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.     

Powerful regression-based quantitative-trait linkage analysis of general pedigrees

We present a new method of quantitative-trait linkage analysis that combines the simplicity and robustness of regression-based methods and the generality and greater power of variance-components models. The new method is based on a regression of estimated identity-by-descent (IBD) sharing between relative pairs on the squared sums and squared differences of trait values of the relative pairs. The method is applicable to pedigrees of arbitrary structure and to pedigrees selected on the basis of trait value, provided that population parameters of the trait distribution can be correctly specified. Ambiguous IBD sharing (due to incomplete marker information) can be accommodated in the method by appropriate specification of the variance-covariance matrix of IBD sharing between relative pairs. We have implemented this regression-based method and have performed simulation studies to assess, under a range of conditions, estimation accuracy, type I error rate, and power. For normally distributed traits and in large samples, the method is found to give the correct type I error rate and an unbiased estimate of the proportion of trait variance accounted for by the additive effects of the locus-although, in cases where asymptotic theory is doubtful, significance levels should be checked by simulations. In large sibships, the new method is slightly more powerful than variance-components models. The proposed method provides a practical and powerful tool for the linkage analysis of quantitative traits.     

DNA markers in primate models of human disease

Nonhuman primates are particularly useful as animal models for common human diseases in which both genetic and environmental factors play important roles. The recent development of DNA markers (restriction fragment length polymorphisms, RFLPs) greatly increases the power of linkage analysis to detect major genes that affect quantitative phenotypes, including those related to diseases. This paper summarizes a strategy for using RFLPs in linkage analysis of baboon pedigrees to identify genes that control lipoprotein phenotype, which in turn is predictive of susceptibility to atherosclerosis. This strategy also can be applied to other common human diseases for which nonhuman primate models exist.     

A robust multiple-locus method for quantitative trait locus analysis of non-normally distributed multiple traits

Linear regression-based quantitative trait loci/association mapping methods such as least squares commonly assume normality of residuals. In genetics studies of plants or animals, some quantitative traits may not follow normal distribution because the data include outlying observations or data that are collected from multiple sources, and in such cases the normal regression methods may lose some statistical power to detect quantitative trait loci. In this work, we propose a robust multiple-locus regression approach for analyzing multiple quantitative traits without normality assumption. In our method, the objective function is least absolute deviation (LAD), which corresponds to the assumption of multivariate Laplace distributed residual errors. This distribution has heavier tails than the normal distribution. In addition, we adopt a group LASSO penalty to produce shrinkage estimation of the marker effects and to describe the genetic correlation among phenotypes. Our LAD-LASSO approach is less sensitive to the outliers and is more appropriate for the analysis of data with skewedly distributed phenotypes. Another application of our robust approach is on missing phenotype problem in multiple-trait analysis, where the missing phenotype items can simply be filled with some extreme values, and be treated as outliers. The efficiency of the LAD-LASSO approach is illustrated on both simulated and real data sets.     

Analysis of quantitative trait loci using hybrid pedigrees: quantitative traits of animals

A method is proposed for analysis of quantitative traits in animal hybrid pedigrees formed by crosses between outbred lines differing in allele frequencies of the genes controlling the trait studied. The method is based on the decomposition of trait variances into components and uses maximization of the likelihood function for estimating model parameters, which allows the estimation of additive and dominance effects of the gene involved in trait determination and its allele frequencies, as well as determination of the chromosomal position of this gene relative to genotyped markers. To test the linkage of this gene with markers, a statistic with the noncentral chi(2) distribution has been chosen. Analytical expressions for the power of this method have been derived. The method has been tested on small model hybrid pedigrees. Phenotypic values of the trait and information on marker genotypes for each individual in hybrid pedigrees are original data for the analysis of a quantitative trait.     

Simulation of pedigree genotypes by random walks.

A random walk method, based on the Metropolis algorithm, is developed for simulating the distribution of trait and linkage marker genotypes in pedigrees where trait phenotypes are already known. The method complements techniques suggested by Ploughman and Boehnke and by Ott that are based on sequential sampling of genotypes within a pedigree. These methods are useful for estimating the power of linkage analysis before complete study of a pedigree is undertaken. We apply the random walk technique to a partially penetrant disease, schizophrenia, and to a recessive disease, ataxia-telangiectasia. In the first case we show that accessory phenotypes with higher penetrance than that of schizophrenia itself may be crucial for effective linkage analysis, and in the second case we show that impressionistic selection of informative pedigrees may be misleading.     

