Towards linkage analysis with markers in linkage disequilibrium by graphical modelling

We review recent developments of MCMC integration methods for computations on graphical models for two applications in statistical genetics: modelling allelic association and pedigree based linkage analysis. We discuss and illustrate estimation of graphical models from haploid and diploid genotypes, and the importance of MCMC updating schemes beyond what is strictly necessary for irreducibility. We then outline an approach combining these methods to compute linkage statistics when alleles at the marker loci are in linkage disequilibrium. Other extensions suitable for analysis of SNP genotype data in pedigrees are also discussed and programs that implement these methods, and which are available from the author's web site, are described. We conclude with a discussion of how this still experimental approach might be further developed.     

Haplotype-based linkage disequilibrium mapping via direct data mining

MOTIVATION: With the availability of large-scale, high-density single-nucleotide polymorphism markers and information on haplotype structures and frequencies, a great challenge is how to take advantage of haplotype information in the association mapping of complex diseases in case-control studies. RESULTS: We present a novel approach for association mapping based on directly mining haplotypes (i.e. phased genotype pairs) produced from case-control data or case-parent data via a density-based clustering algorithm, which can be applied to whole-genome screens as well as candidate-gene studies in small genomic regions. The method directly explores the sharing of haplotype segments in affected individuals that are rarely present in normal individuals. The measure of sharing between two haplotypes is defined by a new similarity metric that combines the length of the shared segments and the number of common alleles around any marker position of the haplotypes, which is robust against recent mutations/genotype errors and recombination events. The effectiveness of the approach is demonstrated by using both simulated datasets and real datasets. The results show that the algorithm is accurate for different population models and for different disease models, even for genes with small effects, and it outperforms some recently developed methods.     

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.     

Mapping trait loci by use of inferred ancestral recombination graphs

Large-scale association studies are being undertaken with the hope of uncovering the genetic determinants of complex disease. We describe a computationally efficient method for inferring genealogies from population genotype data and show how these genealogies can be used to fine map disease loci and interpret association signals. These genealogies take the form of the ancestral recombination graph (ARG). The ARG defines a genealogical tree for each locus, and, as one moves along the chromosome, the topologies of consecutive trees shift according to the impact of historical recombination events. There are two stages to our analysis. First, we infer plausible ARGs, using a heuristic algorithm, which can handle unphased and missing data and is fast enough to be applied to large-scale studies. Second, we test the genealogical tree at each locus for a clustering of the disease cases beneath a branch, suggesting that a causative mutation occurred on that branch. Since the true ARG is unknown, we average this analysis over an ensemble of inferred ARGs. We have characterized the performance of our method across a wide range of simulated disease models. Compared with simpler tests, our method gives increased accuracy in positioning untyped causative loci and can also be used to estimate the frequencies of untyped causative alleles. We have applied our method to Ueda et al.'s association study of CTLA4 and Graves disease, showing how it can be used to dissect the association signal, giving potentially interesting results of allelic heterogeneity and interaction. Similar approaches analyzing an ensemble of ARGs inferred using our method may be applicable to many other problems of inference from population genotype data.     

Performance of single nucleotide polymorphisms versus haplotypes for genome-wide association analysis in barley

Genome-wide association studies (GWAS) may benefit from utilizing haplotype information for making marker-phenotype associations. Several rationales for grouping single nucleotide polymorphisms (SNPs) into haplotype blocks exist, but any advantage may depend on such factors as genetic architecture of traits, patterns of linkage disequilibrium in the study population, and marker density. The objective of this study was to explore the utility of haplotypes for GWAS in barley (Hordeum vulgare) to offer a first detailed look at this approach for identifying agronomically important genes in crops. To accomplish this, we used genotype and phenotype data from the Barley Coordinated Agricultural Project and constructed haplotypes using three different methods. Marker-trait associations were tested by the efficient mixed-model association algorithm (EMMA). When QTL were simulated using single SNPs dropped from the marker dataset, a simple sliding window performed as well or better than single SNPs or the more sophisticated methods of blocking SNPs into haplotypes. Moreover, the haplotype analyses performed better 1) when QTL were simulated as polymorphisms that arose subsequent to marker variants, and 2) in analysis of empirical heading date data. These results demonstrate that the information content of haplotypes is dependent on the particular mutational and recombinational history of the QTL and nearby markers. Analysis of the empirical data also confirmed our intuition that the distribution of QTL alleles in nature is often unlike the distribution of marker variants, and hence utilizing haplotype information could capture associations that would elude single SNPs. We recommend routine use of both single SNP and haplotype markers for GWAS to take advantage of the full information content of the genotype data.     

