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1.  Screening and Replication using the Same Data Set: Testing Strategies for Family-Based Studies in which All Probands Are Affected 
PLoS Genetics  2008;4(9):e1000197.
For genome-wide association studies in family-based designs, we propose a powerful two-stage testing strategy that can be applied in situations in which parent-offspring trio data are available and all offspring are affected with the trait or disease under study. In the first step of the testing strategy, we construct estimators of genetic effect size in the completely ascertained sample of affected offspring and their parents that are statistically independent of the family-based association/transmission disequilibrium tests (FBATs/TDTs) that are calculated in the second step of the testing strategy. For each marker, the genetic effect is estimated (without requiring an estimate of the SNP allele frequency) and the conditional power of the corresponding FBAT/TDT is computed. Based on the power estimates, a weighted Bonferroni procedure assigns an individually adjusted significance level to each SNP. In the second stage, the SNPs are tested with the FBAT/TDT statistic at the individually adjusted significance levels. Using simulation studies for scenarios with up to 1,000,000 SNPs, varying allele frequencies and genetic effect sizes, the power of the strategy is compared with standard methodology (e.g., FBATs/TDTs with Bonferroni correction). In all considered situations, the proposed testing strategy demonstrates substantial power increases over the standard approach, even when the true genetic model is unknown and must be selected based on the conditional power estimates. The practical relevance of our methodology is illustrated by an application to a genome-wide association study for childhood asthma, in which we detect two markers meeting genome-wide significance that would not have been detected using standard methodology.
Author Summary
The current state of genotyping technology has enabled researchers to conduct genome-wide association studies of up to 1,000,000 SNPs, allowing for systematic scanning of the genome for variants that might influence the development and progression of complex diseases. One of the largest obstacles to the successful detection of such variants is the multiple comparisons/testing problem in the genetic association analysis. For family-based designs in which all offspring are affected with the disease/trait under study, we developed a methodology that addresses this problem by partitioning the family-based data into two statistically independent components. The first component is used to screen the data and determine the most promising SNPs. The second component is used to test the SNPs for association, where information from the screening is used to weight the SNPs during testing. This methodology is more powerful than standard procedures for multiple comparisons adjustment (i.e., Bonferroni correction). Additionally, as only one data set is required for screening and testing, our testing strategy is less susceptible to study heterogeneity. Finally, as many family-based studies collect data only from affected offspring, this method addresses a major limitation of previous methodologies for multiple comparisons in family-based designs, which require variation in the disease/trait among offspring.
PMCID: PMC2529406  PMID: 18802462
2.  Bayesian Mixture Models for the Incorporation of Prior Knowledge to Inform Genetic Association Studies 
Genetic epidemiology  2010;34(5):418-426.
In the last decade, numerous genome-wide linkage and association studies of complex diseases have been completed. The critical question remains of how to best use this potentially valuable information to improve study design and statistical analysis in current and future genetic association studies. With genetic effect size for complex diseases being relatively small, the use of all available information is essential to untangling the genetic architecture of complex diseases. One promising approach to incorporating prior knowledge from linkage scans, or other information, is to up- or down-weight p-values resulting from genetic association study in either a frequentist or Bayesian manner. As an alternative to these methods, we propose a fully Bayesian mixture model to incorporate previous knowledge into on-going association analysis. In this approach, both the data and previous information collectively inform the association analysis, in contrast to modifying the association results (p-values) to conform to the prior knowledge. By using a Bayesian framework, one has flexibility in modeling, and is able to comprehensively assess the impact of model specification on posterior inferences. We illustrate use of this method through a genome-wide linkage study of colorectal cancer, and a genome-wide association study of colorectal polyps.
PMCID: PMC2910528  PMID: 20583285
Bayesian; genetic association; linkage; mixture model; prior information
3.  Genomic breeding value prediction using three Bayesian methods and application to reduced density marker panels 
BMC Proceedings  2010;4(Suppl 1):S6.
Bayesian approaches for predicting genomic breeding values (GEBV) have been proposed that allow for different variances for individual markers resulting in a shrinkage procedure that uses prior information to coerce negligible effects towards zero. These approaches have generally assumed application to high-density genotype data on all individuals, which may not be the case in practice. In this study, three approaches were compared for their predictive power in computing GEBV when training at high SNP marker density and predicting at high or low densities: the well- known Bayes-A, a generalization of Bayes-A where scale and degrees of freedom are estimated from the data (Student-t) and a Bayesian implementation of the Lasso method. Twelve scenarios were evaluated for predicting GEBV using low-density marker subsets, including selection of SNP based on genome spacing or size of additive effect and the inclusion of unknown genotype information in the form of genotype probabilities from pedigree and genotyped ancestors.
