Score test for detecting linkage to quantitative traits

The two most popular methods to detect linkage of a quantitative trait to a marker are the Haseman-Elston regression method and the variance components likelihood-ratio test. In the literature, these methods are frequently compared and the relative advantages and disadvantages of each method are well known. In this article, we derive a score test for the variance component attributable to a specific quantitative trait locus and show that for sib-pairs it is mathematically equivalent to a recently proposed version of the Haseman-Elston method that optimally combines the sum squared and the difference squared of the centered phenotype values of the sibs. Because score tests and likelihood-ratio tetsts are equivalent for large sample sizes, the variance components likelihood-ratio test is also asymptotically equivalent to this optimal Haseman-Elston test. This fact gives a theoretical explanation of the empirical observation from simulation studies reporting similar power of the variance components likelihood-ratio test and the optimal Haseman-Elston method. Perhaps more importantly for practical purposes, the score test can also be extended in a natural way to support the simultaneous analysis of more than two subjects and multivariate phenotypes.     

Comparison of methods and sampling designs to test for association between rare variants and quantitative traits

Genome‐wide association studies succeeded in finding genetic variants associated with various phenotypes, but a large portion of the predicted genetic contribution to many traits remains unknown. One plausible explanation is that some missing variation is due to rare variants. Latest sequencing technology facilitates the investigation of such rare variants, but their statistical analysis remains challenging. For quantitative traits, a commonly used approach is to contrast the frequency of putatively functional rare variants between subjects in the two tails of the trait distribution. The contrast is usually performed by Fisher's exact or similar test. These tests are conservative as they discard trait rank information and are most useful under the unrealistic homogeneity assumption (i.e., variants have similar effects). We propose, and investigate via simulations, various designs for resequencing studies and statistical methods that incorporate information about rank, predicted function and allow for heterogeneity of effects. We propose designs which accommodate heterogeneity by sequencing both tails and the middle of the trait and novel statistical tests for trend, for heterogeneity and for a combination of the two. The conclusions of the simulations are four fold: (1) sequencing both tails and the middle of the trait distributions is desirable when heterogeneity is suspected, (2) trend and heterogeneity statistics should be used alongside other methods, (3) using rank information improves power over Fisher's exact test when the number of rare variants is not very large and (4) due to high misclassification rates, incorporating current predictions of a variant's function does not improve power. Genet. Epidemiol. 35: 226‐235, 2011. © 2011 Wiley‐Liss, Inc.     

A more powerful robust sib-pair test of linkage for quantitative traits

A more powerful robust test for linkage is developed from the methodology of Haseman and Elston [Behav Genet 2(1):3-19, 1972]. This new robust test uses weighted least-squares (WLS) methods to detect linkage between a quantitative trait and a polymorphic marker. For comparison, the characteristics of a test for linakge that uses known trait genotypes for the parents are also studied. Sample sizes needed to detect linkage, calculated using asymptotic results, are compared for 1) the usual Haseman-Elston method, 2) the WLS method, and 3) the method that uses parental trait genotype data. The WLS method needs at most twice the number of sib pairs as does the method that uses information on the trait genotypes of the parents. The small sample properties of the Haseman-Elston (H-E) and WLS tests are investigated by simulation. The power calculations for the H-E method are found to be accurate. The power of the WLS method is overestimated when fewer than 300 sib pairs are studied, but the WLS method is nonetheless more powerful than the usual H-E method. In samples of fewer than 300 sib pairs, the WLS test tends to be anticonservative. Treating all sib pairs from sibships of size 3 or 5 as independent does not increase the significance of the tests.     

Power and robustness of linkage tests for quantitative traits in general pedigrees

There are numerous statistical methods for quantitative trait linkage analysis in human studies. An ideal such method would have high power to detect genetic loci contributing to the trait, would be robust to non-normality in the phenotype distribution, would be appropriate for general pedigrees, would allow the incorporation of environmental covariates, and would be appropriate in the presence of selective sampling. We recently described a general framework for quantitative trait linkage analysis, based on generalized estimating equations, for which many current methods are special cases. This procedure is appropriate for general pedigrees and easily accommodates environmental covariates. In this report, we use computer simulations to investigate the power and robustness of a variety of linkage test statistics built upon our general framework. We also propose two novel test statistics that take account of higher moments of the phenotype distribution, in order to accommodate non-normality. These new linkage tests are shown to have high power and to be robust to non-normality. While we have not yet examined the performance of our procedures in the context of selective sampling via computer simulations, the proposed tests satisfy all of the other qualities of an ideal quantitative trait linkage analysis method.     

