MCMC-based linkage analysis for complex traits on general pedigrees: multipoint analysis with a two-locus model and a polygenic component

We describe a new program lm_twoqtl, part of the MORGAN package, for parametric linkage analysis with a quantitative trait locus (QTL) model having one or two QTLs and a polygenic component, which models additional familial correlation from other unlinked QTLs. The program has no restriction on number of markers or complexity of pedigrees, facilitating use of more complex models with general pedigrees. This is the first available program that can handle a model with both two QTLs and a polygenic component. Competing programs use only simpler models: one QTL, one QTL plus a polygenic component, or variance components (VC). Use of simple models when they are incorrect, as for complex traits that are influenced by multiple genes, can bias estimates of QTL location or reduce power to detect linkage. We compute the likelihood with Markov Chain Monte Carlo (MCMC) realization of segregation indicators at the hypothesized QTL locations conditional on marker data, summation over phased multilocus genotypes of founders, and peeling of the polygenic component. Simulated examples, with various sized pedigrees, show that two-QTL analysis correctly identifies the location of both QTLs, even when they are closely linked, whereas other analyses, including the VC approach, fail to identify the location of QTLs with modest contribution. Our examples illustrate the advantage of parametric linkage analysis with two QTLs, which provides higher power for linkage detection and better localization than use of simpler models.     

A variance component approach to dichotomous trait linkage analysis using a threshold model

We developed and utilized a multipoint variance components method to test for linkage between a disease trait and markers on chromosome 5 in the simulated data provided in GAW10 Problem 2. We demonstrated that the discrete trait variance components method recovers unbiased estimates of quantitative trait locus (QTL) location and reasonable estimates of effect size. We also showed that dichotomization of (a continuous trait such as) Q1 diminished the power to detect linkage compared to direct analysis of Q1, and that extended pedigree analyses provided superior power to detect linkage compared to those in nuclear families. © 1997 Wiley-Liss, Inc.     

Ignoring temporal trends in genetic effects substantially reduces power of quantitative trait linkage analysis

Linkage analysis has been one of the most widely used methods for identifying regions of the human genome which contain genes responsible for human diseases. Evidence suggests that the effects of some of the trait causing genes may vary with the age of an individual, giving rise to temporal trends in genetic effects. Linkage analysis routinely tends to ignore such gene-by-age interactions. While linkage analysis methods have been proposed for analysis of longitudinal family data for exploring temporal trends, there are no models to characterize such trends nor methods for analysis of cross-sectional family data. We extend variance component linkage analysis methodology by modeling the variance components due to the quantitative trait locus (QTL) and that due to the polygenic effect to be age dependent. With this model, we investigate the power of linkage analysis in the presence of temporal trends. We show that modeling true temporal trends in QTL effects can substantially increase the power of linkage analysis even when the average locus-specific heritabilities (when trends are ignored) are quite low, thereby demonstrating that, ignoring the gene-by-age interactions, when present, could jeopardize gene discovery.     

Genotype by smoking interaction for leptin levels in the San Antonio Family Heart Study.

Recent studies reported a marked inverse effect of smoking on serum levels of leptin (an adipocyte derived protein), offering a possible explanation for variation in body weight between smokers and non-smokers. The goal of this study was to examine the genetic architecture of the response to smoking in leptin levels using data from the San Antonio Family Heart Study. We employed a variance decomposition analysis using maximum likelihood methods to model genotype by smoking interactions for leptin levels, examined the impact of the exclusion of smokers in a subsequent linkage analysis, and incorporated the QTL identified in the linkage analysis in a model of genotype by smoking interaction. We found significant evidence (P = 0.001) for a genotype by smoking status interaction for serum leptin levels. In the subsequent linkage analysis with smokers excluded, we obtained a maximum LOD score of 3.1 (P = 0.00008) near D8S1102. Using this QTL in a model of genotype by smoking status interaction, we identified significant evidence for an interaction at this specific locus (P = 0.04). Given these results, we hypothesize that a quantitative trait locus in this vicinity of chromosome 8 may have a differential effect on the expression of leptin in smokers versus non-smokers.     