The GEBV accuracy (calculated as correlation between GEBV and traditional breeding values) was highest for Lasso, followed by Student-t and then Bayes-A. When comparing GEBV to true breeding values, Student-t was most accurate, though differences were small. In general the shrinkage applied by the Lasso approach was less conservative than Bayes-A or Student-t, indicating that Lasso may be more sensitive to QTL with small effects. In the reduced-density marker subsets the ranking of the methods was generally consistent. Overall, low-density, evenly-spaced SNPs did a poor job of predicting GEBV, but SNPs selected based on additive effect size yielded accuracies similar to those at high density, even when coverage was low. The inclusion of genotype probabilities to the evenly-spaced subsets showed promising increases in accuracy and may be more useful in cases where many QTL of small effect are expected.
In this dataset the Student-t approach slightly outperformed the other methods when predicting GEBV at both high and low density, but the Lasso method may have particular advantages in situations where many small QTL are expected. When markers were selected at low density based on genome spacing, the inclusion of genotype probabilities increased GEBV accuracy which would allow a single low- density marker panel to be used across traits.
PMCID: PMC2857848  PMID: 20380760
4.  Full Likelihood Analysis of Genetic Risk with Variable Age at Onset Disease—Combining Population-Based Registry Data and Demographic Information 
PLoS ONE  2009;4(8):e6836.
In genetic studies of rare complex diseases it is common to ascertain familial data from population based registries through all incident cases diagnosed during a pre-defined enrollment period. Such an ascertainment procedure is typically taken into account in the statistical analysis of the familial data by constructing either a retrospective or prospective likelihood expression, which conditions on the ascertainment event. Both of these approaches lead to a substantial loss of valuable data.
Methodology and Findings
Here we consider instead the possibilities provided by a Bayesian approach to risk analysis, which also incorporates the ascertainment procedure and reference information concerning the genetic composition of the target population to the considered statistical model. Furthermore, the proposed Bayesian hierarchical survival model does not require the considered genotype or haplotype effects be expressed as functions of corresponding allelic effects. Our modeling strategy is illustrated by a risk analysis of type 1 diabetes mellitus (T1D) in the Finnish population-based on the HLA-A, HLA-B and DRB1 human leucocyte antigen (HLA) information available for both ascertained sibships and a large number of unrelated individuals from the Finnish bone marrow donor registry. The heterozygous genotype DR3/DR4 at the DRB1 locus was associated with the lowest predictive probability of T1D free survival to the age of 15, the estimate being 0.936 (0.926; 0.945 95% credible interval) compared to the average population T1D free survival probability of 0.995.
The proposed statistical method can be modified to other population-based family data ascertained from a disease registry provided that the ascertainment process is well documented, and that external information concerning the sizes of birth cohorts and a suitable reference sample are available. We confirm the earlier findings from the same data concerning the HLA-DR3/4 related risks for T1D, and also provide here estimated predictive probabilities of disease free survival as a function of age.
PMCID: PMC2730012  PMID: 19718441
5.  Maximizing the Power of Genome-Wide Association Studies: A Novel Class of Powerful Family-Based Association Tests 
Statistics in biosciences  2009;1(2):125-143.
For genome-wide association studies in family-based designs, a new, universally applicable approach is proposed. Using a modified Liptak’s method, we combine the p-value of the family-based association test (FBAT) statistic with the p-value for the Van Steen-statistic. The Van Steen-statistic is independent of the FBAT-statistic and utilizes information that is ignored by traditional FBAT-approaches. The new test statistic takes advantages of all available information about the genetic association, while, by virtue of its design, it achieves complete robustness against confounding due to population stratification. The approach is suitable for the analysis of almost any trait type for which FBATs are available, e.g. binary, continuous, time to-onset, multivariate, etc. The efficiency and the validity of the new approach depend on the specification of a nuisance/tuning parameter and the weight parameters in the modified Liptak’s method. For different trait types and ascertainment conditions, we discuss general guidelines for the optimal specification of the tuning parameter and the weight parameters. Our simulation experiments and an application to an Alzheimer study show the validity and the efficiency of the new method, which achieves power levels that are comparable to those of population-based approaches.