Approximating identity-by-descent matrices using multiple haplotype configurations on pedigrees

Identity-by-descent (IBD) matrix calculation is an important step in quantitative trait loci (QTL) analysis using variance component models. To calculate IBD matrices efficiently for large pedigrees with large numbers of loci, an approximation method based on the reconstruction of haplotype configurations for the pedigrees is proposed. The method uses a subset of haplotype configurations with high likelihoods identified by a haplotyping method. The new method is compared with a Markov chain Monte Carlo (MCMC) method (Loki) in terms of QTL mapping performance on simulated pedigrees. Both methods yield almost identical results for the estimation of QTL positions and variance parameters, while the new method is much more computationally efficient than the MCMC approach for large pedigrees and large numbers of loci. The proposed method is also compared with an exact method (Merlin) in small simulated pedigrees, where both methods produce nearly identical estimates of position-specific kinship coefficients. The new method can be used for fine mapping with joint linkage disequilibrium and linkage analysis, which improves the power and accuracy of QTL mapping.     

Power comparison of parametric and nonparametric linkage tests in small pedigrees

When the mode of inheritance of a disease is unknown, the LOD-score method of linkage analysis must take into account uncertainties in model parameters. We have previously proposed a parametric linkage test called "MFLOD," which does not require specification of disease model parameters. In the present study, we introduce two new model-free parametric linkage tests, known as "MLOD" and "MALOD." These tests are defined, respectively, as the LOD score and the admixture LOD score, maximized (subject to the same constraints as MFLOD) over disease-model parameters. We compared the power of these three parametric linkage tests and that of two nonparametric linkage tests, NPLall and NPLpairs, which are implemented in GENEHUNTER. With the use of small pedigrees and a fully informative marker, we found the powers of MLOD, NPLall, and NPLpairs to be almost equivalent to each other and not far below that of a LOD-score analysis performed under the assumption the correct genetic parameters. Thus, linkage analysis is not much hindered by uncertain mode of inheritance. The results also suggest that both parametric and nonparametric methods are suitable for linkage analysis of complex disorders in small pedigrees. However, whether these results apply to large pedigrees remains to be answered.     

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.     

Analysis of quantitative trait loci using hybrid pedigrees: Quantitative traits of animals

A method is proposed for analysis of quantitative traits in animal hybrid pedigrees formed by crosses between outbred lines differing in allele frequencies of the genes controlling the trait studied. The method is based on the decomposition of trait variances into components and uses maximization of the likelihood function for estimating model parameters, which allows the estimation of additive and dominance effects of the gene involved in trait determination and its allele frequencies, as well as determination of the chromosomal position of this gene relative to genotyped markers. To test the linkage of this gene with markers, a statistic with the noncentral x ² distribution has been chosen. Analytical expressions for the power of this method have been derived. The method has been tested on small model hybrid pedigrees. Phenotypic values of the trait and information on marker genotypes for each individual in hybrid pedigrees are initial data for the analysis of a quantitative trait.     

Simultaneous estimation of haplotype frequencies and quantitative trait parameters: applications to the test of association between phenotype and diplotype configuration

The analysis of the haplotype-phenotype relationship has become more and more important. We have developed an algorithm, using individual genotypes at linked loci as well as their quantitative phenotypes, to estimate the parameters of the distribution of the phenotypes for subjects with and without a particular haplotype by an expectation-maximization (EM) algorithm. We assumed that the phenotype for a diplotype configuration follows a normal distribution. The algorithm simultaneously calculates the maximum likelihood (L0max) under the null hypothesis (i.e., nonassociation between the haplotype and phenotype), and the maximum likelihood (Lmax) under the alternative hypothesis (i.e., association between the haplotype and phenotype). Then we tested the association between the haplotype and the phenotype using a test statistic, -2 log(L0max/Lmax). The above algorithm along with some extensions for different modes of inheritance was implemented as a computer program, QTLHAPLO. Simulation studies using single-nucleotide polymorphism (SNP) genotypes have clarified that the estimation was very accurate when the linkage disequilibrium between linked loci was rather high. Empirical power using the simulated data was high enough. We applied QTLHAPLO for the analysis of the real data of the genotypes at the calpain 10 gene obtained from diabetic and control subjects in various laboratories.     

