mapping approach of Fulker et al. to general relative pairs. This multipoint IBD method uses the proportion of alleles shared identical by descent at genotyped loci to estimate IBD sharing at arbitrary points along a chromosome for each relative pair. We have derived correlations in IBD sharing as a function of chromosomal distance for relative pairs in general pedigrees and provide a simple framework whereby these correlations can be easily obtained for any relative pair related by a single line of descent or by multiple independent lines of descent. Once calculated, the multipoint relative-pair IBDs can be utilized in variance-component linkage analysis, which considers the likelihood of the entire pedigree jointly. Examples are given that use simulated data, demonstrating both the accuracy of QTL localization and the increase in power provided by multipoint analysis with 5-, 10-, and 20-cM marker maps. The general pedigree variance component and IBD estimation methods have been implemented in the SOLAR (Sequential Oligogenic Linkage Analysis Routines) computer package.     

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 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.     

Segregation analysis incorporating linkage markers. I. Single-locus models with an application to type I diabetes.

A method is described for segregation analysis that incorporates linkage markers. The model allows for segregation (penetrance), linkage (recombination fraction), and association (linkage disequilibrium) parameters. A single-locus-multiple-allele model underlying the trait phenotype is assumed. When families have been ascertained in a systematic fashion, a joint (markers, phenotypes) likelihood with ascertainment is advocated. When ascertainment correction is not feasible, a conditional (markers given phenotypes) approach is recommended, which is also valid in the presence of reduced fertility and assortative mating. This approach, oriented toward determining mode of inheritance, differs from conventional linkage analysis, which is oriented toward detection of linkage. Therefore, it is more appropriately considered an extension of the affected sib-pair method to arbitrary pedigrees, including association information and allowing for multiple alleles. Incorporation of coupling parameters allows for discrimination between pleiotropy and linkage disequilibrium. The method is demonstrated through a reanalysis of four recently published family studies on type 1 diabetes and HLA. Recessive inheritance is rejected in all four data sets. For three of them, dominant inheritance is not rejected, while in the fourth, all two-allele models are rejected in favor of three alleles. Although association with the DR3 and DR4 alleles is quite strong, pleiotropy with regard to these alleles is unlikely. The results also suggest an additional familial factor(s) (e.g., locus).     

A quantitative-trait locus on chromosome 6p influences different aspects of developmental dyslexia.

Recent application of nonparametric-linkage analysis to reading disability has implicated a putative quantitative-trait locus (QTL) on the short arm of chromosome 6. In the present study, we use QTL methods to evaluate linkage to the 6p25-21.3 region in a sample of 181 sib pairs from 82 nuclear families that were selected on the basis of a dyslexic proband. We have assessed linkage directly for several quantitative measures that should correlate with different components of the phenotype, rather than using a single composite measure or employing categorical definitions of subtypes. Our measures include the traditional IQ/reading discrepancy score, as well as tests of word recognition, irregular-word reading, and nonword reading. Pointwise analysis by means of sib-pair trait differences suggests the presence, in 6p21.3, of a QTL influencing multiple components of dyslexia, in particular the reading of irregular words (P=.0016) and nonwords (P=.0024). A complementary statistical approach involving estimation of variance components supports these findings (irregular words, P=.007; nonwords, P=.0004). Multipoint analyses place the QTL within the D6S422-D6S291 interval, with a peak around markers D6S276 and D6S105 consistently identified by approaches based on trait differences (irregular words, P=.00035; nonwords, P=.0035) and variance components (irregular words, P=.007; nonwords, P=.0038). Our findings indicate that the QTL affects both phonological and orthographic skills and is not specific to phoneme awareness, as has been previously suggested. Further studies will be necessary to obtain a more precise localization of this QTL, which may lead to the isolation of one of the genes involved in developmental dyslexia.     

Comparison of multivariate tests for genetic linkage

OBJECTIVES: Multivariate tests for linkage can provide improved power over univariate tests but the type I error rates and comparative power of commonly used methods have not previously been compared. Here we studied the behavior of bivariate formulations of the variance component (VC) and Haseman-Elston (H-E) approaches. METHODS: We compared through simulation studies the bivariate H-E test with the unconstrained bivariate VC approach and with a VC approach in which the major-gene correlation is constrained to +/-1. We also compared these methods to univariate methods. RESULTS: Bivariate approaches are more powerful than univariate analyses unless the traits are very highly positively correlated. The power of the bivariate H-E test was less than the VC procedures. The constrained test was often less powerful than the unconstrained test. The empirical distributions of the bivariate H-E test and the unconstrained bivariate VC test conformed with asymptotic distributions for samples of 100 or more sibships of size 4. CONCLUSIONS: The unconstrained VC test is valuable for testing for preliminary linkages using multivariate phenotypes. The bivariate H-E test was less powerful than the bivariate VC tests.     

