Association studies of QTL for multi-allele markers by mixed models

In this paper, we extend association study methods of both Fan et al. [Hum Hered 2002;53:130-145], in which a quantitative trait locus (QTL) and a multi-allele marker are considered for trio families, and Fan and Xiong [Biostatistics 2003, in press], in which a QTL and a bi-allelic marker are considered for nuclear families. The objective is to build mixed models for association study between a QTL and a multi-allelic marker for nuclear families with any number of offspring. Two types of nuclear family data are considered: the first is genetic data of offspring from at least one heterozygous parents, and the second is genetic data of offspring of nuclear family. (1) For the data of offspring from at least one heterozygous parents, we assume that at least one parent is heterozygous at the marker locus, and we may infer clearly the transmission of parental marker alleles to the offspring. We show that it can be used in association study in the presence of linkage. The theoretical basis is the difference between the conditional mean of trait value given an allele is transmitted and the conditional mean of trait value given the allele is not transmitted from a heterozygous parent. To build valid models, we calculate the variance covariance structure of trait values of offspring. Besides, the reduction of the number of parameters is discussed under an assumption of tight linkage between the trait locus and the marker. (2) For the data of offspring of nuclear family, we show that it can be used in general association study. In this case, the theoretical basis is the difference between the conditional mean of trait values given an allele is transmitted from a parent and the population mean. Then, we calculate variance-covariance structure of trait values of offspring. (3) Based on the theoretical analysis, mixed models are built for each type of the data, and related test statistics are proposed for association study. By power calculation and comparison, we show that, in some instances, the proposed test statistics have higher power than that by collapsing alleles to be new ones. The proposed models are used to analyze chromosomes 4 and chromosome 16 data of the Oxford asthma data, Genetic Analysis Workshop 12.     

Linkage and association studies of QTL for nuclear families by mixed models

The transmission disequilibrium test (TDT) has been utilized to test the linkage and association between a genetic trait locus and a marker. Spielman et al. (1993) introduced TDT to test linkage between a qualitative trait and a marker in the presence of association. In the presence of linkage, TDT can be applied to test for association for fine mapping (Martin et al., 1997; Spielman and Ewens, 1996). In recent years, extensive research has been carried out on the TDT between a quantitative trait and a marker locus (Allison, 1997; Fan et al., 2002; George et al., 1999; Rabinowitz, 1997; Xiong et al., 1998; Zhu and Elston, 2000, 2001). The original TDT for both qualitative and quantitative traits requires unrelated offspring of heterozygous parents for analysis, and much research has been carried out to extend it to fit for different settings. For nuclear families with multiple offspring, one approach is to treat each child independently for analysis. Obviously, this may not be a valid method since offspring of one family are related to each other. Another approach is to select one offspring randomly from each family for analysis. However, with this method much information may be lost. Martin et al. (1997, 2000) constructed useful statistical tests to analyse the data for qualitative traits. In this paper, we propose to use mixed models to analyse sample data of nuclear families with multiple offspring for quantitative traits according to the models in Amos (1994). The method uses data of all offspring by taking into account their trait mean and variance-covariance structures, which contain all the effects of major gene locus, polygenic loci and environment. A test statistic based on mixed models is shown to be more powerful than the test statistic proposed by George et al. (1999) under moderate disequilibrium for nuclear families. Moreover, it has higher power than the TDT statistic which is constructed by randomly choosing a single offspring from each nuclear family.     

Power studies for the transmission/disequilibrium tests with multiple alleles.

Case-control studies compare marker-allele distributions in affected and unaffected individuals, and significant results suggest linkage but may simply reflect population structure. For markers with m alleles (m > or = 2), a McNemar-like statistic, I, estimates the level of population association between marker and disease loci. To test for linkage after significant case-control tests, within-family tests are performed. These operate on the contingency table, with i, jth element equal to the number of parents that transmit marker allele Mi and do not transmit marker allele Mi to an affected offspring. The dimension of the table is the number of alleles at the marker locus. Three test statistics have recently been proposed in the literature: Tc compares symmetric pairs of cells (i, j) and (j, i), Tm compares row and column totals for the same marker allele, and a likelihood ratio statistic Tl uses all the cells in the table. In addition, we consider a new statistic, Tmhet, that uses only the heterozygous parents and is approximately chi2 with (m - 1) df. We use a Monte Carlo test to guarantee valid tests and to demonstrate the inferiority of Tc and the equality of Tm and Tl in terms of power. The power of the Tmhet test is close but not always equal to the power of the Tm test. We also show that under the alternative hypothesis of linkage, Tm is approximately noncentral chi2 with (m - 1) df and noncentrality parameter 2NT(1 - 2theta)2I*, when data on single affecteds in NT families are used. If the disease has a low population frequency, then I* is estimated using the case-control statistic I. This offers a basis for choosing sample size, or choosing a marker system.     

