Models and tests of linkage and association studies of quantitative trait locus for multi-allele marker Loci

In this paper, we explore models and tests for association and linkage studies of a quantitative trait locus (QTL) linked to a multi-allele marker locus. Based on the difference between an offspring's conditional trait means of receiving and not receiving an allele from a parent at marker locus, we propose three statistics T(m), T(m,row) and T(m,col) to test association or linkage disequilibrium between the marker locus and the QTL. These tests are composite tests, and use the offspring marginal sample means including offspring data of both homozygous and heterozygous parents. For the linkage study, we calculate the offspring's conditional trait mean given the allele transmission status of a heterozygous parent at the marker locus. Based on the difference between the conditional means of a transmitted and a nontransmitted allele from a heterozygous parent, we propose statistics T(parsi), T(satur), T(gen) and T(m,het) to perform composite tests of linkage between the marker locus and the quantitative trait locus in the presence of association. These tests only use the offspring data that are related to the heterozygous parents at the marker locus. T(parsi) is a parsimonious or allele-wise statistic, T(satur) and T(gen )are satured or genotype-wise statistics, and T(m,het) compares the row and column sample means for offspring data of heterozygous parents. After comparing the powers and the sample sizes, we conclude that T(parsi) has higher power than those of the bi-allele tests, T(satur), T(gen), and T(m,het). If there is tight linkage between the marker and the trait locus, T(parsi) is powerful in detecting linkage between the marker and the trait locus in the presence of association. By investigating the goodness-of-fit of T(parsi), we find that T(satur) does not gain much power compared to that of T(parsi). Moreover, T(parsi) takes into account the pattern of the data that is consistent with linkage and linkage disequilibrium. As the number of alleles at the marker locus increases, T(parsi) is very conservative, and can be useful even for sparse data. To illustrate the usefulness and the power of the methods proposed in this paper, we analyze the chromosome 6 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 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.     

Removing the sampling restrictions from family-based tests of association for a quantitative-trait locus

One strategy for localization of a quantitative-trait locus (QTL) is to test whether the distribution of a quantitative trait depends on the number of copies of a specific genetic-marker allele that an individual possesses. This approach tests for association between alleles at the marker and the QTL, and it assumes that association is a consequence of the marker being physically close to the QTL. However, problems can occur when data are not from a homogeneous population, since associations can arise irrespective of a genetic marker being in physical proximity to the QTL-that is, no information is gained regarding localization. Methods to address this problem have recently been proposed. These proposed methods use family data for indirect stratification of a population, thereby removing the effect of associations that are due to unknown population substructure. They are, however, restricted in terms of the number of children per family that can be used in the analysis. Here we introduce tests that can be used on family data with parent and child genotypes, with child genotypes only, or with a combination of these types of families, without size restrictions. Furthermore, equations that allow one to determine the sample size needed to achieve desired power are derived. By means of simulation, we demonstrate that the existing tests have an elevated false-positive rate when the size restrictions are not followed and that a good deal of information is lost as a result of adherence to the size restrictions. Finally, we introduce permutation procedures that are recommended for small samples but that can also be used for extensions of the tests to multiallelic markers and to the simultaneous use of more than one marker.     

Combined linkage and association mapping of quantitative trait loci by multiple markers

