Power of a simplified multivariate test for genetic linkage

In this paper we compare the power of the multivariate Haseman–Elston (MHE) test proposed earlier by Amos et al . (1990) and a computationally rapid new version of the multivariate Haseman–Elston test (NMHE) (Elston et al . 2000). We show that the power of NMHE was, for different simulation setups, identical or higher than that of MHE. In the bivariate case, the power of the NMHE method was somewhat less than that of the computationally intensive maximum likelihood variance components method (Amos et al . 2001). We present comparisons of the empirical distributions of the NMHE test to its limiting distributions for a range of numbers of traits. The distribution of the NMHE test appeared to conform satisfactorily to its limiting asymptotic distribution in large samples. Otherwise, empirical critical values for NMHE are somewhat higher than predicted, i.e. the test proposed by Elston et al . (2000) is non-conservative. The use of empirical critical values is therefore recommended for limited sample sizes (less than several hundred families). We also present the results of a linkage analysis performed by the NMHE method on a set of 4 body size-related traits. The method identified meaningful combinations of traits that showed significant linkage on chromosome 2 and suggestive linkage to regions on chromosomes 16 and 17.     

Evaluation of a restricted likelihood ratio test for mapping quantitative trait loci with extreme discordant sib pairs

Risch and Zhang recently proposed to use extreme discordant sib pairs for mapping quantitative trait loci. Here, it is shown that the set of genetically possible distributions of the number of marker alleles in such sib-pairs is described by two inequalities. Thus, a likelihood ratio test analogous to Holmans's possible triangle test for affected sib pairs can be defined. The performance of this test is compared to the mean test considered by Risch and Zhang. For most of the genetic models considered, the mean test is slightly more powerful than the restricted likelihood ratio test. However, for models with a rare recessive gene (or equivalently a common dominant gene), the restricted likelihood ratio test is much more powerful.     

The impact of genotyping error on family-based analysis of quantitative traits

Errors in genotyping can substantially influence the power to detect linkage using affected sib-pairs, but it is not clear what effect such errors have on quantitative trait analyses. Here we use Monte Carlo simulation to examine the influence of genotyping error on multipoint vs two-point analysis, variable map density, locus effect size and allele frequency in quantitative trait linkage and association studies of sib-pairs. The analyses are conducted using variance components methods. We contrast the effects of error on quantitative trait analyses with those on the affected sib-pair design. The results indicate that genotyping error influences linkage studies of affected sib pairs more severely than studies of quantitative traits in unselected sibs. In situations of modest effect size, 5% genotyping error eliminates all supporting evidence for linkage to a true susceptibility locus in affected pairs, but may only result in a loss of 15% of linkage information in random pairs. Multipoint analysis does not suffer substantially more than two-point analysis; for moderate error rates (< 5%), multipoint analysis with error is more powerful than two-point with no error. Map density does not appear to be an important factor for linkage analysis. QTL association analyses of common alleles are reasonably robust to genotyping error but power can be affected dramatically with rare alleles.     

A permutation test for the robust sib-pair linkage method

The robust sib-pair method introduced by Haseman & Elston (1972) is one of the most widely circulated allele-sharing methods for linkage analysis. The procedure evaluates linkage by significance testing of a regression coefficient and, hence, a standard t -test has traditionally been applied despite known violations of the statistical assumptions underlying the test. We present a permutation based reference distribution for the estimate of the regression coefficient that is motivated by genetic principles rather than by standard regression testing procedures. The permutation test approximates Mendelian co-segregation under the null hypothesis of no linkage, making it a very natural approach. Theory and simulations show that the conventional t -test approximates the permutation test quite well, even when dependent sib pairs are used for analysis. These results thus indirectly address concerns over the t -test. To illustrate the permutation test using real data we applied the procedure to two lipoprotein systems that have been well characterized.     

