Efficient semiparametric estimation of haplotype-disease associations in case-cohort and nested case-control studies

Estimating the effects of haplotypes on the age of onset of a disease is an important step toward the discovery of genes that influence complex human diseases. A haplotype is a specific sequence of nucleotides on the same chromosome of an individual and can only be measured indirectly through the genotype. We consider cohort studies which collect genotype data on a subset of cohort members through case-cohort or nested case-control sampling. We formulate the effects of haplotypes and possibly time-varying environmental variables on the age of onset through a broad class of semiparametric regression models. We construct appropriate nonparametric likelihoods, which involve both finite- and infinite-dimensional parameters. The corresponding nonparametric maximum likelihood estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Consistent variance-covariance estimators are provided, and efficient and reliable numerical algorithms are developed. Simulation studies demonstrate that the asymptotic approximations are accurate in practical settings and that case-cohort and nested case-control designs are highly cost-effective. An application to a major cardiovascular study is provided.     

Haplotype-based association analysis in cohort and nested case-control studies

Genetic epidemiologic studies often collect genotype data at multiple loci within a genomic region of interest from a sample of unrelated individuals. One popular method for analyzing such data is to assess whether haplotypes, i.e., the arrangements of alleles along individual chromosomes, are associated with the disease phenotype or not. For many study subjects, however, the exact haplotype configuration on the pair of homologous chromosomes cannot be derived with certainty from the available locus-specific genotype data (phase ambiguity). In this article, we consider estimating haplotype-specific association parameters in the Cox proportional hazards model, using genotype, environmental exposure, and the disease endpoint data collected from cohort or nested case-control studies. We study alternative Expectation-Maximization algorithms for estimating haplotype frequencies from cohort and nested case-control studies. Based on a hazard function of the disease derived from the observed genotype data, we then propose a semiparametric method for joint estimation of relative-risk parameters and the cumulative baseline hazard function. The method is greatly simplified under a rare disease assumption, for which an asymptotic variance estimator is also proposed. The performance of the proposed estimators is assessed via simulation studies. An application of the proposed method is presented, using data from the Alpha-Tocopherol, Beta-Carotene Cancer Prevention Study.     

Extended Mantel-Haenszel estimating procedure for multivariate logistic regression models

A class of estimating functions is proposed for the estimation of multivariate relative risk in stratified case-control studies. It reduces to the well-known Mantel-Haenszel estimator when there is a single binary risk factor. Large-sample properties of the solutions to the proposed estimating equations are established for two distinct situations. Efficiency calculations suggest that the proposed estimators are nearly fully efficient relative to the conditional maximum likelihood estimator for the parameters considered. Application of the proposed method to family data and longitudinal data, where the conditional likelihood approach fails, is discussed. Two examples from case-control studies and one example from a study on familial aggregation are presented.     

Bias-reduced estimators of conditional odds ratios in matched case-control studies with unmatched confounding

We study bias-reduced estimators of exponentially transformed parameters in general linear models (GLMs) and show how they can be used to obtain bias-reduced conditional (or unconditional) odds ratios in matched case-control studies. Two options are considered and compared: the explicit approach and the implicit approach. The implicit approach is based on the modified score function where bias-reduced estimates are obtained by using iterative procedures to solve the modified score equations. The explicit approach is shown to be a one-step approximation of this iterative procedure. To apply these approaches for the conditional analysis of matched case-control studies, with potentially unmatched confounding and with several exposures, we utilize the relation between the conditional likelihood and the likelihood of the unconditional logit binomial GLM for matched pairs and Cox partial likelihood for matched sets with appropriately setup data. The properties of the estimators are evaluated by using a large Monte Carlo simulation study and an illustration of a real dataset is shown. Researchers reporting the results on the exponentiated scale should use bias-reduced estimators since otherwise the effects can be under or overestimated, where the magnitude of the bias is especially large in studies with smaller sample sizes.     