Type I error rates in association versus joint linkage/association tests in related individuals

Positional gene discovery on pedigree data typically involves initial gross localization by linkage analysis with subsequent finer localization by association analysis in areas that show evidence of linkage. We examine the effect of including linkage information when testing for association in the context of variance-components-based pedigree analysis. We present simulation experiments showing that, at least in the extreme case of a rare private allele, failing to include the linkage variance component in the association model results in excessive Type I error that increases with allele copy number and/or quantitative trait locus (QTL) effect size. Joint estimation of the linkage variance component in the association model reduces Type I error to nominal expectations. This holds whether allele-sharing probabilities are estimated from a polymorphic marker or from the very single-nucleotide polymorphism (SNP) being tested for association, although the latter provides much less power. These results support the idea that an appropriate association analysis must test both the random effect of shared marker alleles (linkage) and the mean effects of the marker genotypes (association).     

Distribution of lod Scores in Oligogenic Linkage Analysis

In variance component oligogenic linkage analysis it can happen that the residual additive genetic variance bounds to zero when estimating the effect of the i^th quantitative trait locus. Using quantitative trait Q1 from the Genetic Analysis Workshop 12 simulated general population data, we compare the observed lod scores from oli‐gogenic linkage analysis with the empirical lod score distribution under a null model of no linkage. We find that zero residual additive genetic variance in the null model alters the usual distribution of the likelihood‐ratio statistic. © 2001 Wiley‐Liss, Inc.     

A method for identifying genes related to a quantitative trait, incorporating multiple siblings and missing parents

When studying either qualitative or quantitative traits, tests of association in the presence of linkage are necessary for fine-mapping. In a previous report, we suggested a polytomous logistic approach to testing linkage and association between a di-allelic marker and a quantitative trait locus, using genotyped triads, consisting of an individual whose quantitative trait has been measured and his or her two parents. Here we extend that approach to incorporate marker information from entire nuclear families. By computing a weighted score function instead of a maximum likelihood test, we allow for both an unspecified correlation structure between siblings and "informative" family size. Both this approach and our original approach allow for population admixture by conditioning on parental genotypes. The proposed method allows for missing parental genotype data through a multiple imputation procedure. We use simulations based on a population with admixture to compare our method to a popular non-parametric family-based association test (FBAT), testing the null of no association in the presence of linkage.     

Replication of Linkage to Quantitative Trait Loci: Variation in Location and Magnitude of the Lod Score

Replication of linkage signals from independent samples is considered an important step toward verifying the significance of linkage signals in studies of complex traits. The purpose of this empirical investigation was to examine the variability in the precision of localizing a quantitative trait locus (QTL) by analyzing multiple replicates of a simulated data set with the use of variance components‐based methods. Specifically, we evaluated across replicates the variation in both the magnitude and the location of the peak lod scores. We analyzed QTLs whose effects accounted for 10–37% of the phenotypic variance in the quantitative traits. Our analyses revealed that the precision of QTL localization was directly related to the magnitude of the QTL effect. For a QTL with effect accounting for > 20% of total phenotypic variation, > 90% of the linkage peaks fall within 10 cM from the true gene location. We found no evidence that, for a given magnitude of the lod score, the presence of interaction influenced the precision of QTL localization. © 2001 Wiley‐Liss, Inc.     

Unified sampling approach for multipoint linkage disequilibrium mapping of qualitative and quantitative traits

Rapid development in biotechnology has enhanced the opportunity to deal with multipoint gene mapping for complex diseases, and association studies using quantitative traits have recently generated much attention. Unlike the conventional hypothesis-testing approach for fine mapping, we propose a unified multipoint method to localize a gene controlling a quantitative trait. We first calculate the sample size needed to detect linkage and linkage disequilibrium (LD) for a quantitative trait, categorized by decile, under three different modes of inheritance. Our results show that sampling trios of offspring and their parents from either extremely low (EL) or extremely high (EH) probands provides greater statistical power than sampling in the intermediate range. We next propose a unified sampling approach for multipoint LD mapping, where the goal is to estimate the map position (tau) of a trait locus and to calculate a confidence interval along with its sampling uncertainty. Our method builds upon a model for an expected preferential transmission statistic at an arbitrary locus conditional on the sampling scheme, such as sampling from EL and EH probands. This approach is valid regardless of the underlying genetic model. The one major assumption for this model is that no more than one quantitative trait locus (QTL) is linked to the region being mapped. Finally we illustrate the proposed method using family data on total serum IgE levels collected in multiplex asthmatic families from Barbados. An unobserved QTL appears to be located at tau; = 41.93 cM with 95% confidence interval of (40.84, 43.02) through the 20-cM region framed by markers D12S1052 and D12S1064 on chromosome 12. The test statistic shows strong evidence of linkage and LD (chi-square statistic = 18.39 with 2 df, P-value = 0.0001).     