A test of transmission/disequilibrium for quantitative traits in pedigree data, by multiple regression.

The transmission/disequilibrium (TD) test (TDT), proposed, by Spielman et al., for binary traits is a powerful method for detection of linkage between a marker locus and a disease locus, in the presence of allelic association. As a test for linkage disequilibrium, the TDT makes the assumption that any allelic association present is due to linkage. Allison proposed a series of TD-type tests for quantitative traits and calculated their power, assuming that the marker locus is the disease locus. All these tests assume that the observations are independent, and therefore they are applicable, as a test for linkage, only for nuclear-family data. In this report, we propose a regression-based TD-type test for linkage between a marker locus and a quantitative trait locus, using information on the parent-to-offspring transmission status of the associated allele at the marker locus. This method does not require independence of observations, thus allowing for analysis of pedigree data as well, and allows adjustment for covariates. We investigate the statistical power and validity of the test by simulating markers at various recombination fractions from the disease locus.     

Power calculations for the transmission/disequilibrium and affected sib pair tests using elementary probability methods

The transmission/disequilibrium test (TDT) and the affected sib pair test (ASP) both test for the association of a marker allele with some conditions. Here, we present methods for calculating the probability of detecting the association (power) for a study examining a fixed number of families for suitability for the study and for calculating the number of such families to be examined. Both calculations use a genetic model for the association. The model considered posits a bi-allelic marker locus that is linked to a bi-allelic disease locus with a possibly nonzero recombination fraction between the loci. The penetrance of the disease is an increasing function of the number of disease alleles. The TDT tests whether the transmission by a heterozygous parent of a particular allele at a marker locus to an affected offspring occurs with probability greater than 0.5. The ASP tests whether transmission of the same allele to two affected sibs occurs with probability greater than 0.5. In either case, evidence that the probability is greater than 0.5 is evidence for association between the marker and the disease. Study inclusion criteria (IC) can greatly affect the necessary sample size of a TDT or ASP study. IC considered by us include a randomly selected parent at least one parent or both parents required to be heterozygous. It also allows a specified minimum number of affected offspring to be required (TDT only). We use elementary probability calculations rather than complex mathematical manipulations or asymptotic methods (large sample size approximations) to compute power and requisite sample size for a proposed study. The advantages of these methods are simplicity and generality.     

Log-linear models for gene mapping with affected sib pair data

In genome-wide screens of genetic marker loci, non-mendelian inheritance of a marker is taken to indicate its vicinity to a disease locus. Heritable complex traits are thought to be under the influence of multiple possibly interacting susceptibility loci yet the most frequently used methods of linkage and association analysis focus on one susceptibility locus at a time. Here we introduce log-linear models for the joint analysis of multiple marker loci and interaction effects between them. Our approach focuses on affected sib pair data and identical by descent (IBD) allele sharing values observed on them. For each heterozygous parent, the IBD values at linked markers represent a sequence of dependent binary variables. We develop log-linear models for the joint distribution of these IBD values. An independence log-linear model is proposed to model the marginal means and the neighboring interaction model is advocated to account for associations between adjacent markers. Under the assumption of conditional independence, likelihood methods are applied to simulated data containing one or two susceptibility loci. It is shown that the neighboring interaction log-linear model is more efficient than the independence model, and incorporating interaction in the two-locus analysis provides increased power and accuracy for mapping of the trait loci.     

Transmission-disequilibrium tests for quantitative traits.