Linkage analysis of extremely discordant and concordant sibling pairs identifies quantitative trait loci influencing variation in human menopausal age

Age at natural menopause may be used as parameter for evaluating the rate of ovarian aging. Environmental factors determine only a small part of the large variation in menopausal age. Studies have shown that genetic factors are likely to be involved in variation in menopausal age. To identify quantitative-trait loci for this trait, we performed a genomewide linkage study with age at natural menopause as a continuous quantitative phenotype in Dutch sister pairs, through use of a selective sampling scheme. A total of 165 families were ascertained using extreme selected sampling and were genotyped for 417 markers. Data were analyzed by Haseman-Elston regression and by an adjusted variance-components analysis. Subgroup analyses for early and late menopausal age were conducted by Haseman-Elston regression. In the adjusted variance-components analysis, 12 chromosomes had a LOD score of > or =1.0. Two chromosomal regions showed suggestive linkage: 9q21.3 (LOD score 2.6) and Xp21.3 (LOD score 3.1). Haseman-Elston regression showed rather similar locations of the peaks but yielded lower LOD scores. A permutation test to obtain empirical P values resulted in a significant peak on the X chromosome. To our knowledge, this is the first study to attempt to identify loci responsible for variability in menopausal age and in which several chromosomal regions were identified with suggestive and significant linkage. Although the finding of the region on the X chromosome comes as no surprise, because of its widespread involvement in premature ovarian failure, the definition of which particular gene is involved is of great interest. The region on chromosome 9 deserves further consideration. Both findings require independent confirmation.     

Mapping quantitative-trait loci in humans by use of extreme concordant sib pairs: selected sampling by parental phenotypes.

In two previous articles, we have considered sample sizes required to detect linkage for mapping quantitative-trait loci in humans, using extreme discordant sib pairs. Here, we examine further the use of extreme concordant sib pairs but consider the effect of parents' phenotypes. Sample sizes necessary to obtain a power of 80% with concordant sib pairs at a significance level of .0001 are given, stratified by parental phenotypes. When there is no residual correlation between sibs, the parental phenotypes have little impact on the sample sizes. When residual correlations between sibs exist, we show, however, that power can be considerably reduced by including extreme sib pairs when the parents also have similarly extreme values. Thus, we recommend the exclusion of such pairs from linkage studies. This recommendation reduces the required sample sizes by 3- to 28-fold. The degree of saving in the required sample sizes varies among different models and allele frequencies. The reduction is most dramatic (a 28-fold reduction) for a rare recessive gene.     

Testing the robustness of the likelihood-ratio test in a variance-component quantitative-trait loci-mapping procedure.

Detection of linkage to genes for quantitative traits remains a challenging task. Recently, variance components (VC) techniques have emerged as among the more powerful of available methods. As often implemented, such techniques require assumptions about the phenotypic distribution. Usually, multivariate normality is assumed. However, several factors may lead to markedly nonnormal phenotypic data, including (a) the presence of a major gene (not necessarily linked to the markers under study), (b) some types of gene x environment interaction, (c) use of a dichotomous phenotype (i.e., affected vs. unaffected), (d) nonnormality of the population within-genotype (residual) distribution, and (e) selective (extreme) sampling. Using simulation, we have investigated, for sib-pair studies, the robustness of the likelihood-ratio test for a VC quantitative-trait locus-detection procedure to violations of normality that are due to these factors. Results showed (a) that some types of nonnormality, such as leptokurtosis, produced type I error rates in excess of the nominal, or alpha, levels whereas others did not; and (b) that the degree of type I error-rate inflation appears to be directly related to the residual sibling correlation. Potential solutions to this problem are discussed. Investigators contemplating use of this VC procedure are encouraged to provide evidence that their trait data are normally distributed, to employ a procedure that allows for nonnormal data, or to consider implementation of permutation tests.