Multipoint oligogenic linkage analysis of quantitative traits

We present an overview of pedigree-based variance component linkage methods and discuss their extension to oligogenic inheritance. As an example, oligogenic linkage analyses were performed using the quantitative trait Q4 from the GAW10 simulated data set. A strategy involving sequential oligogenic analyses was found to have increased power to detect the three quantitative trait loci (QTL) influencing Q4 when compared to the classical marginal approach of requiring each locus to have a lod score ≥ 3. However, it is shown that requiring conditional lod scores ≥ 3 in the sequential analyses may be overly conservative and alternative criteria for the acceptance of multilocus models are discussed. © 1997 Wiley-Liss, Inc     

Genotype-based association mapping of complex diseases: gene-environment interactions with multiple genetic markers and measurement error in environmental exposures

With the advent of dense single nucleotide polymorphism genotyping, population-based association studies have become the major tools for identifying human disease genes and for fine gene mapping of complex traits. We develop a genotype-based approach for association analysis of case-control studies of gene-environment interactions in the case when environmental factors are measured with error and genotype data are available on multiple genetic markers. To directly use the observed genotype data, we propose two genotype-based models: genotype effect and additive effect models. Our approach offers several advantages. First, the proposed risk functions can directly incorporate the observed genotype data while modeling the linkage disequilibrium information in the regression coefficients, thus eliminating the need to infer haplotype phase. Compared with the haplotype-based approach, an estimating procedure based on the proposed methods can be much simpler and significantly faster. In addition, there is no potential risk due to haplotype phase estimation. Further, by fitting the proposed models, it is possible to analyze the risk alleles/variants of complex diseases, including their dominant or additive effects. To model measurement error, we adopt the pseudo-likelihood method by Lobach et al. [2008]. Performance of the proposed method is examined using simulation experiments. An application of our method is illustrated using a population-based case-control study of association between calcium intake with the risk of colorectal adenoma development.     

Multivariate genetic analysis of an oligogenic disease

Joint multivariate segregation and linkage analysis provides a method for simultaneously analyzing data on affection status, correlated phenotypic traits, environmental risk factors, and other covariates. The power of this approach for mapping disease susceptibility loci of small effect (oligogenes) was evaluated by analyzing the GAW9 Problem 2 data set. The program REGRESS, which assumes a pleiotropy model in which one locus influences both affection status (AF) and a quantitative trait, was used to conduct joint segregation and linkage analysis of bivariate phenotypes, each comprising AF and one quantitative trait (Q2,Q3,Q4). A genome-wide search using markers spaced approximately 10 cM apart was conducted and regions on chromosomes 1, 2, and 5 were identified as demonstrating linkage with three respective bivariate phenotypes at the following markers: AF/Q2 - D1G2; AF/Q3 - D2G10; and AF/Q4 - D5G18. The effects of other loci were included in a general model by specifying the quantitative traits they influenced as covariates along with age, sex, and an environmental effect. Use of covariate and quantitative trait data in each analysis resulted in respective χ² values with 1 df of 38.4, 65.4, and 22.0 to reject the no linkage hypothesis at $ {\rm \hat \theta } $ = 0, with respective equivalent lod scores of 8.3, 14.2, and 4.8. Rejection at p < 0.0002 occurred using markers as far away as 20 cM. These loci were not detected when AF alone was analyzed. © 1995 Wiley-Liss, Inc.     