Association and linkage with quantitative traits

This paper examines two approaches for the analysis of quantitative traits: (1) association studies and (2) linkage studies. The trait studied was Q1 from simulated Problem 2 data set in Genetic Analysis Workshop 9. Our purpose was to evaluate associations present in the data, to identify nongenetic and genetic predictors of the trait, and to explore the simulated genome for linkage. Through the association study, we found evidence for the primary major gene associated with this trait. The linkage study found evidence of residual genetic effect acting through other traits. Adjustments of Q1 for Q2 and Q3 led to a failure to find significant effects of MG2 and MG3. This supports the suggestion that adjustment for genetically influenced traits for effects of other genetic traits can reduce the power to detect major gene effects. In summary, we detected the major gene directly associated with the trait of interest through association studies. Linkage analysis detected evidence for two other genes associated to a lesser degree with the trait. © 1995 Wiley-Liss, Inc.     

Robust transmission regression models for linkage and association

We present a general regression model that accounts for both linkage and linkage disequilibrium (LD) when analyzing nuclear family data. The method does not require LD to exist in order to evaluate linkage, but if LD does exist, the power to detect linkage can increase due to improved information on linkage phase. The proposed method is general, allowing for a variety of traits (e.g., binary affection status, categorical and quantitative phenotypes), affecteds only analyses, and covariates. Covariates can be useful to assess heterogeneity of linkage and LD, as well as gene-environment interactions. Other advantages of our methods are that: LD parameters are not defined without linkage, so that population stratification cannot bias the analyses; a combined test for linkage and LD can be used to test for linkage; given the existence of linkage, an adjusted LD test useful for fine-mapping can be constructed; covariate effects can be flexibly modeled; and families containing a single child and families containing multiple offspring can be combined for a single analysis (capitalizing on the LD information provided by single-child families and the combined linkage and LD information provided by multiple offspring). The basic features of the regression model are presented, as well as discussions of potential applications and critical statistical issues.     

Power and robustness of a score test for linkage analysis of quantitative traits using identity by descent data on sib pairs

Identification of genes involved in complex traits by traditional (lod score) linkage analysis is difficult due to many complicating factors. An unfortunate drawback of non-parametric procedures in general, though, is their low power to detect genetic effects. Recently, Dudoit and Speed [2000] proposed using a (likelihood-based) score test for detecting linkage with IBD data on sib pairs. This method uses the likelihood for theta, the recombination fraction between a trait locus and a marker locus, conditional on the phenotypes of the two sibs to test the null hypothesis of no linkage (theta = (1/2)). Although a genetic model must be specified, the approach offers several advantages. This paper presents results of simulation studies characterizing the power and robustness properties of this score test for linkage, and compares the power of the test to the Haseman-Elston and modified Haseman-Elston tests. The score test is seen to have impressively high power across a broad range of true and assumed models, particularly under multiple ascertainment. Assuming an additive model with a moderate allele frequency, in the range of p = 0.2 to 0.5, along with heritability H = 0.3 and a moderate residual correlation rho = 0.2 resulted in a very good overall performance across a wide range of trait-generating models. Generally, our results indicate that this score test for linkage offers a high degree of protection against wrong assumptions due to its strong robustness when used with the recommended additive model.     

New joint covariance- and marginal-based tests for association and linkage for quantitative traits for random and non-random sampling

We develop novel statistical tests for transmission disequilibrium testing (tests of linkage in the presence of association) for quantitative traits using parents and offspring. These joint tests utilize information in both the covariance (or more generally, dependency) between genotype and phenotype and the marginal distribution of genotype. Using computer simulation we test the validity (Type I error rate control) and power of the proposed methods, for additive, dominant, and recessive modes of inheritance, locus-specific heritability of the trait 0.05, 0.1, 0.2 with allele frequencies of P=0.2 and 0.4, and sample sizes of 500, 200, and 100 trios. Both random sampling and extreme sampling schemes were investigated. A multinomial logistic joint test provides the highest overall power irrespective of sample size, allele frequency, heritability, and modes of inheritance.     