PMCID: PMC3349940  PMID: 22582089
FBAT; Liptak’s method; Tuning parameter
6.  Correcting for Population Stratification in Genomewide Association Studies 
Genomewide association studies have become the primary tool for discovering the genetic basis of complex human diseases. Such studies are susceptible to the confounding effects of population stratification, in that the combination of allele-frequency heterogeneity with disease-risk heterogeneity among different ancestral subpopulations can induce spurious associations between genetic variants and disease. This article provides a statistically rigorous and computationally feasible solution to this challenging problem of unmeasured confounders. We show that the odds ratio of disease with a genetic variant is identifiable if and only if the genotype is independent of the unknown population substructure conditional on a set of observed ancestry-informative markers in the disease-free population. Under this condition, the odds ratio of interest can be estimated by fitting a semiparametric logistic regression model with an arbitrary function of a propensity score relating the genotype probability to ancestry-informative markers. Approximating the unknown function of the propensity score by B-splines, we derive a consistent and asymptotically normal estimator for the odds ratio of interest with a consistent variance estimator. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. An application to the well-known Wellcome Trust Case-Control Study is presented. Supplemental materials are available online.
PMCID: PMC3314247  PMID: 22467997
B-spline; Case-control study; Principal components; Propensity score; Semiparametric logistic regression; Single nucleotide polymorphism
7.  Extension of the bayesian alphabet for genomic selection 
BMC Bioinformatics  2011;12:186.
Two Bayesian methods, BayesCπ and BayesDπ, were developed for genomic prediction to address the drawback of BayesA and BayesB regarding the impact of prior hyperparameters and treat the prior probability π that a SNP has zero effect as unknown. The methods were compared in terms of inference of the number of QTL and accuracy of genomic estimated breeding values (GEBVs), using simulated scenarios and real data from North American Holstein bulls.
Estimates of π from BayesCπ, in contrast to BayesDπ, were sensitive to the number of simulated QTL and training data size, and provide information about genetic architecture. Milk yield and fat yield have QTL with larger effects than protein yield and somatic cell score. The drawback of BayesA and BayesB did not impair the accuracy of GEBVs. Accuracies of alternative Bayesian methods were similar. BayesA was a good choice for GEBV with the real data. Computing time was shorter for BayesCπ than for BayesDπ, and longest for our implementation of BayesA.
Collectively, accounting for computing effort, uncertainty as to the number of QTL (which affects the GEBV accuracy of alternative methods), and fundamental interest in the number of QTL underlying quantitative traits, we believe that BayesCπ has merit for routine applications.
PMCID: PMC3144464  PMID: 21605355
8.  Power, Validity, Bias and Robustness of Family-based Association Analysis Methods in the Presence of Linkage for Late Onset Diseases 
Annals of Human Genetics  2008;72(6):793-800.
This simulation-based report compares the performance of five methods of association analysis in the presence of linkage using extended sibships: the Family-Based Association Test (FBAT), Empirical Variance FBAT (EV-FBAT), Conditional Logistic Regression (CLR), Robust CLR (R-CLR) and Sibship Disequilibrium Test (SDT). The two tests accounting for residual familial correlation (EV-FBAT and R-CLR) and the model-free SDT showed correct test size in all simulated designs, while FBAT and CLR were only valid for small effect sizes. SDT had the lowest power, while CLR had the highest power, generally similar to FBAT and the robust variance analogues. The power of all model-dependent tests dropped when the model was misspecified, although often not substantially. Estimates of genetic effect with CLR and R-CLR were unbiased when the disease locus was analysed but biased when a nearby marker was analysed. This study demonstrates that the genetic effect does not need to be extreme to invalidate tests that ignore familial correlation and confirms that analogous methods using robust variance estimation provide a valid alternative at little cost to power. Overall R-CLR is the best-performing method among these alternatives for the analysis of extended sibship data.
PMCID: PMC2659381  PMID: 18782299
Extended sibships; conditional logistic regression; robust variance; simulation
9.  Impact of prior specifications in ashrinkage-inducing Bayesian model for quantitative trait mapping and genomic prediction 
In quantitative trait mapping and genomic prediction, Bayesian variable selection methods have gained popularity in conjunction with the increase in marker data and computational resources. Whereas shrinkage-inducing methods are common tools in genomic prediction, rigorous decision making in mapping studies using such models is not well established and the robustness of posterior results is subject to misspecified assumptions because of weak biological prior evidence.
Here, we evaluate the impact of prior specifications in a shrinkage-based Bayesian variable selection method which is based on a mixture of uniform priors applied to genetic marker effects that we presented in a previous study. Unlike most other shrinkage approaches, the use of a mixture of uniform priors provides a coherent framework for inference based on Bayes factors. To evaluate the robustness of genetic association under varying prior specifications, Bayes factors are compared as signals of positive marker association, whereas genomic estimated breeding values are considered for genomic selection. The impact of specific prior specifications is reduced by calculation of combined estimates from multiple specifications. A Gibbs sampler is used to perform Markov chain Monte Carlo estimation (MCMC) and a generalized expectation-maximization algorithm as a faster alternative for maximum a posteriori point estimation. The performance of the method is evaluated by using two publicly available data examples: the simulated QTLMAS XII data set and a real data set from a population of pigs.