Mapping a quantitative trait locus via the EM algorithm and Bayesian classification

Mapping a locus controlling a quantitative genetic trait (e.g., blood pressure) to a specific genomic region is of considerable interest. Data on the quantitative trait under consideration and several codominant genetic markers with known genomic locations are collected from members of families and statistically analyzed to draw inferences on the genomic position of the trait locus. The vector of parameters of interest comprises the pairwise recombination fractions, theta, between the putative quantitative trait locus and the marker loci. One of the major complications in estimating theta for a quantitative trait in humans is the lack of haplotype information on members of families. The purpose of this study was to devise a computationally simple and efficient method of estimation of theta in the absence of haplotype information. We have proposed a two-stage estimation procedure using the expectation-maximization (EM) algorithm. In the first stage, parameters of the QTL are estimated based on data of a sample of unrelated individuals. From estimates thus obtained, we have used a Bayes' rule to infer QTL genotypes of parents in families. Finally, in the second stage of the procedure, we have proposed an EM algorithm for obtaining the maximum likelihood estimate of theta based on data of informative families (which are identified upon inferring parental QTL genotypes performed in the first stage). We have shown, using simulated data, that the proposed procedure is cost-effective, computationally simple, and statistically efficient. As expected, analysis of data on multiple markers jointly is more efficient than the analysis based on single markers.     

Linkage Analysis of Quantitative Traits in Randomly Ascertained Pedigrees: Comparison of Penetrance‐Based and Variance Component Analysis

Penetrance‐based linkage analysis and variance component linkage analysis are two methods that are widely used to localize genes influencing quantitative traits. Using computer programs PAP and SOLAR as representative software implementations, we have conducted an empirical comparison of both methods' power to map quantitative trait loci in extended, randomly ascertained pedigrees, using simulated data. Two‐point linkage analyses were conducted on several quantitative traits of different genetic and environmental etiology using both programs, and the lod scores were compared. The two methods appear to have similar power when the underlying quantitative trait locus is diallelic, with one or the other method being slightly more powerful depending on the characteristics of the quantitative trait and the quantitative trait locus. In the case of a multiallelic quantitative trait locus, however, the variance component approach has much greater power. These findings suggest that one should give careful thought to the likely allelic architecture of the quantitative trait to be analyzed when choosing between these two analytical approaches. It may be the case in general that linkage methods which explicitly or implicitly rely on the assumption of a diallelic trait locus fare poorly when this assumption is incorrect. © 2001 Wiley‐Liss, Inc.     

Limits of fine-mapping a quantitative trait

Once a significant linkage is found, an important goal is reducing the error in the estimated location of the linked locus. A common approach to reducing location error, called fine-mapping, is the genotyping of additional markers in the linked region to increase the genetic information. The utility of fine-mapping for quantitative trait linkage analysis is largely unknown. To explore this issue, we performed a fine-mapping simulation in which the region containing a significant linkage at a 10-centiMorgan (cM) resolution was fine-mapped at 2, 1, and 0.5 cM. We simulated six quantitative trait models in which the proportion of variation due to the quantitative trait locus (QTL) ranged from 0.20-0.90. We used four sampling designs that were all combinations of 100 and 200 families of sizes 5 and 7. Variance components linkage analysis (Genehunter) was performed until 1,000 replicates were found with a maximum lodscore greater than 3.0. For each of these 1,000 replications, we repeated the linkage analysis three times: once for each of the fine-map resolutions. For the most realistic model, reduction in the average location error ranged from 3-15% for 2-cM fine-mapping and from 3-18% for 1-cM fine-mapping, depending on the number of families and family size. Fine-mapping at 0.5 cM did not differ from the 1-cM results. Thus, if the QTL accounts for a small proportion of the variation, as is the case for realistic traits, fine-mapping has little value.     

An empirical test of the significance of an observed quantitative trait locus effect that preserves additive genetic variation

We propose a constrained permutation test that assesses the significance of an observed quantitative trait locus effect against a background of genetic and environmental variation. Permutations of phenotypes are not selected at random, but rather are chosen in a manner that attempts to maintain the additive genetic variability in phenotypes. Such a constraint maintains the nonindependence among observations under the null hypothesis of no linkage. The empirical distribution of the lod scores calculated using permuted phenotypes is compared to that obtained using phenotypes simulated from the assumed underlying multivariate normal model. We make comparisons of univariate analyses for both a quantitative phenotype that appears consistent with a multivariate normal model and a quantitative phenotype containing pronounced outliers. An example of a bivariate analysis is also presented.     