Using multiple diallelic markers, variance component models are proposed for high-resolution combined linkage and association mapping of quantitative trait loci (QTL) based on nuclear families. The objective is to build a model that may fully use marker information for fine association mapping of QTL in the presence of prior linkage. The measures of linkage disequilibrium and the genetic effects are incorporated in the mean coefficients and are decomposed into orthogonal additive and dominance effects. The linkage information is modeled in variance-covariance matrices. Hence, the proposed methods model both association and linkage in a unified model. On the basis of marker information, a multipoint interval mapping method is provided to estimate the proportion of allele sharing identical by descent (IBD) and the probability of sharing two alleles IBD at a putative QTL for a sib-pair. To test the association between the trait locus and the markers, both likelihood-ratio tests and F-tests can be constructed on the basis of the proposed models. In addition, analytical formulas of noncentrality parameter approximations of the F-test statistics are provided. Type I error rates of the proposed test statistics are calculated to show their robustness. After comparing with the association between-family and association within-family (AbAw) approach by Abecasis and Fulker et al., it is found that the method proposed in this article is more powerful and advantageous based on simulation study and power calculation. By power and sample size comparison, it is shown that models that use more markers may have higher power than models that use fewer markers. The multiple-marker analysis can be more advantageous and has higher power in fine mapping QTL. As an application, the Genetic Analysis Workshop 12 German asthma data are analyzed using the proposed methods.     

Fractioned DNA pooling: a new cost-effective strategy for fine mapping of quantitative trait loci

Selective DNA pooling (SDP) is a cost-effective means for an initial scan for linkage between marker and quantitative trait loci (QTL) in suitable populations. The method is based on scoring marker allele frequencies in DNA pools from the tails of the population trait distribution. Various analytical approaches have been proposed for QTL detection using data on multiple families with SDP analysis. This article presents a new experimental procedure, fractioned-pool design (FPD), aimed to increase the reliability of SDP mapping results, by "fractioning" the tails of the population distribution into independent subpools. FPD is a conceptual and structural modification of SDP that allows for the first time the use of permutation tests for QTL detection rather than relying on presumed asymptotic distributions of the test statistics. For situations of family and cross mapping design we propose a spectrum of new tools for QTL mapping in FPD that were previously possible only with individual genotyping. These include: joint analysis of multiple families and multiple markers across a chromosome, even when the marker loci are only partly shared among families; detection of families segregating (heterozygous) for the QTL; estimation of confidence intervals for the QTL position; and analysis of multiple-linked QTL. These new advantages are of special importance for pooling analysis with SNP chips. Combining SNP microarray analysis with DNA pooling can dramatically reduce the cost of screening large numbers of SNPs on large samples, making chip technology readily applicable for genomewide association mapping in humans and farm animals. This extension, however, will require additional, nontrivial, development of FPD analytical tools.     

Optimal use of regression models in genome-wide association studies

The performance of linear regression models in genome-wide association studies is influenced by how marker information is parameterized in the model. Considering the impact of parameterization is especially important when using information from multiple markers to test for association. Properties of the population, such as linkage disequilibrium (LD) and allele frequencies, will also affect the ability of a model to provide statistical support for an underlying quantitative trait locus (QTL). Thus, for a given location in the genome, the relationship between population properties and model parameterization is expected to influence the performance of the model in providing evidence for the position of a QTL. As LD and allele frequencies vary throughout the genome and between populations, understanding the relationship between these properties and model parameterization is of considerable importance in order to make optimal use of available genomic data. Here, we evaluate the performance of regression-based association models using genotype and haplotype information across the full spectrum of allele frequency and LD scenarios. Genetic marker data from 200 broiler chickens were used to simulate genomic conditions by selecting individual markers to act as surrogate QTL (sQTL) and then investigating the ability of surrounding markers to estimate sQTL genotypes and provide statistical support for their location. The LD and allele frequencies of markers and sQTL are shown to have a strong effect on the performance of models relative to one another. Our results provide an indication of the best choice of model parameterization given certain scenarios of marker and QTL LD and allele frequencies. We demonstrate a clear advantage of haplotype-based models, which account for phase uncertainty over other models tested, particularly for QTL with low minor allele frequencies. We show that the greatest advantage of haplotype models over single-marker models occurs when LD between markers and the causal locus is low. Under these situations, haplotype models have a greater accuracy of predicting the location of the QTL than other models tested.     

