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.     

Estimation of Linkage and Association from Allele Transmission Data

The TDT provides a hypothesis test for the presence of linkage or association (linkage disequilibrium). However, since the TDT is a single test statistic, it cannot be used to separate association and linkage. The importance of this difficulty, following a significant TDT result, has been recently emphasized by Whittaker , Denham and Morris (2000), who alert the community to the possibility that a significant TDT may result from loose linkage and strong association, or from tight linkage and weak association. To attack this problem we start with the parametric model for family‐based allele transmission data of Sham and Curtis (1995) (or Sham (1998)) and find that the parameters in the model are not always identifiable. So we introduce a reparameterization that resolves the identifiability issues and leads to a valid likelihood ratio (LR) test for linkage. Since the linkage and association parameters are both of interest, we next introduce and apply an integrated likelihood (IL) approach to provide separate point estimates and confidence intervals for these parameters. The estimates are shown to have generally small bias and mean square error, while the confidence intervals have good average length and coverage probabilities. We compare the power of the IL approach for testing linkage and, separately association, with the TDT and LR.     

The quantitative trait linkage disequilibrium test: a more powerful alternative to the quantitative transmission disequilibrium test for use in the absence of population stratification

Linkage analysis based on identity-by-descent allele-sharing can be used to identify a chromosomal region harboring a quantitative trait locus (QTL), but lacks the resolution required for gene identification. Consequently, linkage disequilibrium (association) analysis is often employed for fine-mapping. Variance-components based combined linkage and association analysis for quantitative traits in sib pairs, in which association is modeled as a mean effect and linkage is modeled in the covariance structure has been extended to general pedigrees (quantitative transmission disequilibrium test, QTDT). The QTDT approach accommodates data not only from parents and siblings, but also from all available relatives. QTDT is also robust to population stratification. However, when population stratification is absent, it is possible to utilize even more information, namely the additional information contained in the founder genotypes. In this paper, we introduce a simple modification of the allelic transmission scoring method used in the QTDT that results in a more powerful test of linkage disequilibrium, but is only applicable in the absence of population stratification. This test, the quantitative trait linkage disequilibrium (QTLD) test, has been incorporated into a new procedure in the statistical genetics computer package SOLAR. We apply this procedure in a linkage/association analysis of an electrophysiological measurement previously shown to be related to alcoholism. We also demonstrate by simulation the increase in power obtained with the QTLD test, relative to the QTDT, when a true association exists between a marker and a QTL.     

Multilocus linkage analysis in humans: detection of linkage and estimation of recombination.

Multilocus linkage analysis is investigated from the viewpoint of the efficiency of recombination estimates under different strategies for detecting linkage and determining gene order within a linkage group. We consider the appropriateness of assuming no interference with data available in human genetic studies. Examples are given to show the significance of multilocus analysis in humans. A computer program package, LINKAGE, for multilocus linkage analysis is described.     

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.     

Association between a genetic trait and a marker: Discrimination between epistasis and gametic disequilibrium

Summary We reconsider the method of Ott and Falk (1982) for the analysis of genetic linkage and of epistasis in the presence of phenotypic association. Their approach is extended to allow for gametic disequilibrium between marker and trait loci. We show that epistasis and tight linkage with gametic disequilibrium may be indistinguishable as explanations of association even in a very large pedigree.     

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.     

Doubled haploids for estimating genetic variances in presence of linkage and gene association

Summary The effect of gene association (or dispersion) and linkage on the estimation of genetic variances in a diallel experiment involving doubled haploid lines is evaluated. It is shown that the estimates of the additive and the additive X additive genetic variances, as obtained by Choo et al. (1979), are biased if genes are linked or are not independently distributed in the parents. However, this bias only occurs in the presence of interaction between homozygous loci. Gene association (or dispersion) and linkage, if present, can be detected by comparing the parental vs the crosses mean, the parental vs the doubled haploid lines variance, and the among vs the within crosses variance.     

Testing for Genetic Association in the Presence of Linkage and Gene–Covariate Interactions

In order to study family‐based association in the presence of linkage, we extend a generalized linear mixed model proposed for genetic linkage analysis (Lebrec and van Houwelingen (2007), Human Heredity 64 , 5–15) by adding a genotypic effect to the mean. The corresponding score test is a weighted family‐based association tests statistic, where the weight depends on the linkage effect and on other genetic and shared environmental effects. For testing of genetic association in the presence of gene–covariate interaction, we propose a linear regression method where the family‐specific score statistic is regressed on family‐specific covariates. Both statistics are straightforward to compute. Simulation results show that adjusting the weight for the within‐family variance structure may be a powerful approach in the presence of environmental effects. The test statistic for genetic association in the presence of gene–covariate interaction improved the power for detecting association. For illustration, we analyze the rheumatoid arthritis data from GAW15. Adjusting for smoking and anti‐cyclic citrullinated peptide increased the significance of the association with the DR locus.     