Combined estimates of Bayes factors were very successful in identifying quantitative trait loci, and the ranking of Bayes factors was fairly stable among markers with positive signals of association under varying prior assumptions, but their magnitudes varied considerably. Genomic estimated breeding values using the mixture of uniform priors compared well to other approaches for both data sets and loss of accuracy with the generalized expectation-maximization algorithm was small as compared to that with MCMC.
Since no error-free method to specify priors is available for complex biological phenomena, exploring a wide variety of prior specifications and combining results provides some solution to this problem. For this purpose, the mixture of uniform priors approach is especially suitable, because it comprises a wide and flexible family of distributions and computationally intensive estimation can be carried out in a reasonable amount of time.
PMCID: PMC3750442  PMID: 23834140
10.  Ranking Bias in Association Studies 
Human Heredity  2009;67(4):267-275.
It is widely appreciated that genomewide association studies often yield overestimates of the association of a marker with disease when attention focuses upon the marker showing the strongest relationship. For example, in a case-control setting the largest (in absolute value) estimated odds ratio has been found to typically overstate the association as measured in a second, independent set of data. The most common reason given for this observation is that the choice of the most extreme test statistic is often conditional upon first observing a significant p value associated with the marker. A second, less appreciated reason is described here. Under common circumstances it is the multiple testing of many markers and subsequent focus upon those with most extreme test statistics (i.e. highly ranked results) that leads to bias in the estimated effect sizes.
This bias, termed ranking bias, is separate from that arising from conditioning on a significant p value and may often be a more important factor in generating bias. An analytic description of this bias, simulations demonstrating its extent, and identification of some factors leading to its exacerbation are presented.
PMCID: PMC2880722  PMID: 19172085
Estimation bias; Multiple comparisons
11.  The Bayesian Covariance Lasso 
Statistics and its interface  2013;6(2):243-259.
Estimation of sparse covariance matrices and their inverse subject to positive definiteness constraints has drawn a lot of attention in recent years. The abundance of high-dimensional data, where the sample size (n) is less than the dimension (d), requires shrinkage estimation methods since the maximum likelihood estimator is not positive definite in this case. Furthermore, when n is larger than d but not sufficiently larger, shrinkage estimation is more stable than maximum likelihood as it reduces the condition number of the precision matrix. Frequentist methods have utilized penalized likelihood methods, whereas Bayesian approaches rely on matrix decompositions or Wishart priors for shrinkage. In this paper we propose a new method, called the Bayesian Covariance Lasso (BCLASSO), for the shrinkage estimation of a precision (covariance) matrix. We consider a class of priors for the precision matrix that leads to the popular frequentist penalties as special cases, develop a Bayes estimator for the precision matrix, and propose an efficient sampling scheme that does not precalculate boundaries for positive definiteness. The proposed method is permutation invariant and performs shrinkage and estimation simultaneously for non-full rank data. Simulations show that the proposed BCLASSO performs similarly as frequentist methods for non-full rank data.
PMCID: PMC3925647  PMID: 24551316
Bayesian covariance lasso; non-full rank data; Network exploration; Penalized likelihood; Precision matrix
12.  Calibration of prior variance in the Bayesian Continual Reassessment Method 
Statistics in medicine  2011;30(17):2081-2089.
The continual reassessment method (CRM) is an adaptive model-based design used to estimate the maximum tolerated dose in phase I clinical trials. Asymptotically, the method has been shown to select the correct dose given that certain conditions are satisfied. When sample size is small, specifying a reasonable model is important. While an algorithm has been proposed for the calibration of the initial guesses of the probabilities of toxicity, the calibration of the prior distribution of the parameter for the Bayesian CRM has not been addressed. In this paper, we introduce the concept of least informative prior variance for a normal prior distribution. We also propose two systematic approaches to jointly calibrate the prior variance and the initial guesses of the probability of toxicity at each dose. The proposed calibration approaches are compared with existing approaches in the context of two examples via simulations. The new approaches and the previously proposed methods yield very similar results since the latter used appropriate vague priors. However, the new approaches yield a smaller interval of toxicity probabilities in which a neighboring dose may be selected.