A Statistical Method for Identification of Polymorphisms That Explain a Linkage Result

Suppose that many polymorphic sites have been identified and genotyped in a region showing strong linkage with a trait. A key question of interest is which site (or combination of sites) in the region influences susceptibility to the trait. We have developed a novel statistical approach to this problem, in the context of qualitative-trait mapping, in which we use linkage data to identify the polymorphic sites whose genotypes could fully explain the observed linkage to the region. The information provided by this analysis is different from that provided by tests of either linkage or association. Our approach is based on the observation that if a particular site is the only site in the region that influences the trait, then-conditional on the genotypes at that site for the affected relatives-there should be no unexplained oversharing in the region among affected individuals. We focus on the affected sib-pair study design and develop test statistics that are variations on the usual allele-sharing methods used in linkage studies. We perform hypothesis tests and derive a confidence set for the true causal polymorphic site, under the assumption that there is only one site in the region influencing the trait. Our method is appropriate under a very general model for how the site influences the trait, including epistasis with unlinked loci, correlated environmental effects within families, and gene-environment interaction. We extend our method to larger sibships and apply it to an NIDDM1 data set.     

Construction of the model for the Genetic Analysis Workshop 14 simulated data: genotype-phenotype relationships,gene interaction,linkage, association,disequilibrium, and ascertainment effects for a complex phenotype

The Genetic Analysis Workshop 14 simulated dataset was designed 1) To test the ability to find genes related to a complex disease (such as alcoholism). Such a disease may be given a variety of definitions by different investigators, have associated endophenotypes that are common in the general population, and is likely to be not one disease but a heterogeneous collection of clinically similar, but genetically distinct, entities. 2) To observe the effect on genetic analysis and gene discovery of a complex set of gene x gene interactions. 3) To allow comparison of microsatellite vs. large-scale single-nucleotide polymorphism (SNP) data. 4) To allow testing of association to identify the disease gene and the effect of moderate marker x marker linkage disequilibrium. 5) To observe the effect of different ascertainment/disease definition schemes on the analysis. Data was distributed in two forms. Data distributed to participants contained about 1,000 SNPs and 400 microsatellite markers. Internet-obtainable data consisted of a finer 10,000 SNP map, which also contained data on controls. While disease characteristics and parameters were constant, four "studies" used varying ascertainment schemes based on differing beliefs about disease characteristics. One of the studies contained multiplex two- and three-generation pedigrees with at least four affected members. The simulated disease was a psychiatric condition with many associated behaviors (endophenotypes), almost all of which were genetic in origin. The underlying disease model contained four major genes and two modifier genes. The four major genes interacted with each other to produce three different phenotypes, which were themselves heterogeneous. The population parameters were calibrated so that the major genes could be discovered by linkage analysis in most datasets. The association evidence was more difficult to calibrate but was designed to find statistically significant association in 50% of datasets. We also simulated some marker x marker linkage disequilibrium around some of the genes and also in areas without disease genes. We tried two different methods to simulate the linkage disequilibrium.     

An Efficient Genome-Wide Multilocus Epistasis Search

There has been a continuing interest in approaches that analyze pairwise locus-by-locus (epistasis) interactions using multilocus association models in genome-wide data sets. In this paper, we suggest an approach that uses sure independence screening to first lower the dimension of the problem by considering the marginal importance of each interaction term within the huge loop. Subsequent multilocus association steps are executed using an extended Bayesian least absolute shrinkage and selection operator (LASSO) model and fast generalized expectation-maximization estimation algorithms. The potential of this approach is illustrated and compared with PLINK software using data examples where phenotypes have been simulated conditionally on marker data from the Quantitative Trait Loci Mapping and Marker Assisted Selection (QTLMAS) Workshop 2008 and real pig data sets.     

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

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

Haplotype-sharing analysis using Mantel statistics for combined genetic effects

We applied a new approach based on Mantel statistics to analyze the Genetic Analysis Workshop 14 simulated data with prior knowledge of the answers. The method was developed in order to improve the power of a haplotype sharing analysis for gene mapping in complex disease. The new statistic correlates genetic similarity and phenotypic similarity across pairs of haplotypes from case-control studies. The genetic similarity is measured as the shared length between haplotype pairs around a genetic marker. The phenotypic similarity is measured as the mean corrected cross-product based on the respective phenotypes. Cases with phenotype P1 and unrelated controls were drawn from the population of Danacaa. Power to detect main effects was compared to the X2-test for association based on 3-marker haplotypes and a global permutation test for haplotype association to test for main effects. Power to detect gene x gene interaction was compared to unconditional logistic regression. The results suggest that the Mantel statistics might be more powerful than alternative tests.     