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

Joint modeling of linkage and association: identifying SNPs responsible for a linkage signal

Once genetic linkage has been identified for a complex disease, the next step is often association analysis, in which single-nucleotide polymorphisms (SNPs) within the linkage region are genotyped and tested for association with the disease. If a SNP shows evidence of association, it is useful to know whether the linkage result can be explained, in part or in full, by the candidate SNP. We propose a novel approach that quantifies the degree of linkage disequilibrium (LD) between the candidate SNP and the putative disease locus through joint modeling of linkage and association. We describe a simple likelihood of the marker data conditional on the trait data for a sample of affected sib pairs, with disease penetrances and disease-SNP haplotype frequencies as parameters. We estimate model parameters by maximum likelihood and propose two likelihood-ratio tests to characterize the relationship of the candidate SNP and the disease locus. The first test assesses whether the candidate SNP and the disease locus are in linkage equilibrium so that the SNP plays no causal role in the linkage signal. The second test assesses whether the candidate SNP and the disease locus are in complete LD so that the SNP or a marker in complete LD with it may account fully for the linkage signal. Our method also yields a genetic model that includes parameter estimates for disease-SNP haplotype frequencies and the degree of disease-SNP LD. Our method provides a new tool for detecting linkage and association and can be extended to study designs that include unaffected family members.     

Associative overdominance: Evaluating the effects of inbreeding and linkage disequilibrium

Expressions are obtained for the expected phenotypic values of homozygous and heterozygous genotypes for a neutral marker locus linked to a locus segregating for a recessive deleterious gene. The phenotypic values are functions of the allele frequencies at the marker locus, the inbreeding coefficient and the degree of association of the deleterious gene with the marker alleles. The analysis is extended to more than two alleles at the marker locus. Either linkage disequilibrium or inbreeding alone can produce an apparent superiority of heterozygotes for the marker locus (unless specified otherwise, the terms ‘homozygote’ and ‘heterozygote’ will refer to the marker locus). The effect of linkage disequilibrium on the difference between the heterozygote and homozygote values can be positive (associative overdominance) or negative (associative underdominance), depending on the frequencies of the marker alleles and the degree of their association with the deleterious gene. Inbreeding has always a positive effect. In general, the expected value of a homozygote is a positive function of its allele frequency. When the various homozygous genotypes are combined into one class and the various heterozygous genotypes into another, the phenotypic difference of the two classes is a function of the evenness of the allelic frequency distribution. Inbreeding is a more likely explanation of associative overdominance if the frequency of the deleterious gene is low, but its effect on the character high. Conversely, linkage disequilibrium is more likely if the frequency is high and the effect low. The degrees of association between marker alleles and the deleterious gene can, in principle, be estimated from the observed phenotypic scores and used to calculate expected multi-locus genotype scores. This could provide the basis for statistical tests of the associative overdominance hypothesis as an explanation of observed correlations between multi-locus heterozygosity and phenotypic traits.     

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.     

渐近混合群体的混合连锁不平衡及其遗传连锁推断

在渐近混合模型中,混合现象发生在每一世代,通过对其混合连锁不平衡的理论分析,发现混合连锁不平衡与两个子群体间的基因频率差成正比。基于这一点,构造了一个对重组率严格单调的函数（△ker=△／（p1-p2）,其中△代表连锁不平衡）,进而据此推断标记基因座与疾病基因座的遗传连锁。应用人类基因组上不连锁的标记基因提供的连锁不平衡信息,基于病人组数据构造了一个准似然比统计量。模拟结果显示,此检验可用于精确的基因定位。文章亦讨论了参数对检验的影响。     

Characterisation of the virescent locus controlling a recessive phenotype in apple rootstocks (Malus pumila Mill.)

Certain progenies of Malling apple rootstocks (Malus pumila) have been reported to segregate for a virescent trait: leaves are chlorotic at germination or bud break but turn green as the season progresses. The M432 rootstock mapping progeny, from which a linkage map has recently been elaborated with 323 simple sequence repeat (SSR) markers and 3,069 single nucleotide polymorphism (SNP) markers, also segregates for this phenotype. In this investigation, 188 seedlings were scored and, on the basis of a 3:1 segregation, virescence was attributed to the recessive gene (vir) for which the two parents, M.27 and M.116, are heterozygous. At least seven of 28 Malling rootstocks are heterozygous for this apparently deleterious trait. With the published marker data the gene was mapped to linkage group 12, tightly flanked by the SSR CH01g12 and the SNP marker 475880474, and was located in a physical interval of 2.36 Mb on the Golden Delicious genome sequence. A PCR-based marker was developed from the SNP and along with the SSR was scored in a set of Malus rootstock accessions. The screening of this collection demonstrated that those accessions known to be heterozygous at the vir locus all carried the 152 allele of the SSR and the G allele of the SNP, whilst a virescent accession was homozygous for the alleles. The results we present here could help predict the genotype of apple rootstocks at the vir locus, assist in the fine mapping of the vir locus to identify potential candidate genes for the trait and also aid rootstock breeding.     