Bivariate association analyses for the mixture of continuous and binary traits with the use of extended generalized estimating equations

Genome-wide association (GWA) study is becoming a powerful tool in deciphering genetic basis of complex human diseases/traits. Currently, the univariate analysis is the most commonly used method to identify genes associated with a certain disease/phenotype under study. A major limitation with the univariate analysis is that it may not make use of the information of multiple correlated phenotypes, which are usually measured and collected in practical studies. The multivariate analysis has proven to be a powerful approach in linkage studies of complex diseases/traits, but it has received little attention in GWA. In this study, we aim to develop a bivariate analytical method for GWA study, which can be used for a complex situation in which continuous trait and a binary trait are measured under study. Based on the modified extended generalized estimating equation (EGEE) method we proposed herein, we assessed the performance of our bivariate analyses through extensive simulations as well as real data analyses. In the study, to develop an EGEE approach for bivariate genetic analyses, we combined two different generalized linear models corresponding to phenotypic variables using a seemingly unrelated regression model. The simulation results demonstrated that our EGEE-based bivariate analytical method outperforms univariate analyses in increasing statistical power under a variety of simulation scenarios. Notably, EGEE-based bivariate analyses have consistent advantages over univariate analyses whether or not there exists a phenotypic correlation between the two traits. Our study has practical importance, as one can always use multivariate analyses as a screening tool when multiple phenotypes are available, without extra costs of statistical power and false-positive rate. Analyses on empirical GWA data further affirm the advantages of our bivariate analytical method.     

Two-level Haseman-Elston regression for general pedigree data analysis

The Haseman-Elston (HE) (Haseman and Elston [1972] Behav Genet 2:3-19) method is widely used in genetic linkage studies for quantitative traits. We propose a new version of the HE regression model, a two-level HE regression model (tHE) in which the variance-covariance structure of family data is modeled under the framework of multiple-level regression. An iterative generalized least squares (IGLS) algorithm is adopted to handle the varying variance-covariance structures across families in a simple fashion. In this way, the tHE can compete favorably with any current version of HE in that it can naturally make use of all the trait information available in any general pedigree, simultaneously incorporate individual-level and pedigree-level covariates, marker genotypes for linkage (i.e., the number of allele shared identically by descent [IBD]), and marker alleles for association. Under the assumption of normality, the method is asymptotically equivalent to the usual variance component model for detecting linkage. For the situation where the assumption of normality is critical, a robust globally consistent estimator of the quantitative trait locus (QTL) variance is available. Complex genetic mechanisms, including gene-gene interaction, gene-environmental interaction, and imprinting, can be directly modeled in this version of HE regression.     

Detecting Genotype × Age Interaction

This paper explores several extensions to the variance component method, which incorporate genotype × age interaction effects. We evaluate the performance of these methods for detecting genotype × age interaction in quantitative genetic analyses of a quantitative trait (Q4), contrasting this with false positive detection rates obtained from a phenotype influenced by the same genes but without genotype × age interaction effects (Q3). We then assess the impact on linkage power and false positive rate of allowing a QTL‐specific genotype × age interaction in linkage analysis of these same traits. © 2001 Wiley‐Liss, Inc.     

Genetic analysis of personality traits and alcoholism using a mixed discrete continuous trait variance component model

Bivariate analyses can improve power to detect linkage. This paper describes one application of a bivariate variance component method for estimating joint likelihoods of a continuous and a discrete trait. This method is applied to the Collaborative Study on the Genetics of Alcoholism data set to investigate the relationship between personality traits derived from the tridimensional personality questionnaire (TPQ) and alcoholism. The results indicate that the novelty-seeking subscale of the TPQ and alcoholism share a strong and significant genetic correlation (rho G = 0.83) and modest environmental correlation (rho E = 0.31). When both traits are considered jointly in a multipoint linkage model compared with the alcoholism trait alone, there is an improvement in the ability to detect and localize a quantitative trait locus on chromosome 4.