Visinets: A Web-Based Pathway Modeling and Dynamic Visualization Tool

In this report we describe a novel graphically oriented method for pathway modeling and a software package that allows for both modeling and visualization of biological networks in a user-friendly format. The Visinets mathematical approach is based on causal mapping (CMAP) that has been fully integrated with graphical interface. Such integration allows for fully graphical and interactive process of modeling, from building the network to simulation of the finished model. To test the performance of Visinets software we have applied it to: a) create executable EGFR-MAPK pathway model using an intuitive graphical way of modeling based on biological data, and b) translate existing ordinary differential equation (ODE) based insulin signaling model into CMAP formalism and compare the results. Our testing fully confirmed the potential of the CMAP method for broad application for pathway modeling and visualization and, additionally, showed significant advantage in computational efficiency. Furthermore, we showed that Visinets web-based graphical platform, along with standardized method of pathway analysis, may offer a novel and attractive alternative for dynamic simulation in real time for broader use in biomedical research. Since Visinets uses graphical elements with mathematical formulas hidden from the users, we believe that this tool may be particularly suited for those who are new to pathway modeling and without the in-depth modeling skills often required when using other software packages.     

Genome-wide linkage analysis of systolic blood pressure slope using the Genetic Analysis Workshop 13 data sets

Systolic blood pressure (SBP) is an age-dependent complex trait for which both environmental and genetic factors may play a role in explaining variability among individuals. We performed a genome-wide scan of the rate of change in SBP over time on the Framingham Heart Study data and one randomly selected replicate of the simulated data from the Genetic Analysis Workshop 13. We used a variance-component model to carry out linkage analysis and a Markov chain Monte Carlo-based multiple imputation approach to recover missing information. Furthermore, we adopted two selection strategies along with the multiple imputation to deal with subjects taking antihypertensive treatment. The simulated data were used to compare these two strategies, to explore the effectiveness of the multiple imputation in recovering varying degrees of missing information, and its impact on linkage analysis results. For the Framingham data, the marker with the highest LOD score for SBP slope was found on chromosome 7. Interestingly, we found that SBP slopes were not heritable in males but were for females; the marker with the highest LOD score was found on chromosome 18. Using the simulated data, we found that handling treated subjects using the multiple imputation improved the linkage results. We conclude that multiple imputation is a promising approach in recovering missing information in longitudinal genetic studies and hence in improving subsequent linkage analyses.     

Detecting susceptibility genes in case-control studies using set association

Complex diseases are generally caused by intricate interactions of multiple genes and environmental factors. Most available linkage and association methods are developed to identify individual susceptibility genes assuming a simple disease model blind to any possible gene - gene and gene - environmental interactions. We used a set association method that uses single-nucleotide polymorphism markers to locate genetic variation responsible for complex diseases in which multiple genes are involved. Here we extended the set association method from bi-allelic to multiallelic markers. In addition, we studied the type I error rates and power for both approaches using simulations based on the coalescent process. Both bi-allelic set association (BSA) and multiallelic set association (MSA) tests have the correct type I error rates. In addition, BSA and MSA can have more power than individual marker analysis when multiple genes are involved in a complex disease. We applied the MSA approach to the simulated data sets from Genetic Analysis Workshop 13. High cholesterol level was used as the definitive phenotype for a disease. MSA failed to detect markers with significant linkage disequilibrium with genes responsible for cholesterol level. This is due to the wide spacing between the markers and the lack of association between the marker loci and the simulated phenotype.     

A Graphical Weighted Power Improving Multiplicity Correction Approach for SNP Selections

Controlling for the multiplicity effect is an essential part of determining statistical significance in large-scale single-locus association genome scans on Single Nucleotide Polymorphisms (SNPs). Bonferroni adjustment is a commonly used approach due to its simplicity, but is conservative and has low power for large-scale tests. The permutation test, which is a powerful and popular tool, is computationally expensive and may mislead in the presence of family structure. We propose a computationally efficient and powerful multiple testing correction approach for Linkage Disequilibrium (LD) based Quantitative Trait Loci (QTL) mapping on the basis of graphical weighted-Bonferroni methods. The proposed multiplicity adjustment method synthesizes weighted Bonferroni-based closed testing procedures into a powerful and versatile graphical approach. By tailoring different priorities for the two hypothesis tests involved in LD based QTL mapping, we are able to increase power and maintain computational efficiency and conceptual simplicity. The proposed approach enables strong control of the familywise error rate (FWER). The performance of the proposed approach as compared to the standard Bonferroni correction is illustrated by simulation and real data. We observe a consistent and moderate increase in power under all simulated circumstances, among different sample sizes, heritabilities, and number of SNPs. We also applied the proposed method to a real outbred mouse HDL cholesterol QTL mapping project where we detected the significant QTLs that were highlighted in the literature, while still ensuring strong control of the FWER.     