Tests for linkage and association in nuclear families.

The transmission/disequilibrium test (TDT) originally was introduced to test for linkage between a genetic marker and a disease-susceptibility locus, in the presence of association. Recently, the TDT has been used to test for association in the presence of linkage. The motivation for this is that linkage analysis typically identifies large candidate regions, and further refinement is necessary before a search for the disease gene is begun, on the molecular level. Evidence of association and linkage may indicate which markers in the region are closest to a disease locus. As a test of linkage, transmissions from heterozygous parents to all of their affected children can be included in the TDT; however, the TDT is a valid chi2 test of association only if transmissions to unrelated affected children are used in the analysis. If the sample contains independent nuclear families with multiple affected children, then one procedure that has been used to test for association is to select randomly a single affected child from each sibship and to apply the TDT to those data. As an alternative, we propose two statistics that use data from all of the affected children. The statistics give valid chi2 tests of the null hypothesis of no association or no linkage and generally are more powerful than the TDT with a single, randomly chosen, affected child from each family.     

Combined segregation and linkage analysis of Graves disease with a thyroid autoantibody diathesis.

Combined segregation and linkage analysis is a powerful technique for modeling linkage to diseases whose etiology is more complex than the effect of a well-described single genetic locus and for investigating the influence of single genes on various aspects of the disease phenotype. Graves disease is familial and is associated with human leukocyte antigen (HLA) allele DR3. Probands with Graves disease, as well as close relatives, have raised levels of thyroid autoantibodies. This phenotypic information additional to affection status may be considered by the computer program COMDS for combined segregation and linkage analysis, when normals are classified into diathesis classes of increasing thyroid autoantibody titer. The ordinal model considers the cumulative odds of lying in successive classes, and a single additional parameter is introduced for each gene modeled. Distributional assumptions are avoided by providing estimates of the population frequencies of each class. Evidence for linkage was increased by considering the thyroid autoantibody diathesis and by testing two-locus models. The analysis revealed evidence for linkage to HLA-DR when the strong coupling of the linked locus to allele DR3 was considered (lod score of 6.6). Linkage analysis of the residual variation revealed no evidence of linkage to Gm, but a suggestion of linkage to Km.     

Linkage analysis of "necessary" disease loci versus "susceptibility" loci.

The association of some diseases with specific alleles of certain genetic markers has been difficult to explain. Several explanations have been proposed for the phenomenon of association, e.g. the existence of multiple, interacting genes (epistasis) or a disease locus in linkage disequilibrium with the marker locus. One might suppose that when marker data from families with associated diseases are analyzed for linkage, the existence of the association would assure that linkage will be found, and found at a tight recombination fraction. In fact, however, linkage analyses of some diseases associated with HLA, as well as diseases associated with alleles at other loci located throughout the genome, show significant evidence against linkage, and others show loose linkage, to the puzzlement of many researchers. In part, the puzzlement arises because linkage analysis is ideal for looking for loci that are necessary, even if not sufficient, for disease expression but may be much less useful for finding loci that are neither necessary nor sufficient for disease expression (so-called susceptibility loci). This work explores what happens when one looks for linkage to susceptibility loci. A susceptibility locus in this case means that the allele increases risk but is neither necessary nor sufficient for disease expression. It might be either an allele at the marker locus itself that is increasing susceptibility or an allele at a locus in linkage disequilibrium with the marker. This work uses computer simulation to examine how linkage analyses behave when confronted with data from such a model.(ABSTRACT TRUNCATED AT 250 WORDS)     

Sample sizes required to detect linkage disequilibrium between two or three loci.

Estimates of the degree of nonrandom association among genes (linkage disequilibrium) can provide evidence of the role of natural selection in maintaining allozyme polymorphisms in natural populations. This paper outlines the maximum likelihood procedures for such estimates based on gametic or zygotic frequencies at the level of two loci. The analysis is extended to estimating disequilibrium between three loci. In particular, the question of the sampling requirements to detect different intensities of disequilibrium is considered. It is found that relatively large samples are required to detect nonrandom association, unless gene frequencies are intermediate and disequilibrium is relatively intense. This might be one reason why cases of linkage disequilibrium have so far proved to be the exception, rather than the rule, in population studies.     