Linkage analysis of quantitative trait loci in the presence of heterogeneity

Variance component modeling for linkage analysis of quantitative traits is a powerful tool for detecting and locating genes affecting a trait of interest, but the presence of genetic heterogeneity will decrease the power of a linkage study and may even give biased estimates of the location of the quantitative trait loci. Many complex diseases are believed to be influenced by multiple genes and therefore genetic heterogeneity is likely to be present for many real applications of linkage analysis. We consider a mixture of multivariate normals to model locus heterogeneity by allowing only a proportion of the sampled pedigrees to segregate trait-influencing allele(s) at a specific locus. However, for mixtures of normals the classical asymptotic distribution theory of the maximum likelihood estimates does not hold, so tests of linkage and/or heterogeneity are evaluated using resampling methods. It is shown that allowing for genetic heterogeneity leads to an increase in power to detect linkage. This increase is more prominent when the genetic effect of the locus is small or when the percentage of pedigrees not segregating trait-influencing allele(s) at the locus is high.     

The probabilistic determination of identity-by-descent sharing for pairs of relatives from pedigrees.

Methods for detecting genetic linkage are more powerful when they fully use all of the data collected from pedigrees. We first discuss a method for obtaining the probability that a pedigree member has a given genotype, conditional on the phenotypes of his relatives. We then develop a rapid method to obtain the conditional probabilities of identity-by-descent sharing of marker alleles for all related pairs of individuals from extended pedigrees. The method assumes that the individuals are noninbred and that the relationship between genotype and phenotype is known for the marker locus studied. The probabilities of identity-by-descent sharing among relative pairs, conditional on marker phenotype information, can then be used in any of the model free tests for linkage between a trait locus and a marker locus.     

Sample-size guidelines for linkage analysis of a dominant locus for a quantitative trait by the method of lod scores.

Sample-size guidelines for linkage studies of quantitative traits partially determined by a dominant major locus are needed to provide a rough estimate of the amount of pedigree material that should be sampled to map the loci that influence such traits. After pedigrees are sampled, a specific power calculation can be carried out to evaluate the linkage information provided by the sampled pedigrees. Using computer simulation, I provide sample-size guidelines for linkage studies by the method of lod scores of quantitative traits partially determined by a dominant major locus. I consider the effects of a trait model, marker characteristics, and sampling strategy, with particular attention to sampling strategy because it is the one factor which the investigator can fully control. My results suggest that linkage studies of quantitative traits are practical, particularly if the investigator chooses an efficient sampling design and an efficient strategy to select pedigrees for linkage analysis.     

Linkage analysis of alpha 1-antitrypsin deficiency: lessons for complex diseases.

OBJECTIVES: Severe alpha 1-antitrypsin (A1AT) deficiency is the one proven genetic risk factor for chronic obstructive pulmonary disease (COPD). Familial aggregation has been demonstrated for COPD among individuals who do not have A1AT deficiency, but linkage analysis of COPD has not been reported. To investigate the optimal phenotype definitions and analytical methods for the linkage analysis of COPD, we examined a set of 28 A1AT- deficient families containing 155 individuals. We have used the protease inhibitor (PI) type as a genetic marker rather than a disease gene, and we have performed linkage analysis between PI type and serum A1AT level and spirometry-related phenotypes. METHODS: Linkage analysis was performed on the quantitative phenotypes forced expiratory volume at 1 s (FEV(1) as % predicted), the ratio of FEV(1) to forced vital capacity (FEV(1)/FVC as % predicted), and serum A1AT level using the variance component approach in SOLAR, the generalized estimating equation approach in RELPAL, and the model-based classical lod score method in LINKAGE. Linkage analysis with qualitative A1AT and spirometry phenotypes was performed using a model-based method (LINKAGE) and a model-free method (GENEHUNTER). Adjustments for smoking effects were investigated under each method. RESULTS: All of the methods demonstrated linkage of PI type to serum A1AT level. Interestingly, however, the other quantitative phenotypes provided only weak evidence for linkage of PI type to lung disease. Better evidence for linkage of lung disease to PI type was found using a moderate or a mild threshold for the definition of airflow obstruction. CONCLUSIONS: For linkage analysis of spirometry phenotypes in A1AT deficiency, qualitative phenotypes provided stronger evidence for linkage than quantitative phenotypes. Possible contributors to the stronger evidence for linkage to qualitative spirometry phenotypes include the ascertainment scheme and the nonnormality of the pulmonary function data in PI Z subjects. This study provides guidelines for studies of the genetics of COPD unrelated to A1AT deficiency.     