Maximum-likelihood estimation of relatedness

Relatedness between individuals is central to many studies in genetics and population biology. A variety of estimators have been developed to enable molecular marker data to quantify relatedness. Despite this, no effort has been given to characterize the traditional maximum-likelihood estimator in relation to the remainder. This article quantifies its statistical performance under a range of biologically relevant sampling conditions. Under the same range of conditions, the statistical performance of five other commonly used estimators of relatedness is quantified. Comparison among these estimators indicates that the traditional maximum-likelihood estimator exhibits a lower standard error under essentially all conditions. Only for very large amounts of genetic information do most of the other estimators approach the likelihood estimator. However, the likelihood estimator is more biased than any of the others, especially when the amount of genetic information is low or the actual relationship being estimated is near the boundary of the parameter space. Even under these conditions, the amount of bias can be greatly reduced, potentially to biologically irrelevant levels, with suitable genetic sampling. Additionally, the likelihood estimator generally exhibits the lowest root mean-square error, an indication that the bias in fact is quite small. Alternative estimators restricted to yield only biologically interpretable estimates exhibit lower standard errors and greater bias than do unrestricted ones, but generally do not improve over the maximum-likelihood estimator and in some cases exhibit even greater bias. Although some nonlikelihood estimators exhibit better performance with respect to specific metrics under some conditions, none approach the high level of performance exhibited by the likelihood estimator across all conditions and all metrics of performance.     

Influence of Feature Encoding and Choice of Classifier on Disease Risk Prediction in Genome-Wide Association Studies

Various attempts have been made to predict the individual disease risk based on genotype data from genome-wide association studies (GWAS). However, most studies only investigated one or two classification algorithms and feature encoding schemes. In this study, we applied seven different classification algorithms on GWAS case-control data sets for seven different diseases to create models for disease risk prediction. Further, we used three different encoding schemes for the genotypes of single nucleotide polymorphisms (SNPs) and investigated their influence on the predictive performance of these models. Our study suggests that an additive encoding of the SNP data should be the preferred encoding scheme, as it proved to yield the best predictive performances for all algorithms and data sets. Furthermore, our results showed that the differences between most state-of-the-art classification algorithms are not statistically significant. Consequently, we recommend to prefer algorithms with simple models like the linear support vector machine (SVM) as they allow for better subsequent interpretation without significant loss of accuracy.     

Bias-reduced estimators and confidence intervals for odds ratios in genome-wide association studies

Genome-wide association studies (GWAS) provide an important approach to identifying common genetic variants that predispose to human disease. A typical GWAS may genotype hundreds of thousands of single nucleotide polymorphisms (SNPs) located throughout the human genome in a set of cases and controls. Logistic regression is often used to test for association between a SNP genotype and case versus control status, with corresponding odds ratios (ORs) typically reported only for those SNPs meeting selection criteria. However, when these estimates are based on the original data used to detect the variant, the results are affected by a selection bias sometimes referred to the "winner's curse" (Capen and others, 1971). The actual genetic association is typically overestimated. We show that such selection bias may be severe in the sense that the conditional expectation of the standard OR estimator may be quite far away from the underlying parameter. Also standard confidence intervals (CIs) may have far from the desired coverage rate for the selected ORs. We propose and evaluate 3 bias-reduced estimators, and also corresponding weighted estimators that combine corrected and uncorrected estimators, to reduce selection bias. Their corresponding CIs are also proposed. We study the performance of these estimators using simulated data sets and show that they reduce the bias and give CI coverage close to the desired level under various scenarios, even for associations having only small statistical power.     

A haplotype-based test of association using data from cohort and nested case-control epidemiologic studies

Haplotype-based risk models can lead to powerful methods for detecting the association of a disease with a genomic region of interest. In population-based studies of unrelated individuals, however, the haplotype status of some subjects may not be discernible without ambiguity from available locus-specific genotype data. A score test for detecting haplotype-based association using genotype data has been developed in the context of generalized linear models for analysis of data from cross-sectional and retrospective studies. In this article, we develop a test for association using genotype data from cohort and nested case-control studies where subjects are prospectively followed until disease incidence or censoring (end of follow-up) occurs. Assuming a proportional hazard model for the haplotype effects, we derive an induced hazard function of the disease given the genotype data, and hence propose a test statistic based on the associated partial likelihood. The proposed test procedure can account for differential follow-up of subjects, can adjust for possibly time-dependent environmental co-factors and can make efficient use of valuable age-at-onset information that is available on cases. We provide an algorithm for computing the test statistic using readily available statistical software. Utilizing simulated data in the context of two genomic regions GPX1 and GPX3, we evaluate the validity of the proposed test for small sample sizes and study its power in the presence and absence of missing genotype data.     