An entropy-based measure for QTL mapping using extreme samples of population

Quantitative trait locus (QTL) mapping can be accomplished through the method of selective genotyping, which is based on the differences of frequencies between an upper sample and a lower sample in population. However, amplifying the differences in marker allele frequencies in extreme samples may increase the probability for QTL mapping. Shannon entropy, which is a nonlinear function of allele frequencies, can be used to amplify the differences in marker allele frequencies. In this paper, we present a novel measure for linkage disequilibrium (LD) between a marker and single QTL, that is based on the comparison of the entropy and conditional entropy in a marker in extreme samples of population. This measure of LD between the marker and the trait locus can be used when the marker allele frequencies are known in the extreme samples of a population. We investigate the mapping performance in both analytic and simulation scenarios of a single QTL linked to a single marker. Our results show that the measure has very reasonable performance. In addition, a simulation study is performed on the basis of the haplotype frequencies of 10 SNPs of angiotensin-I converting enzyme (ACE) genes.     

How do homozygous parents affect TDT as a test for association?

The traditional transmission disequilibrium test (TDT) (Spielman et al., 1993) is a powerful test for association only in the presence of linkage. Since allele transmissions from homozygous parents do not carry any information on linkage, the TDT statistic uses data only on heterozygous parents. However, homozygous parents carry information on association between alleles at a marker locus and a disease locus. In this article, we explore whether inclusion of homozygous parents increases the power to detect association. The resultant test statistic follows a chi(2) distribution with 2 degrees of freedom. Monte-Carlo simulations are included to compare the performance of this test with the traditional TDT under different disease models.     

Linkage between amylase-1 locus and a major gene for milk fat content in cattle

Summary
Linkage between the amylase-1 ( Am-1 ) locus and a quantitative trait locus influencing fat content in milk was studied in offspring from heterozygous sires of the Swedish Red and White dairy breed. The effect on bull breeding values for fat content was estimated as interactions between sire and paternal Am-1 allele using a model eliminating the direct effects of sire and Am-1 allele. There were strong indications of linkage, confirming results of previous studies. The interaction was caused by strong associations in 7 out of 14 sire families. A test for within-family variance heterogeneity performed on the whole population of breeding bulls also supported the presence of a major gene for fat content in milk. The results indicate that there is genetic linkage between the Am-1 locus and a locus with large effect on milk fat content.     

Accounting for linkage in family-based tests of association with missing parental genotypes

In studies of complex diseases, a common paradigm is to conduct association analysis at markers in regions identified by linkage analysis, to attempt to narrow the region of interest. Family-based tests for association based on parental transmissions to affected offspring are often used in fine-mapping studies. However, for diseases with late onset, parental genotypes are often missing. Without parental genotypes, family-based tests either compare allele frequencies in affected individuals with those in their unaffected siblings or use siblings to infer missing parental genotypes. An example of the latter approach is the score test implemented in the computer program TRANSMIT. The inference of missing parental genotypes in TRANSMIT assumes that transmissions from parents to affected siblings are independent, which is appropriate when there is no linkage. However, using computer simulations, we show that, when the marker and disease locus are linked and the data set consists of families with multiple affected siblings, this assumption leads to a bias in the score statistic under the null hypothesis of no association between the marker and disease alleles. This bias leads to an inflated type I error rate for the score test in regions of linkage. We present a novel test for association in the presence of linkage (APL) that correctly infers missing parental genotypes in regions of linkage by estimating identity-by-descent parameters, to adjust for correlation between parental transmissions to affected siblings. In simulated data, we demonstrate the validity of the APL test under the null hypothesis of no association and show that the test can be more powerful than the pedigree disequilibrium test and family-based association test. As an example, we compare the performance of the tests in a candidate-gene study in families with Parkinson disease.     