Linkage disequilibrium mapping of quantitative trait loci under truncation selection

As a dense map of single nucleotide polymorphism (SNP) markers are available, population-based linkage disequilibrium (LD) mapping or association study is becoming one of the major tools for identifying quantitative trait loci (QTL) and for fine gene mapping. However, in many cases, LD between the marker and trait locus is not very strong. Approaches that maximize the potential of detecting LD will be essential for the success of LD mapping of QTL. In this paper, we propose two strategies for increasing the probability of detecting LD: (1) phenotypic selection and (2) haplotype LD mapping. To provide the foundations for LD mapping of QTL under selection, we develop analytic tools for assessing the impact of phenotypic selection on allele and haplotype frequencies, and LD under three trait models: single trait locus, two unlinked trait loci, and two linked trait loci with or without epistasis. In addition to a traditional chi(2) test, which compares the difference in allele or haplotype frequencies in the selected sample and population sample, we present multiple regression methods for LD mapping of QTL, and investigate which methods are effective in employing phenotypic selection for QTL mapping. We also develop a statistical framework for investigating and comparing the power of the single marker and multilocus haplotype test for LD mapping of QTL. Finally, the proposed methods are applied to mapping QTL influencing variation in systolic blood pressure in an isolated Chinese population.     

Power of two and three-generation QTL mapping experiments in an outbred population containing full-sib or half-sib families

QTL mapping experiments involve many animals to be genotyped and performance tested. Consequently, experimental designs need to be optimized to minimize the costs of data collection and genotyping. The present study has analyzed the power and efficiency of experiments with two or three-generation family structures containing full-sib families, half-sib families, or both. The focus was on data from one outbred population because the main interest is to locate genes that can be used for within-line selection. For a two generation experiment more animals had to be typed for marker loci to obtain a certain power than for a three generation experiment. Fewer trait values, however, had to be obtained for a two-generation experiment than for a three-generation experiment. A two or three-generation family structure with full-sib offspring was more efficient than a two or three-generation family structure with half-sib offspring. A family structure with full-sib grand-offspring, however, was less efficient than a family structure with half-sib grand-offspring. For the most efficient family structure each pair of parents had full-sib offspring that were genotyped for the marker. For the most-efficient family structure each full-sib offspring had half-sib grand-offspring for which trait values were obtained. For equal power with a heritability of 0.1 and 100 grand-offspring per full-sib offspring, 30-times less marker typings were required for this most efficient family structure than for a two-generation half-sib structure in which marker genotypes and trait values were obtained for half-sib offspring. The effect of heritability and the type of analysis (single marker or interval analysis) on the efficiency of a family structure is described. The results of this study should help to design QTL mapping experiments in an outbred population.     

Disequilibrium mapping of a quantitative-trait locus in an expanding population.

Linkage disequilibrium (LD) mapping can be successful if there is strong nonrandom association between marker alleles and an allele affecting a trait of interest. The principles of LD mapping of dichotomous traits are well understood, but less is known about LD mapping of a quantitative-trait locus (QTL). It is shown in this report that selective genotyping can increase the power to detect and map a rare allele of large effect at a QTL. Two statistical tests of the association between an allele and a quantitative character are proposed. These tests are approximately independent, so information from them can be combined. Analytic theory is developed to show that these two tests are effective in detecting the presence of a low-frequency allele with a relatively large effect on the character when the QTL is either already a candidate locus or closely linked to a marker locus that is in strong LD with the QTL. The latter situation is expected in a rapidly growing population in which the allele of large effect was present initially in one copy. Therefore, the proposed tests are useful under the same conditions as those for successful LD mapping of a dichotomous trait or disease. Simulations show that, for detection of the presence of a QTL, these tests are more powerful than a simple t-test. The tests also provide a basis for defining a measure of association, gamma, between a low-frequency allele at a putative QTL and a low-frequency allele at a marker locus.     