Maximum likelihood estimation of linkage disequilibrium in half-sib families

Maximum likelihood methods for the estimation of linkage disequilibrium between biallelic DNA-markers in half-sib families (half-sib method) are developed for single and multifamily situations. Monte Carlo computer simulations were carried out for a variety of scenarios regarding sire genotypes, linkage disequilibrium, recombination fraction, family size, and number of families. A double heterozygote sire was simulated with recombination fraction of 0.00, linkage disequilibrium among dams of δ=0.10, and alleles at both markers segregating at intermediate frequencies for a family size of 500. The average estimates of δ were 0.17, 0.25, and 0.10 for Excoffier and Slatkin (1995), maternal informative haplotypes, and the half-sib method, respectively. A multifamily EM algorithm was tested at intermediate frequencies by computer simulation. The range of the absolute difference between estimated and simulated δ was between 0.000 and 0.008. A cattle half-sib family was genotyped with the Illumina 50K BeadChip. There were 314,730 SNP pairs for which the sire was a homo-heterozygote with average estimates of r2 of 0.115, 0.067, and 0.111 for half-sib, Excoffier and Slatkin (1995), and maternal informative haplotypes methods, respectively. There were 208,872 SNP pairs for which the sire was double heterozygote with average estimates of r2 across the genome of 0.100, 0.267, and 0.925 for half-sib, Excoffier and Slatkin (1995), and maternal informative haplotypes methods, respectively. Genome analyses for all possible sire genotypes with 829,042 tests showed that ignoring half-sib family structure leads to upward biased estimates of linkage disequilibrium. Published inferences on population structure and evolution of cattle should be revisited after accommodating existing half-sib family structure in the estimation of linkage disequilibrium.     

X-APL: an improved family-based test of association in the presence of linkage for the X chromosome

Family-based association methods have been developed primarily for autosomal markers. The X-linked sibling transmission/disequilibrium test (XS-TDT) and the reconstruction-combined TDT for X-chromosome markers (XRC-TDT) are the first association-based methods for testing markers on the X chromosome in family data sets. These are valid tests of association in family triads or discordant sib pairs but are not theoretically valid in multiplex families when linkage is present. Recently, XPDT and XMCPDT, modified versions of the pedigree disequilibrium test (PDT), were proposed. Like the PDT, XPDT compares genotype transmissions from parents to affected offspring or genotypes of discordant siblings; however, the XPDT can have low power if there are many missing parental genotypes. XMCPDT uses a Monte Carlo sampling approach to infer missing parental genotypes on the basis of true or estimated population allele frequencies. Although the XMCPDT was shown to be more powerful than the XPDT, variability in the statistic due to the use of an estimate of allele frequency is not properly accounted for. Here, we present a novel family-based test of association, X-APL, a modification of the test for association in the presence of linkage (APL) test. Like the APL, X-APL can use singleton or multiplex families and properly infers missing parental genotypes in linkage regions by considering identity-by-descent parameters for affected siblings. Sampling variability of parameter estimates is accounted for through a bootstrap procedure. X-APL can test individual marker loci or X-chromosome haplotypes. To allow for different penetrances in males and females, separate sex-specific tests are provided. Using simulated data, we demonstrated validity and showed that the X-APL is more powerful than alternative tests. To show its utility and to discuss interpretation in real-data analysis, we also applied the X-APL to candidate-gene data in a sample of families with Parkinson disease.     

Improvement of mapping accuracy by unifying linkage and association analysis

It is well known that pedigree/family data record information on the coexistence in founder haplotypes of alleles at nearby loci and the cotransmission from parent to offspring that reveal different, but complementary, profiles of the genetic architecture. Either conventional linkage analysis that assumes linkage equilibrium or family-based association tests (FBATs) capture only partial information, leading to inefficiency. For example, FBATs will fail to detect even very tight linkage in the case where no allelic association exists, while a violation of the assumption of linkage equilibrium will result in biased estimation and reduced efficiency in linkage mapping. In this article, by using a data augmentation technique and the EM algorithm, we propose a likelihood-based approach that embeds both linkage and association analyses into a unified framework for general pedigree data. Relative to either linkage or association analysis, the proposed approach is expected to have greater estimation accuracy and power. Monte Carlo simulations support our theoretical expectations and demonstrate that our new methodology: (1) is more powerful than either FBATs or classic linkage analysis; (2) can unbiasedly estimate genetic parameters regardless of whether association exists, thus remedying the bias and less precision of traditional linkage analysis in the presence of association; and (3) is capable of identifying tight linkage alone. The new approach also holds the theoretical advantage that it can extract statistical information to the maximum extent and thereby improve mapping accuracy and power because it integrates multilocus population-based association study and pedigree-based linkage analysis into a coherent framework. Furthermore, our method is numerically stable and computationally efficient, as compared to existing parametric methods that use the simplex algorithm or Newton-type methods to maximize high-order multidimensional likelihood functions, and also offers the computation of Fisher's information matrix. Finally, we apply our methodology to a genetic study on bone mineral density (BMD) for the vitamin D receptor (VDR) gene and find that VDR is significantly linked to BMD at the one-third region of the wrist.     