Identification of genes for complex disease using longitudinal phenotypes

Using the simulated data set from Genetic Analysis Workshop 13, we explored the advantages of using longitudinal data in genetic analyses. The weighted average of the longitudinal data for each of seven quantitative phenotypes were computed and analyzed. Genome screen results were then compared for these longitudinal phenotypes and the results obtained using two cross-sectional designs: data collected near a single age (45 years) and data collected at a single time point. Significant linkage was obtained for nine regions (LOD scores ranging from 5.5 to 34.6) for six of the phenotypes. Using cross-sectional data, LOD scores were slightly lower for the same chromosomal regions, with two regions becoming nonsignificant and one additional region being identified. The magnitude of the LOD score was highly correlated with the heritability of each phenotype as well as the proportion of phenotypic variance due to that locus. There were no false-positive linkage results using the longitudinal data and three false-positive findings using the cross-sectional data. The three false positive results appear to be due to the kurtosis in the trait distribution, even after removing extreme outliers. Our analyses demonstrated that the use of simple longitudinal phenotypes was a powerful means to detect genes of major to moderate effect on trait variability. In only one instance was the power and heritability of the trait increased by using data from one examination. Power to detect linkage can be improved by identifying the most heritable phenotype, ensuring normality of the trait distribution and maximizing the information utilized through novel longitudinal designs for genetic analysis.     

Selecting pedigrees for linkage analysis of a quantitative trait: the expected number of informative meioses.

With evidence of segregation at a major locus for a quantitative trait having been found, a logical next step is to select a subset of the pedigrees to include in a linkage study to map the major locus. Ideally this subset should include much of the linkage information in the sample but include only a fraction of the pedigrees. We previously described a strategy for selecting pedigrees for linkage analysis of a quantitative trait on the basis of a pedigree likelihood-ratio statistic. For quantitative traits controlled by a major locus with a rare dominant allele, the likelihood-ratio strategy extracted nearly all the information for linkage while typically requiring marker data on only about one-third of the pedigrees. Here, we describe a new strategy to select pedigrees for linkage analysis on the basis of the expected number of potentially informative meioses in each pedigree. We demonstrate that this informative-meioses strategy provides an efficient and more general means to select pedigrees for a linkage study of a quantitative trait.     

Estimating the power of a proposed linkage study for a complex genetic trait.

Many genetic traits have complex modes of inheritance; they may exhibit incomplete or age-dependent penetrance or fail to show any clear Mendelian inheritance pattern. As primary linkage maps for the human genome near completion, it is becoming increasingly possible to map these traits. Prior to undertaking a linkage study, it is important to consider whether the pedigrees available for the proposed study are likely to provide sufficient information to demonstrate linkage, assuming a linked marker is tested. In the current paper, we describe a computer simulation method to estimate the power of a proposed study to detect linkage for a complex genetic trait, given a hypothesized genetic model for the trait. Our method simulates trait locus genotypes consistent with observed trait phenotypes, in such a way that the probability to detect linkage can be estimated by sample statistics of the maximum lod score distribution. The method uses terms available when calculating the likelihood of the trait phenotypes for the pedigree and is applicable to any trait determined by one or a few genetic loci; individual-specific environmental effects can also be dealt with. Our method provides an objective answer to the question, Will these pedigrees provide sufficient information to map this complex genetic trait?     

Linkage studies on Gilles de la Tourette syndrome: what is the strategy of choice?

For a linkage study it is important to ascertain family material that is sufficiently informative. The statistical power of a linkage sample can be determined via computer simulation. For complex traits uncertain parameters such as incomplete penetrance, frequency of phenocopies, gene frequency and variable expression have to be taken into account. One can either include only the most severe phenotype in the analysis or apply multiple linkage tests for a gradually broadened disease phenotype. Gilles de la Tourette syndrome (GTS) is a chronic neurological disorder characterized by multiple, intermittent motor and vocal tics. Segregation analyses suggest that GTS and milder phenotypes are caused by a single dominant gene. We report here the results of an extensive simulation study on a large set of families. We compared the effectiveness of linkage tests with only the GTS phenotype versus multiple tests that included various milder phenotypes and different gene frequencies. The scenario of multiple tests yielded superior power. Our results show that computer simulation can indicate the strategy of choice in linkage studies of multiple, complex phenotypes.     

Smoothing of the bivariate LOD score for non-normal quantitative traits

Variance component analysis provides an efficient method for performing linkage analysis for quantitative traits. However, type I error of variance components-based likelihood ratio testing may be affected when phenotypic data are non-normally distributed (especially with high values of kurtosis). This results in inflated LOD scores when the normality assumption does not hold. Even though different solutions have been proposed to deal with this problem with univariate phenotypes, little work has been done in the multivariate case. We present an empirical approach to adjust the inflated LOD scores obtained from a bivariate phenotype that violates the assumption of normality. Using the Collaborative Study on the Genetics of Alcoholism data available for the Genetic Analysis Workshop 14, we show how bivariate linkage analysis with leptokurtotic traits gives an inflated type I error. We perform a novel correction that achieves acceptable levels of type I error.