Effects of age at onset on the power of the affected sib pair and transmission/disequilibrium tests

Information on the age of a patient at disease onset, an important feature of complex diseases, is often collected in studies designed to map the disease genes. Penetrance-model-free methods, requiring no specification of penetrance functions, have been used extensively for detecting linkage and association between marker and disease loci. In this paper, we conduct an analytical study to examine the effects of incorporation of age at onset information on the power of two commonly used penetrance-model-free methods, the affected sib-pair (ASP) and transmission/disequilibrium tests (TDT). Assuming a Cox model with a major gene effect for the age at onset, we quantify analytically how age at onset affects the identity by descent (IBD) probabilities, the mean IBD values, and the expected numbers of alleles transmitted from heterozygous parents to affected children under various genetic models. We show that the power of the mean IBD test and the TDT can be greatly affected by the ages at onset of affected siblings or children used in the study. Generally, the most powerful test for ASPs is that based on affected sib pairs both having early disease onset and for TDT analyses is that based on trios with early-onset children. Naively combining affected sib pairs with different ages at onset or parent-children trios with different ages at onset of affected children can result in reduced power for detecting linkage or association. These results may be used to guide collection and analysis of sib pairs or families for diseases with variable age at onset.     

European Mathematical Genetics Meeting, Heidelberg, Germany, 12th–13th April 2007

Saurabh Ghosh ¹
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 contribution of a common QTL 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 sharing at the marker locus. The null hypothesis cannot be rejected unless there exists a common QTL. 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 the Collaborative Study On The Genetics Of Alcoholism project.     

Statistically robust approaches for sib-pair linkage analysis

Many traits that distinguish one individual from another, such as height or weight, are clearly heritable and yet vary continuously in populations. Continuous, heritable variation in trait levels presumably reflects the segregation of multiple genes, but elucidation of the genetic architecture of quantitative traits has been limited. Haseman & Elston (1972) developed a genetically robust method (HE) for detecting linkage to quantitative trait loci using sib-pairs. The method is based on a simple linear regression of the squared sib-pairs trait difference on the proportion of alleles shared identical by descent at a marker locus. Linkage is detected by a negative slope which has been traditionally assessed by a standard t -test. Wan, Cohen & Guerra (1997) have shown that the standard t -test is robust to the violations of the stochastic assumptions underlying the test. In practice, however, the standard t -test, based on least-squares regression, is sensitive to outliers. The presence of outliers in the data can lead to false positive and false negative linkage results. Accordingly we have developed and evaluated a statistically robust procedure for the HE approach to linkage. The procedure is based on robust regression. Simulation studies show that this robust procedure has greater power than the standard t -test in the presence of outliers, and has similar power to the standard t -test in the absence of outliers. This robust procedure also shows greater power than rank-based approaches either in the absence or presence of outliers. To illustrate the methods using real data we reanalyse data from two lipoprotein systems that motivated this work.     

Extension of the Haseman–Elston method to multiple alleles and multiple loci: theory and practice for candidate genes

The Haseman & Elston (1972) sibling-pair regression method has been used to detect and estimate the variance contribution to observed values of a quantitative trait by allelic variation in specific candidate genes. The procedure was developed under a model with a single biallelic trait locus. This assumption does not hold for several known systems. In this paper we prove that for candidate gene analysis the Haseman–Elston procedure extends to the case of multiple trait loci, each possibly having more than two alleles. Simulation experiments comparing single-locus to two-locus models show that fitting the extended regression equations maintains nominal significance levels, but the power to detect linkage to trait variation is not improved by including additional loci. These results indicate that the original proposal is statistically robust to violations of the underlying genetic model. Practical issues associated with quantifying the relative variance contribution by individual loci are also discussed. Applications of the extended regression equations to lipoprotein(a) and high density lipoprotein cholesterol are given for illustration.     

Linkage of multilocus components of variance to polymorphic markers

Haseman & Elston (1972) introduced a sib pair method using classical regression analysis to detect linkage between a polymorphic marker locus and any quantitative trait locus. Most of the diseases mapped to date follow simple Mendelian, single locus transmission. But there are many familial diseases that do not follow simple Mendelian segregation, for example diabetes, several forms of cancer, etc. In this paper, we extend Haseman and Elston's sib pair method to two unlinked quantitative trait loci each linked to one of two unlinked polymorphic marker loci. For the two-locus epistatic model, we give a general formulation of the complete regression model and details of the regression coefficients in terms of variance components.     