Family-based tests of association in the presence of linkage

Linkage analysis may not provide the necessary resolution for identification of the genes underlying phenotypic variation. This is especially true for gene-mapping studies that focus on complex diseases that do not exhibit Mendelian inheritance patterns. One positional genomic strategy involves application of association methodology to areas of identified linkage. Detection of association in the presence of linkage localizes the gene(s) of interest to more-refined regions in the genome than is possible through linkage analysis alone. This strategy introduces a statistical complexity when family-based association tests are used: the marker genotypes among siblings are correlated in linked regions. Ignoring this correlation will compromise the size of the statistical hypothesis test, thus clouding the interpretation of test results. We present a method for computing the expectation of a wide range of association test statistics under the null hypothesis that there is linkage but no association. To standardize the test statistic, an empirical variance-covariance estimator that is robust to the sibling marker-genotype correlation is used. This method is widely applicable: any type of phenotypic measure or family configuration can be used. For example, we analyze a deletion in the A2M gene at the 5' splice site of "exon II" of the bait region in Alzheimer disease (AD) discordant sibships. Since the A2M gene lies in a chromosomal region (chromosome 12p) that consistently has been linked to AD, association tests should be conducted under the null hypothesis that there is linkage but no association.     

A sibship test for linkage in the presence of association: the sib transmission/disequilibrium test.

Linkage analysis with genetic markers has been successful in the localization of genes for many monogenic human diseases. In studies of complex diseases, however, tests that rely on linkage disequilibrium (the simultaneous presence of linkage and association) are often more powerful than those that rely on linkage alone. This advantage is illustrated by the transmission/disequilibrium test (TDT). The TDT requires data (marker genotypes) for affected individuals and their parents; for some diseases, however, data from parents may be difficult or impossible to obtain. In this article, we describe a method, called the "sib TDT" (or "S-TDT"), that overcomes this problem by use of marker data from unaffected sibs instead of from parents, thus allowing application of the principle of the TDT to sibships without parental data. In a single collection of families, there might be some that can be analyzed only by the TDT and others that are suitable for analysis by the S-TDT. We show how all the data may be used jointly in one overall TDT-type procedure that tests for linkage in the presence of association. These extensions of the TDT will be valuable for the study of diseases of late onset, such as non-insulin-dependent diabetes, cardiovascular diseases, and other diseases associated with aging.     

Combining the meiosis Gibbs sampler with the random walk approach for linkage and association studies with a general complex pedigree and multimarker loci

A linkage analysis for finding inheritance states and haplotype configurations is an essential process for linkage and association mapping. The linkage analysis is routinely based upon observed pedigree information and marker genotypes for individuals in the pedigree. It is not feasible for exact methods to use all such information for a large complex pedigree especially when there are many missing genotypic data. Proposed Markov chain Monte Carlo approaches such as a single-site Gibbs sampler or the meiosis Gibbs sampler are able to handle a complex pedigree with sparse genotypic data; however, they often have reducibility problems, causing biased estimates. We present a combined method, applying the random walk approach to the reducible sites in the meiosis sampler. Therefore, one can efficiently obtain reliable estimates such as identity-by-descent coefficients between individuals based on inheritance states or haplotype configurations, and a wider range of data can be used for mapping of quantitative trait loci within a reasonable time.     

HLA and the inheritance of multiple sclerosis: linkage analysis of 72 pedigrees.

Linkage analysis of 72 pedigrees by the maximum-likelihood method provides evidence of linkage between HLA and the hypothesized multiple sclerosis susceptibility gene (MSSG) for both the dominant and recessive models of inheritance and for penetrance values ranging from 5%--65% (or higher). This MSSG, if it exists, is most likely located at 15%--20% recombination units from the HLA complex, probably on the B-D side. The analysis also shows that there is no heterogeneity in the estimates of linkage, and lod scores are not artifically inflated because of the association of multiple sclerosis (MS) with HLA-B7.