Diagnostic tools in linkage analysis for quantitative traits

Diagnostic methods are key components in any good statistical analysis. Because of the similarities between the variance components approach and regression analysis with respect to the normality assumption, when performing quantitative genetic linkage analysis using variance component methods, one must check the normality assumption of the quantitative trait and outliers. Thus, the main purposes of this paper are to describe methods for testing the normality assumption, to describe various diagnostic methods for identifying outliers, and to discuss the issues that may arise when outliers are present when using variance components models in quantitative trait linkage analysis. Data from the Rochester Family Heart Study are used to illustrate the various diagnostic methods and related issues.     

Power of a score test for quantitative trait linkage analysis of relative pairs

The score test of Dudoit and Speed [(2000) Biostatistics 1:1-26] to detect linkage between a trait locus and a marker locus, using identity by descent data on sib pairs, is extended to other types of relative pairs (grandparent/grandchild, avuncular, and half-sib relationships). The test is based on the likelihood of the recombination fraction theta between trait and marker loci, conditional on phenotypes of the relatives. We present results of simulation studies characterizing power and robustness properties of this linkage score test, and compare the power of the score test to that of the classical and modified Haseman-Elston tests. The score test has considerable power, particularly under sampling schemes where selection is on double probands. Use of a generic additive model [Goldstein et al., submitted] with allele frequency p = 0.2, heritability H = 0.3, and a moderate residual correlation of rho = 0.2 resulted in a very good overall performance across a wide range of trait-generating models.     

Variance‐components methods for linkage and association analysis of ordinal traits in general pedigrees

Many complex human diseases such as alcoholism and cancer are rated on ordinal scales. Well‐developed statistical methods for the genetic mapping of quantitative traits may not be appropriate for ordinal traits. We propose a class of variance‐component models for the joint linkage and association analysis of ordinal traits. The proposed models accommodate arbitrary pedigrees and allow covariates and gene‐environment interactions. We develop efficient likelihood‐based inference procedures under the proposed models. The maximum likelihood estimators are approximately unbiased, normally distributed, and statistically efficient. Extensive simulation studies demonstrate that the proposed methods perform well in practical situations. An application to data from the Collaborative Study on the Genetics of Alcoholism is provided. Genet. Epidemiol. 34: 232–237, 2010. © 2009 Wiley‐Liss, Inc.     

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     

Score test for detecting linkage to complex traits in selected samples

We present a unified approach to selection and linkage analysis of selected samples, for both quantitative and dichotomous complex traits. It is based on the score test for the variance attributable to the trait locus and applies to general pedigrees. The method is equivalent to regressing excess IBD sharing on a function of the traits. It is shown that when population parameters for the trait are known, such inversion does not entail any loss of information. For dichotomous traits, pairs of pedigree members of different phenotypic nature (e.g., affected sib pairs and discordant sib pairs) can easily be combined as well as populations with different trait prevalences.     

Multipoint linkage analysis of quantitative traits on sex-chromosomes

Variance component models form a powerful and flexible tool for multipoint linkage analysis of quantitative traits. Estimates of genetic similarity are needed for the variance component model to detect linkage and to locate genes, and two methods are commonly used to calculate multipoint identity-by-descent (IBD) estimates for autosomes. Fulker et al. ([1995] Am. J. Hum. Genet. 56: 1229-1233) and Almasy and Blangero ([1998] Am. J. Hum. Genet. 62: 119-121) used multiple regression to estimate the IBD sharing along a chromosome, while the approach of Kruglyak and Lander ([1995] Am. J. Hum. Genet. 57: 439-454) is based on a hidden Markov model. In this paper, we modify the variance component model to accommodate sex-chromosomes, and we extend both multipoint IBD estimation methods to accommodate sex-linked loci. Simulation studies demonstrate the power and precision of the variance component model to detect QTLs located on the sex-chromosome. The two multipoint IBD estimation methods have the same accuracy to identify QTL position, but the hidden Markov model yields a larger average maximum LOD score to detect linkage than the regression model. The extension of the multipoint IBD estimation methods and the variance component model to the X chromosome shows that the variance component model is a powerful and flexible tool for linkage analysis of quantitative traits on both autosomes and sex-chromosomes.     