Indication of linkage and genetic heterogeneity for asthma and atopy on chromosomes 8p and 12q in 107 French EGEA families

Using the sample of 107 families with at least two asthmatic siblings, as part of the EGEA study, we have investigated linkage to asthma (or atopy) and genetic heterogeneity according to the presence/absence of atopy (or asthma) using two approaches: (1) the triangle test statistic (TTS), which considers the identical by descent (IBD) distribution among affected sib-pairs discordant for another associated phenotype (eg asthmatic sib-pairs discordant for atopy) and (2) the predivided sample test (PST), which compares the IBD distribution of marker alleles between affected sib-pairs concordant and discordant for the associated phenotype. Two regions, 8p and 12q, already reported to be linked to both asthma and atopy, were examined here. A total of 20 asthmatic sib-pairs discordant for atopy and 24 atopic pairs discordant for asthma were analyzed by both TTS and PST methods and 83 pairs with atopic asthma by PST. Some evidence for linkage was observed for two markers in the 8p23.3-p23.2 region; D8S504 for asthma with genetic heterogeneity according to the presence/absence of atopy and D8S503 for atopy with genetic heterogeneity according to the presence/absence of asthma. In the 12q14.2-q21.33 region, there was also some evidence of linkage to two markers, D12S83 and D12S95, for atopy and asthma, respectively, with genetic heterogeneity according to the presence/absence of the associated trait. Provided the small distance between the two markers on either 8p (16 cM) or 12q (21 cM), it is unclear whether one or two genetic factors are involved in either region.     

Integrating sibship data for mapping quantitative trait loci

Sibship methods have been shown to be more powerful than traditional sib-pair methods in mapping quantative trait loci. We propose a statistical procedure which integrates data on sibships into a so-called 'contrast function', a natural extension of the classical squared sib-pair trait difference proposed by Haseman & Elston (1972). We also develop a combined mean and contrast function which provides more information on linkage compared to the contrast function. Our method is extended to multiple, epistatically interacting trait loci. Monte-Carlo simulations are included to compare the efficiencies of the proposed procedures with some currently used methods. An application of our proposed method is presented using data on alcohol dependence.     

Quantitative trait locus for reading disability on chromosome 6p is pleiotropic for attention-deficit/hyperactivity disorder

Comorbidity is pervasive among both adult and child psychiatric disorders; however, the etiological mechanisms underlying the majority of comorbidities are unknown. This study used genetic linkage analysis to assess the etiology of comorbidity between reading disability (RD) and attention-deficit hyperactivity disorder (ADHD), two common childhood disorders that frequently co-occur. Sibling pairs (N = 85) were ascertained initially because at least one individual in each pair exhibited a history of reading difficulties. Univariate linkage analyses in sibling pairs selected for ADHD from within this RD-ascertained sample suggested that a quantitative trait locus (QTL) on chromosome 6p is a susceptibility locus for ADHD. Because this QTL is in the same region as a well-replicated QTL for reading disability, subsequent bivariate analyses were conducted to test if this QTL contributed to comorbidity between the two disorders. Analyses of data from sib pairs selected for reading deficits revealed suggestive bivariate linkage for ADHD and three measures of reading difficulty, indicating that comorbidity between RD and ADHD may be due at least in part to pleiotropic effects of a QTL on chromosome 6p.     