Joint-multiple family linkage analysis predicts within-family variation better than single-family analysis of the maize nested association mapping population

Quantitative trait locus (QTL) mapping has been used to dissect the genetic architecture of complex traits and predict phenotypes for marker-assisted selection. Many QTL mapping studies in plants have been limited to one biparental family population. Joint analysis of multiple biparental families offers an alternative approach to QTL mapping with a wider scope of inference. Joint-multiple population analysis should have higher power to detect QTL shared among multiple families, but may have lower power to detect rare QTL. We compared prediction ability of single-family and joint-family QTL analysis methods with fivefold cross-validation for 6 diverse traits using the maize nested association mapping population, which comprises 25 biparental recombinant inbred families. Joint-family QTL analysis had higher mean prediction abilities than single-family QTL analysis for all traits at most significance thresholds, and was always better at more stringent significance thresholds. Most robust QTL (detected in >50% of data samples) were restricted to one family and were often not detected at high frequency by joint-family analysis, implying substantial genetic heterogeneity among families for complex traits in maize. The superior predictive ability of joint-family QTL models despite important genetic differences among families suggests that joint-family models capture sufficient smaller effect QTL that are shared across families to compensate for missing some rare large-effect QTL.     

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.     

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.     

Oil palm (Elaeis guineensis Jacq.) linkage map,and quantitative trait locus analysis for sex ratio and related traits

Sex ratio and shell-thickness type are among the main components determining yield in oil palm. An integrated linkage map of oil palm was constructed based on 208 offspring derived from a cross between two tenera palms differing in inherited sex ratio. The map consisted of 210 genomic simple sequence repeats (SSRs), 28 expressed sequence tag SSRs, 185 amplified fragment length polymorphism markers, and the Sh locus, which controls shell-thickness phenotype, distributed across 16 linkage groups covering 1,931 cM, with an average marker distance of 4.6 cM. Quantitative trait locus (QTL) analysis identified eight QTLs across six linkage groups associated with sex ratio and related traits. These QTLs explained 8.1–13.1 % of the total phenotypic variance. The QTL for sex ratio on linkage group 8 overlapped with a QTL for number of male inflorescences. In most cases a specific QTL allele combination was responsible for genotype class mean differences, suggesting that most QTLs in heterozygous oil palm are likely to be segregating for multiple alleles with different degrees of dominance. In addition, two new SSRs were shown to flank the major Sh locus controlling the fruit variety type in oil palm.     

An improved procedure of mapping a quantitative trait locus via the EM algorithm using posterior probabilities

Mapping a locus controlling a quantitative genetic trait (e.g. blood pressure) to a specific genomic region is of considerable contemporary 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 analysed to estimate the recombination fraction, θ, between the putative quantitative trait locus and a genetic marker. One of the major complications in estimating θ for a quantitative trait in humans is the lack of haplotype information on members of families. We have devised a computationally simple two-stage method of estimation of θ in the absence of haplotypic information using the expectation-maximization (EM) algorithm. In the first stage, parameters of the quantitative trait locus (QTL) are estimated on the basis of data of a sample of unrelated individuals and a Bayes’s rule is used to classify each parent into a QTL genotypic class. In the second stage, we have proposed an EM algorithm for obtaining the maximum-likelihood estimate of θ based on data of informative families (which are identified upon inferring parental QTL genotypes performed in the first stage). The purpose of this paper is to investigate whether, instead of using genotypically ‘classified’ data of parents, the use of posterior probabilities of QT genotypes of parents at the second stage yields better estimators. We show, using simulated data, that the proposed procedure using posterior probabilities is statistically more efficient than our earlier classification procedure, although it is computationally heavier.     

Unequivocal determination of sire allele origin for multiallelic microsatellites when only the sire and progeny are genotyped

The effect of a segregating economic trait locus (ETL) can be detected with the aid of a linked genetic marker, if specific alleles of each locus are in association among the individuals genotyped for the genetic marker. For dairy cattle this can be achieved by application of the ‘granddaughter design’. If only the sires and their sons are genotyped for the genetic markers, then the allele origin of sons having the same genotypes as their sires cannot be determined. Seven sires and 101 sons were genotyped for five microsatellites. The mean frequency of heterozygous sires was 77%. The mean number of alleles per locus was 8.2. Frequency of informative sons per locus ranged from 60% to 80% with a mean of 72%. With highly polymorphic microsatellites, at least 60% more grandsire families can be included in the analysis, and the number of sons assayed can be reduced by 40%, as compared to diallelic markers.     