On the utility of gene set methods in genomewide association studies of quantitative traits

In genomewide genetic association studies, prior biological knowledge may help distinguish variation that is truly associated with a quantitative trait from the vast majority of unassociated variation that may be significant in hypothesis testing due to chance. However, formal methods for integrating prior biological knowledge into association studies have only been proposed recently, and their potential utility has not been thoroughly evaluated. Herein, gene set methods from genomewide analysis of gene expression data are adapted for application to genomewide genetic analysis of quantitative traits. The proposed gene set method was tested in simulations with gene sets that included up to 500 total variants, among which up to 20 collectively explained 5% of the variance. In a population of 1,000 individuals, the gene set method was largely more efficient at detecting truly associated variants in these gene sets than a comparably calibrated conventional approach relying on P-values alone. While extremely strong associations remain best identified by conventional methods, the gene set approach may provide a complementary mode of analysis for revealing the full spectrum of genes that influence a quantitative trait.     

Sequential sib-pair and association studies to detect genes in quantitative traits

We applied sib-pair and association methods to a GAW data set of nuclear families with quantitative traits. Our approaches included 1) preliminary statistical studies including correlations and linear regressions, 2) sib-pair methods, and 3) association studies. We used a single data set to screen for linkage and association and, subsequently, additional data sets to confirm the preliminary results. Using this sequential approach, sib-pair analysis provided evidence for the genes influencing Q1, Q2, and Q4. We correctly predicted MG1 for Q1, MG2 for Q2, and MG4 for Q4. We did not find any false positives using this approach. Association studies identified chromosomes 8 and 9 to be associated with Q4; however these are assumed to be false positives as no associations were modeled into the data. © 1997 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).     

Search for faster methods of fitting the regressive models to quantitative traits

The regressive models describe familial patterns of dependence of quantitative measures by specifying regression relationships among a person's phenotype and genotype and the phenotypes and genotypes of antecedents. When the number of sibs in the pattern of dependence increases, as in the class D regressive model, computation of the likelihood becomes time consuming, since the Elston-Stewart algorithm cannot be used generally. On the other hand, the simpler class A regressive model, which imposes a restriction on the sib-sib correlation, may lead to inference of a spurious major gene, as already observed in some instances. A simulation study is performed to explore the robustness of class A model with respect to false inference of a major gene and to search for faster methods of computing the likelihood under class D model. The class A model is not robust against the presence of a sib-sib correlation exceeding that specified by the model, unless tests on transmission probabilities are performed carefully: false detection of a major gene is reduced from a number of 26-30 to between 0 and 4 data sets out of 30 replicates after testing both the Mendelian transmission and the absence of transmission of a major effect against the general transmission model. Among various approximations of the likelihood formulation of the class D model, approximations 6 and 8 are found to work appropriately in terms of both the estimation of all parameters and hypothesis testing, for each generating model. These approximations lessen the computer time by allowing use of the Elston-Stewart algorithm.     

Removing phenotypic distribution assumptions from tests of linkage disequilibrium for quantitative traits

The transmission disequilibrium test (TDT) has become a family-based method of reference to search for linkage disequilibrium (LD). Although it was first developed for dichotomous traits, numerous approaches have extended the TDT to quantitative phenotypes that either rely on regression or variance component techniques. Both of these approaches are based on some phenotypic distribution assumptions, the violation of which can lead to inflation of type I error rates, and derive information from phenotypic variability, so that their power is very low under some selection schemes (e.g., one-tailed selection). We propose a new family-based test of association for quantitative traits, denoted maximum-likelihood-binomial (MLB)-QTDT, which addresses the two previous issues by incorporating a latent binary variable that captures the LD information between the marker allele and the quantitative phenotype. The method can be understood as a classical TDT for binary traits that would include pure affected and pure unaffected children, and the probability for a child to be affected or unaffected depends on his/her quantitative phenotypic value. Simulation studies under the null hypothesis show that the MLB-QTDT provides very consistent type I errors even in small and/or selected samples. Under the alternative hypothesis, the MLB-QTDT has good power to analyze one-tailed selected samples, and performs as well as a classical approach in other designs. The MLB-QTDT is a flexible distribution-free method to test for LD with quantitative phenotypes in nuclear families, and can easily incorporate previous extensions developed in the context of family-based association studies with binary traits.