Differential genetic etiology of reading component processes as a function of IQ

Results obtained from previous studies of word recognition, reading component skills, and reading composite measures suggest that genetic factors may be more important as a cause for reading disability among children with higher IQ scores than among those with lower IQ scores. To investigate the genetic etiology of reading disability further, measures of word recognition, phonological decoding, orthographic coding, and phoneme awareness were obtained from a total of 465 twin pairs with a positive school history of reading problems. The basic and extended DeFries and Fulker (DF) multiple regression models for the analysis of selected twin data were employed to investigate the etiology of group deficits in reading and language skills, as well as to assess differential genetic etiology for the reading-related measures as a function of IQ. Data from 168 sibling pairs (drawn from the twins' families), including fraternal twin pairs and their siblings, as well as non-twin siblings of identical twins were subjected to single-marker analyses using the DF basic linkage model to examine evidence for linkage of a quantitative-trait locus (QTL) for reading and language deficits to the short arm of chromosome 6. Lastly, to investigate the possible differential influence of this QTL as a function of IQ, the sibling pair data were fitted to an extension of the DF basic linkage model. Results indicated that reading and language deficits are significantly heritable and that differential genetic influences as a function of IQ are evident for measures of word recognition and phonological decoding. Results obtained from linkage analyses confirmed the presence of a QTL on chromosome 6p that influences phonological and orthographic skills, as well as phoneme awareness measures, and suggest that this QTL may influence phoneme awareness differentially as a function of IQ: however, future analyses with considerably larger samples are needed to test the hypothesis of differential QTL influence more rigorously.     

Assessing genetic linkage and association with robust components of variance approaches

Simulation studies are used to explore the properties of procedures for estimating components of variance and constructing test statistics in genetic linkage studies of quantitative traits. We evaluated the bias and median squared error of estimates of the linked additive genetic variance obtained by regression, maximum likelihood and quasilikelihood estimation procedures. The quasilikelihood and regression procedures provided unbiased estimates of the additive component of variance. Maximum likelihood procedures that assumed multivariate normality were biased for most sample sizes considered but had more precision for most generating models than regression or quasilikelihood methods did. Wald tests derived from quasilikelihood procedures had similar or greater power than Wald tests based upon estimators from maximum likelihood analyses. Quasilikelihood estimation may therefore be preferable whenever there is uncertainty about the generating distribution for the error variance, but the robustness of this approach is offset by its required computational complexity.     

Optimal Selection of Sib Pairs from Random Samples for Linkage Analysis of a QTL Using the EDAC Test

Percentages of extremely concordant and extremely discordant sib pairs are calculated that maximize the power to detect a quantitative trait locus (QTL) under a variety of circumstances using the EDAC test. We assume a large fixed number of randomly sampled sib pairs, such as one would hope to find in the large twin registries, and limited resources to genotype a certain number of selected sib pairs. Our aim is to investigate whether optimal selection can be achieved when prior knowledge concerning the QTL gene action, QTL allele frequency, QTL effect size, and background (residual) sib correlation is limited or absent. To this end we calculate the best selection percentages for a large number of models, which differ in QTL gene action allele frequency, background correlation, and QTL effect size. By averaging these percentages over gene action, over allele frequency, over gene action, and over allele frequencies, we arrive at general recommendations concerning selection percentages. The soundness of these recommendations is subsequently in a number of test cases.     

The additive genetic gamma frailty model for linkage analysis of age-of-onset variation

Age of onset is a key factor in the linkage analysis of many complex diseases. Current methods in nonparametric linkage analysis are mainly concentrated on the affected relative pairs or affected family members with age of onset information either ignored or taken into account by specifying age-dependent penetrances for liability classes. On the other hand, gamma frailty models were developed in the biostatistics literature to model familial aggregation of age of onset. However, these frailty models cannot be used directly for linkage analysis. This paper extends the gamma frailty model by incorporating inheritance vector information and provides a semiparametric approach for linkage testing. For a given inheritance vector at the putative disease locus, we construct an additive genetic gamma frailty for each individual within a nuclear family and use the Cox proportional hazard model to model age of onset. We derive the conditional hazard ratio parameter for sib pairs and define a likelihood ratio based LOD score statistic under our model. The EM algorithm is used for estimating the parameters and the maximum likelihood functions. Simulated data sets are used to illustrate these new statistical methods.     