Family-based association tests for genomewide association scans

With millions of single-nucleotide polymorphisms (SNPs) identified and characterized, genomewide association studies have begun to identify susceptibility genes for complex traits and diseases. These studies involve the characterization and analysis of very-high-resolution SNP genotype data for hundreds or thousands of individuals. We describe a computationally efficient approach to testing association between SNPs and quantitative phenotypes, which can be applied to whole-genome association scans. In addition to observed genotypes, our approach allows estimation of missing genotypes, resulting in substantial increases in power when genotyping resources are limited. We estimate missing genotypes probabilistically using the Lander-Green or Elston-Stewart algorithms and combine high-resolution SNP genotypes for a subset of individuals in each pedigree with sparser marker data for the remaining individuals. We show that power is increased whenever phenotype information for ungenotyped individuals is included in analyses and that high-density genotyping of just three carefully selected individuals in a nuclear family can recover >90% of the information available if every individual were genotyped, for a fraction of the cost and experimental effort. To aid in study design, we evaluate the power of strategies that genotype different subsets of individuals in each pedigree and make recommendations about which individuals should be genotyped at a high density. To illustrate our method, we performed genomewide association analysis for 27 gene-expression phenotypes in 3-generation families (Centre d'Etude du Polymorphisme Humain pedigrees), in which genotypes for ~860,000 SNPs in 90 grandparents and parents are complemented by genotypes for ~6,700 SNPs in a total of 168 individuals. In addition to increasing the evidence of association at 15 previously identified cis-acting associated alleles, our genotype-inference algorithm allowed us to identify associated alleles at 4 cis-acting loci that were missed when analysis was restricted to individuals with the high-density SNP data. Our genotype-inference algorithm and the proposed association tests are implemented in software that is available for free.     

Haplotype block structure and its applications to association studies: power and study designs

Recent studies have shown that the human genome has a haplotype block structure, such that it can be divided into discrete blocks of limited haplotype diversity. In each block, a small fraction of single-nucleotide polymorphisms (SNPs), referred to as "tag SNPs," can be used to distinguish a large fraction of the haplotypes. These tag SNPs can potentially be extremely useful for association studies, in that it may not be necessary to genotype all SNPs; however, this depends on how much power is lost. Here we develop a simulation study to quantitatively assess the power loss for a variety of study designs, including case-control designs and case-parental control designs. First, a number of data sets containing case-parental or case-control samples are generated on the basis of a disease model. Second, a small fraction of case and control individuals in each data set are genotyped at all the loci, and a dynamic programming algorithm is used to determine the haplotype blocks and the tag SNPs based on the genotypes of the sampled individuals. Third, the statistical power of tests was evaluated on the basis of three kinds of data: (1) all of the SNPs and the corresponding haplotypes, (2) the tag SNPs and the corresponding haplotypes, and (3) the same number of randomly chosen SNPs as the number of tag SNPs and the corresponding haplotypes. We study the power of different association tests with a variety of disease models and block-partitioning criteria. Our study indicates that the genotyping efforts can be significantly reduced by the tag SNPs, without much loss of power. Depending on the specific haplotype block-partitioning algorithm and the disease model, when the identified tag SNPs are only 25% of all the SNPs, the power is reduced by only 4%, on average, compared with a power loss of approximately 12% when the same number of randomly chosen SNPs is used in a two-locus haplotype analysis. When the identified tag SNPs are approximately 14% of all the SNPs, the power is reduced by approximately 9%, compared with a power loss of approximately 21% when the same number of randomly chosen SNPs is used in a two-locus haplotype analysis. Our study also indicates that haplotype-based analysis can be much more powerful than marker-by-marker analysis.     