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.     

Robust indices of Hardy-Weinberg disequilibrium for QTL fine mapping

Hardy-Weinberg disequilibrium (HWD) measures have been proposed using dense markers to fine map a quantitative trait locus (QTL) to regions < approximately 1 cM. Earlier HWD measures may introduce bias in the fine mapping because they are dependent on marker allele frequencies across loci. Hence, HWD indices that do not depend on marker allele frequencies are desired for fine mapping. Based on our earlier work, here we present four new HWD indices that do not depend on marker allele frequencies. Two are for use when marker allele frequencies in a study population are known, and two are for use when marker allele frequencies in a study population are not known and are only known in the extreme samples. The new measures are a function of the genetic distance between the marker locus and a QTL. Through simulations, we investigated and compared the fine mapping performance of the new HWD measures with that of the earlier ones. Our results show that when marker allele frequencies vary across loci, the new measures presented here are more robust and powerful.     

High-resolution association mapping of quantitative trait loci: a population-based approach

In this article, population-based regression models are proposed for high-resolution linkage disequilibrium mapping of quantitative trait loci (QTL). Two regression models, the "genotype effect model" and the "additive effect model," are proposed to model the association between the markers and the trait locus. The marker can be either diallelic or multiallelic. If only one marker is used, the method is similar to a classical setting by Nielsen and Weir, and the additive effect model is equivalent to the haplotype trend regression (HTR) method by Zaykin et al. If two/multiple marker data with phase ambiguity are used in the analysis, the proposed models can be used to analyze the data directly. By analytical formulas, we show that the genotype effect model can be used to model the additive and dominance effects simultaneously; the additive effect model takes care of the additive effect only. On the basis of the two models, F-test statistics are proposed to test association between the QTL and markers. By a simulation study, we show that the two models have reasonable type I error rates for a data set of moderate sample size. The noncentrality parameter approximations of F-test statistics are derived to make power calculation and comparison. By a simulation study, it is found that the noncentrality parameter approximations of F-test statistics work very well. Using the noncentrality parameter approximations, we compare the power of the two models with that of the HTR. In addition, a simulation study is performed to make a comparison on the basis of the haplotype frequencies of 10 SNPs of angiotensin-1 converting enzyme (ACE) genes.     

Detection of parent-of-origin effects based on complete and incomplete nuclear families with multiple affected children

Parent-of-origin effects are important in studying genetic traits. More than 1% of all mammalian genes are believed to show parent-of-origin effects. Some statistical methods may be ineffective or fail to detect linkage or association for a gene with parent-of-origin effects. Based on case-parents trios, the parental-asymmetry test (PAT) is simple and powerful in detecting parent-of-origin effects. However, it is common in practice to collect nuclear families with both parents as well as nuclear families with only one parent. In this paper, when only one parent is available for each family with an arbitrary number of affected children, we firstly develop a new test statistic 1-PAT to test for parent-of-origin effects in the presence of association between an allele at the marker locus under study and a disease gene. Then we extend the PAT to accommodate complete nuclear families each with one or more affected children. Combining families with both parents and families with only one parent, the C-PAT is proposed to detect parent-of-origin effects. The validity of the test statistics is verified by simulation in various scenarios of parameter values. A power study shows that using the additional information from incomplete nuclear families in the analysis greatly improves the power of the tests, compared to that based on only complete nuclear families. Also, utilizing all affected children in each family, the proposed tests have a higher power than when only one affected child from each family is selected. Additional power comparison also demonstrates that the C-PAT is more powerful than a number of other tests for detecting parent-of-origin effects.