Quantitative trait locus for reading disability on chromosome 6p is pleiotropic for attention‐deficit/hyperactivity disorder

Comorbidity is pervasive among both adult and child psychiatric disorders; however, the etiological mechanisms underlying the majority of comorbidities are unknown. This study used genetic linkage analysis to assess the etiology of comorbidity between reading disability (RD) and attention‐deficit hyperactivity disorder (ADHD), two common childhood disorders that frequently co‐occur. Sibling pairs (N = 85) were ascertained initially because at least one individual in each pair exhibited a history of reading difficulties. Univariate linkage analyses in sibling pairs selected for ADHD from within this RD‐ascertained sample suggested that a quantitative trait locus (QTL) on chromosome 6p is a susceptibility locus for ADHD. Because this QTL is in the same region as a well‐replicated QTL for reading disability, subsequent bivariate analyses were conducted to test if this QTL contributed to comorbidity between the two disorders. Analyses of data from sib pairs selected for reading deficits revealed suggestive bivariate linkage for ADHD and three measures of reading difficulty, indicating that comorbidity between RD and ADHD may be due at least in part to pleiotropic effects of a QTL on chromosome 6p. © 2002 Wiley‐Liss, Inc.     

Optimal weighting scheme for affected sib-pair analysis of sibship data

Application of the affected sib-pair method of linkage analysis to sibships with variable numbers of affected and unaffected members requires a scheme for weighting the contributions from the sibships in the calculation of an overall test statistic. Currently accepted weighting schemes are based on the concepts of independent pairs and Shannon information content. Here, we show that the weighting scheme with maximum power to detect linkage can be determined from the theoretical means and variances of the sibship contributions under the null and alternative hypotheses. We derive the theoretical means and variances of the contributions from different types of sibships under a generalized single locus model. We use these theoretical means and variances to obtain the optimal weights for a variety of single locus models, and compare the power of existing weighting schemes relative to the optimum. The results suggest that, under a range of plausible assumptions, optimal power is nearly obtained by weighting to all sibpairs equally, except those occuring in sibships with so many affected siblings (usually five or more) that the probability of parental homozygosity for the disease allele becomes substantial. A corollary is that, in affected sib-pair analysis, the 'informativeness' of a sibship is more nearly proportional to the number of affected sib-pairs than to the number of affected siblings, up to about five affected siblings.     

Power of a simplified multivariate test for genetic linkage

In this paper we compare the power of the multivariate Haseman-Elston (MHE) test proposed earlier by Amos et al. (1990) and a computationally rapid new version of the multivariate Haseman-Elston test (NMHE) (Elston et al. 2000). We show that the power of NMHE was, for different simulation setups, identical or higher than that of MHE. In the bivariate case, the power of the NMHE method was somewhat less than that of the computationally intensive maximum likelihood variance components method (Amos et al. 2001). We present comparisons of the empirical distributions of the NMHE test to its limiting distributions for a range of numbers of traits. The distribution of the NMHE test appeared to conform satisfactorily to its limiting asymptotic distribution in large samples. Otherwise, empirical critical values for NMHE are somewhat higher than predicted, i.e. the test proposed by Elston et al. (2000) is non-conservative. The use of empirical critical values is therefore recommended for limited sample sizes (less than several hundred families). We also present the results of a linkage analysis performed by the NMHE method on a set of 4 body size-related traits. The method identified meaningful combinations of traits that showed significant linkage on chromosome 2 and suggestive linkage to regions on chromosomes 16 and 17.     

Diagnosis as a covariate in sib-pair linkage analysis.

A novel technique to detect significant covariates in linkage analysis using a logistic regression approach is illustrated. An overall test of linkage is first performed to determine whether there is significant perturbation from the expected 50% sharing in sib pairs. Covariates may include variables defined on the sib pair (multiple levels of diagnosis), covariates defined on the parents (parental diagnosis or gender of the transmitting parent), or indicators of study when analyzing multiple datasets to examine heterogeneity. A detailed example using a hierarchical diagnosis of severe, intermediate, mild, and unaffected is provided. This permits testing whether sib allele sharing differs by level of diagnosis and allows the use of discordant pairs to protect against marker allele misspecification.