Aneuploidy in dizygotic twin sheep detected using genome-wide single nucleotide polymorphism data from two commonly used commercial vendors

Early detection of karyotype abnormalities, including aneuploidy, could aid producers in identifying animals which, for example, would not be suitable candidate parents. Genome-wide genetic marker data in the form of single nucleotide polymorphisms (SNPs) are now being routinely generated on animals. The objective of the present study was to describe the statistics that could be generated from the allele intensity values from such SNP data to diagnose karyotype abnormalities; of particular interest was whether detection of aneuploidy was possible with both commonly used genotyping platforms in agricultural species, namely the Applied Biosystems^TM Axiom^TM and the Illumina platform. The hypothesis was tested using a case study of a set of dizygotic X-chromosome monosomy 53,X sheep twins. Genome-wide SNP data were available from the Illumina platform (11 082 autosomal and 191 X-chromosome SNPs) on 1848 male and 8954 female sheep and available from the Axiom^TM platform (11 128 autosomal and 68 X-chromosome SNPs) on 383 female sheep. Genotype allele intensity values, either as their original raw values or transformed to logarithm intensity ratio (LRR), were used to accurately diagnose two dizygotic (i.e. fraternal) twin 53,X sheep, both of which received their single X chromosome from their sire. This is the first reported case of 53,X dizygotic twins in any species. Relative to the X-chromosome SNP genotype mean allele intensity values of normal females, the mean allele intensity value of SNP genotypes on the X chromosome of the two females monosomic for the X chromosome was 7.45 to 12.4 standard deviations less, and were easily detectable using either the Axiom^TM or Illumina genotype platform; the next lowest mean allele intensity value of a female was 4.71 or 3.3 standard deviations less than the population mean depending on the platform used. Both 53,X females could also be detected based on the genotype LRR although this was more easily detectable when comparing the mean LRR of the X chromosome of each female to the mean LRR of their respective autosomes. On autopsy, the ovaries of the two sheep were small for their age and evidence of prior ovulation was not appreciated. In both sheep, the density of primordial follicles in the ovarian cortex was lower than normally found in ovine ovaries and primary follicle development was not observed. Mammary gland development was very limited. Results substantiate previous studies in other species that aneuploidy can be readily detected using SNP genotype allele intensity values generally already available, and the approach proposed in the present study was agnostic to genotype platform.     

A maximum-likelihood estimation of pairwise relatedness for autopolyploids

Relatedness between individuals is central to ecological genetics. Multiple methods are available to quantify relatedness from molecular data, including method-of-moment and maximum-likelihood estimators. We describe a maximum-likelihood estimator for autopolyploids, and quantify its statistical performance under a range of biologically relevant conditions. The statistical performances of five additional polyploid estimators of relatedness were also quantified under identical conditions. When comparing truncated estimators, the maximum-likelihood estimator exhibited lower root mean square error under some conditions and was more biased for non-relatives, especially when the number of alleles per loci was low. However, even under these conditions, this bias was reduced to be statistically insignificant with more robust genetic sampling. We also considered ambiguity in polyploid heterozygote genotyping and developed a weighting methodology for candidate genotypes. The statistical performances of three polyploid estimators under both ideal and actual conditions (including inbreeding and double reduction) were compared. The software package POLYRELATEDNESS is available to perform this estimation and supports a maximum ploidy of eight.     

Conditional likelihood methods for haplotype-based association analysis using matched case-control data

Genetic epidemiologists routinely assess disease susceptibility in relation to haplotypes, that is, combinations of alleles on a single chromosome. We study statistical methods for inferring haplotype-related disease risk using single nucleotide polymorphism (SNP) genotype data from matched case-control studies, where controls are individually matched to cases on some selected factors. Assuming a logistic regression model for haplotype-disease association, we propose two conditional likelihood approaches that address the issue that haplotypes cannot be inferred with certainty from SNP genotype data (phase ambiguity). One approach is based on the likelihood of disease status conditioned on the total number of cases, genotypes, and other covariates within each matching stratum, and the other is based on the joint likelihood of disease status and genotypes conditioned only on the total number of cases and other covariates. The joint-likelihood approach is generally more efficient, particularly for assessing haplotype-environment interactions. Simulation studies demonstrated that the first approach was more robust to model assumptions on the diplotype distribution conditioned on environmental risk variables and matching factors in the control population. We applied the two methods to analyze a matched case-control study of prostate cancer.     