Association studies under general disease models

There is great expectation that the levels of association found between genetic markers and disease status will play a role in the location of disease genes. This expectation follows from regarding association as being proportional to linkage disequilibrium and therefore inversely related to recombination value. For disease genes with more than two alleles, the association measure is instead a weighted average of linkage disequilibria, with the weights depending on allele frequencies and genotype susceptibilities at the disease loci. There is no longer a simple relationship, even in expectation, with recombination. We adopt a general framework to examine association mapping methods which helps to clarify the nature of case-control and transmission/disequilibrium-type tests and reveals the relationship between measures of association and coefficients of linkage disequilibrium. In particular, we can show the consequences of additive and nonadditive effects at the trait locus on the behavior of these tests. These concepts have a natural extension to marker haplotypes. The association of two-locus marker haplotypes with disease phenotype depends on a weighted average of three-locus disequilibria (two markers with each disease locus). It is likely that these two-marker analyses will provide additional information in association mapping studies.     

Absence of evidence for linkage between Booroola gene and genetic markers at 11 sheep blood polymorphic loci

Summary. Genetic linkage between the Booroola locus ( Fec ) and 11 sheep blood polymorphic loci (i.e. Tf, Hb, CA, OLA, and A, B, C, D, M, R, F41 red cell blood groups) was investigated in six large sire families (163 informative female offspring). The six sires tested were heterozygous for the Booroola allele ( Fec^B ) and for several genetic markers. No evidence in favour of linkage was found. Moreover, depending on the marker locus considered, linkage closer than or as close as the recombination frequency of 10–30% was excluded.     

Transmission test for linkage disequilibrium: the insulin gene region and insulin-dependent diabetes mellitus (IDDM).

A population association has consistently been observed between insulin-dependent diabetes mellitus (IDDM) and the "class 1" alleles of the region of tandem-repeat DNA (5'' flanking polymorphism [5''FP]) adjacent to the insulin gene on chromosome 11p. This finding suggests that the insulin gene region contains a gene or genes contributing to IDDM susceptibility. However, several studies that have sought to show linkage with IDDM by testing for cosegregation in affected sib pairs have failed to find evidence for linkage. As means for identifying genes for complex diseases, both the association and the affected-sib-pairs approaches have limitations. It is well known that population association between a disease and a genetic marker can arise as an artifact of population structure, even in the absence of linkage. On the other hand, linkage studies with modest numbers of affected sib pairs may fail to detect linkage, especially if there is linkage heterogeneity. We consider an alternative method to test for linkage with a genetic marker when population association has been found. Using data from families with at least one affected child, we evaluate the transmission of the associated marker allele from a heterozygous parent to an affected offspring. This approach has been used by several investigators, but the statistical properties of the method as a test for linkage have not been investigated. In the present paper we describe the statistical basis for this "transmission test for linkage disequilibrium" (transmission/disequilibrium test [TDT]). We then show the relationship of this test to tests of cosegregation that are based on the proportion of haplotypes or genes identical by descent in affected sibs. The TDT provides strong evidence for linkage between the 5''FP and susceptibility to IDDM. The conclusions from this analysis apply in general to the study of disease associations, where genetic markers are usually closely linked to candidate genes. When a disease is found to be associated with such a marker, the TDT may detect linkage even when haplotype-sharing tests do not.     

Association tests in nuclear families

We present a conditional likelihood approach for testing linkage disequilibrium in nuclear families having multiple affected offspring. The likelihood, conditioned on the identity-by-descent (IBD) structure of the sibling genotypes, is unaffected by familial correlation in disease status that arises from linkage between a marker locus and the unobserved trait locus. Two such conditional likelihoods are compared: one that conditions on IBD and phase of the transmitted alleles and a second which conditions only on IBD of the transmitted alleles. Under the log-additive model, the first likelihood is equivalent to the allele-counting methods proposed in the literature. The second likelihood is valid under the added assumption of equal male and female recombination fractions. In a simulation study, we demonstrated that in sibships having two or three affected siblings the score test from each likelihood had the correct test size for testing disequilibrium. They also led to equivalent power to detect linkage disequilibrium at the 5% significance level.