Whole-genome association mapping in elite inbred crop varieties

We have previously shown that linkage disequilibrium (LD) in the elite cultivated barley (Hordeum vulgare) gene pool extends, on average, for <1-5 cM. Based on this information, we have developed a platform for whole genome association studies that comprises a collection of elite lines that we have characterized at 3060 genome-wide single nucleotide polymorphism (SNP) marker loci. Interrogating this data set shows that significant population substructure is present within the elite gene pool and that diversity and LD vary considerably across each of the seven barley chromosomes. However, we also show that a subpopulation comprised of only the two-rowed spring germplasm is less structured and well suited to whole genome association studies without the need for extensive statistical intervention to account for structure. At the current marker density, the two-rowed spring population is suited for fine mapping simple traits that are located outside of the genetic centromeres with a resolution that is sufficient for candidate gene identification by exploiting conservation of synteny with fully sequenced model genomes and the emerging barley physical map.     

How to perform meaningful estimates of genetic effects

Although the genotype-phenotype map plays a central role both in Quantitative and Evolutionary Genetics, the formalization of a completely general and satisfactory model of genetic effects, particularly accounting for epistasis, remains a theoretical challenge. Here, we use a two-locus genetic system in simulated populations with epistasis to show the convenience of using a recently developed model, NOIA, to perform estimates of genetic effects and the decomposition of the genetic variance that are orthogonal even under deviations from the Hardy-Weinberg proportions. We develop the theory for how to use this model in interval mapping of quantitative trait loci using Halley-Knott regressions, and we analyze a real data set to illustrate the advantage of using this approach in practice. In this example, we show that departures from the Hardy-Weinberg proportions that are expected by sampling alone substantially alter the orthogonal estimates of genetic effects when other statistical models, like F₂ or G2A, are used instead of NOIA. Finally, for the first time from real data, we provide estimates of functional genetic effects as sets of effects of natural allele substitutions in a particular genotype, which enriches the debate on the interpretation of genetic effects as implemented both in functional and in statistical models. We also discuss further implementations leading to a completely general genotype-phenotype map.     

Association between Stress Response Genes and Features of Diurnal Cortisol Curves in the Multi-Ethnic Study of Atherosclerosis: A New Multi-Phenotype Approach for Gene-Based Association Tests

The hormone cortisol is likely to be a key mediator of the stress response that influences multiple physiologic systems that are involved in common chronic disease, including the cardiovascular system, the immune system, and metabolism. In this paper, a candidate gene approach was used to investigate genetic contributions to variability in multiple correlated features of the daily cortisol profile in a sample of European Americans, African Americans, and Hispanic Americans from the Multi-Ethnic Study of Atherosclerosis (MESA). We proposed and applied a new gene-level multiple-phenotype analysis and carried out a meta-analysis to combine the ethnicity specific results. This new analysis, instead of a more routine single marker-single phenotype approach identified a significant association between one gene (ADRB2) and cortisol features (meta-analysis p-value=0.0025), which was not identified by three other commonly used existing analytic strategies: 1. Single marker association tests involving each single cortisol feature separately; 2. Single marker association tests jointly testing for multiple cortisol features; 3. Gene-level association tests separately carried out for each single cortisol feature. The analytic strategies presented consider different hypotheses regarding genotype-phenotype association and imply different costs of multiple testing. The proposed gene-level analysis integrating multiple cortisol features across multiple ethnic groups provides new insights into the gene-cortisol association.     

Testing Hardy-Weinberg proportions in a frequency-matched case-control genetic association study

In case-control genetic association studies, cases are subjects with the disease and controls are subjects without the disease. At the time of case-control data collection, information about secondary phenotypes is also collected. In addition to studies of primary diseases, there has been some interest in studying genetic variants associated with secondary phenotypes. In genetic association studies, the deviation from Hardy-Weinberg proportion (HWP) of each genetic marker is assessed as an initial quality check to identify questionable genotypes. Generally, HWP tests are performed based on the controls for the primary disease or secondary phenotype. However, when the disease or phenotype of interest is common, the controls do not represent the general population. Therefore, using only controls for testing HWP can result in a highly inflated type I error rate for the disease- and/or phenotype-associated variants. Recently, two approaches, the likelihood ratio test (LRT) approach and the mixture HWP (mHWP) exact test were proposed for testing HWP in samples from case-control studies. Here, we show that these two approaches result in inflated type I error rates and could lead to the removal from further analysis of potential causal genetic variants associated with the primary disease and/or secondary phenotype when the study of primary disease is frequency-matched on the secondary phenotype. Therefore, we proposed alternative approaches, which extend the LRT and mHWP approaches, for assessing HWP that account for frequency matching. The goal was to maintain more (possible causative) single-nucleotide polymorphisms in the sample for further analysis. Our simulation results showed that both extended approaches could control type I error probabilities. We also applied the proposed approaches to test HWP for SNPs from a genome-wide association study of lung cancer that was frequency-matched on smoking status and found that the proposed approaches can keep more genetic variants for association studies.