Numerical equivalence of imputing scores and weighted estimators in regression analysis with missing covariates

Imputation, weighting, direct likelihood, and direct Bayesian inference (Rubin, 1976) are important approaches for missing data regression. Many useful semiparametric estimators have been developed for regression analysis of data with missing covariates or outcomes. It has been established that some semiparametric estimators are asymptotically equivalent, but it has not been shown that many are numerically the same. We applied some existing methods to a bladder cancer case-control study and noted that they were the same numerically when the observed covariates and outcomes are categorical. To understand the analytical background of this finding, we further show that when observed covariates and outcomes are categorical, some estimators are not only asymptotically equivalent but also actually numerically identical. That is, although their estimating equations are different, they lead numerically to exactly the same root. This includes a simple weighted estimator, an augmented weighted estimator, and a mean-score estimator. The numerical equivalence may elucidate the relationship between imputing scores and weighted estimation procedures.     

A family-based likelihood ratio test for general pedigree structures that allows for genotyping error and missing data

The purpose of this work is the development of a family-based association test that allows for random genotyping errors and missing data and makes use of information on affected and unaffected pedigree members. We derive the conditional likelihood functions of the general nuclear family for the following scenarios: complete parental genotype data and no genotyping errors; only one genotyped parent and no genotyping errors; no parental genotype data and no genotyping errors; and no parental genotype data with genotyping errors. We find maximum likelihood estimates of the marker locus parameters, including the penetrances and population genotype frequencies under the null hypothesis that all penetrance values are equal and under the alternative hypothesis. We then compute the likelihood ratio test. We perform simulations to assess the adequacy of the central chi-square distribution approximation when the null hypothesis is true. We also perform simulations to compare the power of the TDT and this likelihood-based method. Finally, we apply our method to 23 SNPs genotyped in nuclear families from a recently published study of idiopathic scoliosis (IS). Our simulations suggest that this likelihood ratio test statistic follows a central chi-square distribution with 1 degree of freedom under the null hypothesis, even in the presence of missing data and genotyping errors. The power comparison shows that this likelihood ratio test is more powerful than the original TDT for the simulations considered. For the IS data, the marker rs7843033 shows the most significant evidence for our method (p = 0.0003), which is consistent with a previous report, which found rs7843033 to be the 2nd most significant TDTae p value among a set of 23 SNPs.     

Estimating pairwise relatedness between individuals with different levels of ploidy

Estimates of relatedness coefficients, based on genetic marker data, are often necessary for studies of genetics and ecology. Whilst many estimates based on method‐of‐moment or maximum‐likelihood methods exist for diploid organisms, no such estimators exist for organisms with multiple ploidy levels, which occur in some insect and plant species. Here, we extend five estimators to account for different levels of ploidy: one relatedness coefficient estimator, three coefficients of coancestry estimators and one maximum‐likelihood estimator. We use arrhenotoky (when unfertilized eggs develop into haploid males) as an example in evaluations of estimator performance by Monte Carlo simulation. Also, three virtual sex‐determination systems are simulated to evaluate their performances for higher levels of ploidy. Additionally, we used two real data sets to test the robustness of these estimators under actual conditions. We make available a software package, PolyRelatedness , for other researchers to apply to organisms that have various levels of ploidy.     

Single nucleotide polymorphisms (SNPs) are inherited from parents and they measure heritable events

Single nucleotide polymorphisms (SNPs) are extensively used in case-control studies of practically all cancer types. They are used for the identification of inherited cancer susceptibility genes and those that may interact with environmental factors. However, being genetic markers, they are applicable only on heritable conditions, which is often a neglected fact. Based on the data in the nationwide Swedish Family-Cancer Database, we review familial risks for all main cancers and discuss the evidence for a heritable component in cancer. The available evidence is not conclusive but it is consistent in pointing to a minor heritable etiology in cancer, which will hamper the success of SNP-based association studies. Empirical familial risks should be used as guidance for the planning of SNP studies. We provide calculations for the assessment of familial risks for assumed allele frequencies and gene effects (odds ratios) for different modes of inheritance. Based on these data, we discuss the gene effects that could account for the unexplained proportion of familial breast and lung cancer. As a conclusion, we are concerned about the indiscriminate use of a genetic tool to cancers, which are mainly environmental in origin. We consider the likelihood of a successful application of SNPs in gene-environment studies small, unless established environmental risk factors are tested on proven candidate genes.     

Identifying the susceptibility gene(s) in a set of trait-linked genes using genotype data

There are generally three steps to isolate a disease linkage-susceptibility gene: genome-wide scan, fine mapping, and, last, positional cloning. The last step is time consuming and involves intensive laboratory work. In some cases, fine mapping cannot proceed further on a set of markers because they are tightly linked. For years, genetic statisticians have been trying different ways to narrow the fine-mapping results to provide some guidance for the next step of laboratory work. Although these methods are practical and efficient, most of them are based on IBD data, which usually can be inferred only from the genotype data with some uncertainty. The corresponding methods thus have no greater power than one using genotype data directly. Also, IBD-based methods apply only to relative pair data. Here, using genotype data, we have developed a statistical hypothesis-testing method to pinpoint a SNP, or SNPs, suspected of responsibility for a disease trait linkage among a set of SNPs tightly linked in a region. Our method uses genotype data of affected individuals or case-control studies, which are widely available in the laboratory. The testing statistic can be constructed using any genotype-based disease-marker disequilibrium measure and is asymptotically distributed as a chi-square mixture. This method can be used for singleton data, relative pair data, or general pedigree data. We have applied the method to simulated data as well as a real data set; it gives satisfactory results.     

Robust best linear estimation for regression analysis using surrogate and instrumental variables

We investigate methods for regression analysis when covariates are measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies the classical measurement error model, but it may not have repeated measurements. In addition to the surrogate variables that are available among the subjects in the calibration sample, we assume that there is an instrumental variable (IV) that is available for all study subjects. An IV is correlated with the unobserved true exposure variable and hence can be useful in the estimation of the regression coefficients. We propose a robust best linear estimator that uses all the available data, which is the most efficient among a class of consistent estimators. The proposed estimator is shown to be consistent and asymptotically normal under very weak distributional assumptions. For Poisson or linear regression, the proposed estimator is consistent even if the measurement error from the surrogate or IV is heteroscedastic. Finite-sample performance of the proposed estimator is examined and compared with other estimators via intensive simulation studies. The proposed method and other methods are applied to a bladder cancer case-control study.     

Statistical inference for net benefit measures in biomarker validation studies

Referral strategies based on risk scores and medical tests are commonly proposed. Direct assessment of their clinical utility requires implementing the strategy and is not possible in the early phases of biomarker research. Prior to late-phase studies, net benefit measures can be used to assess the potential clinical impact of a proposed strategy. Validation studies, in which the biomarker defines a prespecified referral strategy, are a gold standard approach to evaluating biomarker potential. Uncertainty, quantified by a confidence interval, is important to consider when deciding whether a biomarker warrants an impact study, does not demonstrate clinical potential, or that more data are needed. We establish distribution theory for empirical estimators of net benefit and propose empirical estimators of variance. The primary results are for the most commonly employed estimators of net benefit: from cohort and unmatched case-control samples, and for point estimates and net benefit curves. Novel estimators of net benefit under stratified two-phase and categorically matched case-control sampling are proposed and distribution theory developed. Results for common variants of net benefit and for estimation from right-censored outcomes are also presented. We motivate and demonstrate the methodology with examples from lung cancer research and highlight its application to study design.