PMCID: PMC3129459  PMID: 21413054
Dose finding; indifference interval; least informative prior; phase I clinical trials
13.  The null distribution of stochastic search gene suggestion: a Bayesian approach to gene mapping 
BMC Proceedings  2007;1(Suppl 1):S113.
Bayesian methods continue to permeate genetic epidemiology investigations of genetic markers associated with or linked to causal genes for complex diseases. The attraction of these methods is an ability to capitalize on Bayesian priors to model additional complexity and information about the disease outside the specific data analyzed. It is well known that the larger the sample size, the more the Bayesian method with uninformative priors can be approximated by its Frequentist analogue. However, what is not known is how much impact the priors have on a Bayesian method when analyzing a null region of the chromosome. Here, we look at the impact of various prior values on stochastic search gene suggestion (SSGS) when analyzing a region of simulated chromosome 6 known to be unassociated with the simulated disease. SSGS is a recently developed Bayesian variable selection method tailored to investigate disease-gene association using case-parent triads. Our findings indicate that the prior probability values do affect false positives, and this study suggests values to calibrate the prior. Also, the sensitivity of the results to the prior probability values depends on two factors: the linkage disequilibrium between the marker loci examined, and whether this dependence is included in the model. In order to assess the null distribution we used the simulated data with the "answers" known.
PMCID: PMC2367497  PMID: 18466454
14.  Tests of Association for Quantitative Traits in Nuclear Families Using Principal Components to Correct for Population Stratification 
Annals of human genetics  2009;73(Pt 6):601-613.
Traditional transmission disequilibrium test (TDT) based methods for genetic association analyses are robust to population stratification at the cost of a substantial loss of power. We here describe a novel method for family-based association studies that corrects for population stratification with the use of an extension of principal component analysis (PCA). Specifically, we adopt PCA on unrelated parents in each family. We then infer principal components for children from those for their parents through a TDT-like strategy. Two test statistics within variance-components model are proposed for association tests. Simulation results show that the proposed tests have correct type I error rates regardless of population stratification, and have greatly improved power over two popular TDT-based methods: QTDT and FBAT. The application to the Genetic Analysis Workshop 16 (GAW16) data sets attests to the feasibility of the proposed method.
PMCID: PMC2764806  PMID: 19702646
Family Based Association Tests (FBATs); Transmission Disequilibrium Test (TDT); Principal Component Analysis (PCA); Variance-Components
15.  Variable selection for large p small n regression models with incomplete data: Mapping QTL with epistases 
BMC Bioinformatics  2008;9:251.
Identifying quantitative trait loci (QTL) for both additive and epistatic effects raises the statistical issue of selecting variables from a large number of candidates using a small number of observations. Missing trait and/or marker values prevent one from directly applying the classical model selection criteria such as Akaike's information criterion (AIC) and Bayesian information criterion (BIC).
We propose a two-step Bayesian variable selection method which deals with the sparse parameter space and the small sample size issues. The regression coefficient priors are flexible enough to incorporate the characteristic of "large p small n" data. Specifically, sparseness and possible asymmetry of the significant coefficients are dealt with by developing a Gibbs sampling algorithm to stochastically search through low-dimensional subspaces for significant variables. The superior performance of the approach is demonstrated via simulation study. We also applied it to real QTL mapping datasets.
The two-step procedure coupled with Bayesian classification offers flexibility in modeling "large p small n" data, especially for the sparse and asymmetric parameter space. This approach can be extended to other settings characterized by high dimension and low sample size.
PMCID: PMC2435550  PMID: 18510743
16.  Regularized regression method for genome-wide association studies 
BMC Proceedings  2011;5(Suppl 9):S67.
We use a novel penalized approach for genome-wide association study that accounts for the linkage disequilibrium between adjacent markers. This method uses a penalty on the difference of the genetic effect at adjacent single-nucleotide polymorphisms and combines it with the minimax concave penalty, which has been shown to be superior to the least absolute shrinkage and selection operator (LASSO) in terms of estimator bias and selection consistency. Our method is implemented using a coordinate descent algorithm. The value of the tuning parameters is determined by extended Bayesian information criteria. The leave-one-out method is used to compute p-values of selected single-nucleotide polymorphisms. Its applicability to a simulated data from Genetic Analysis Workshop 17 replication one is illustrated. Our method selects three SNPs (C13S522, C13S523, and C13S524), whereas the LASSO method selects two SNPs (C13S522 and C13S523).
PMCID: PMC3287906  PMID: 22373491
17.  Bayesian flux balance analysis applied to a skeletal muscle metabolic model 
Journal of theoretical biology  2007;248(1):91-110.
In this article, the steady state condition for the multi-compartment models for cellular metabolism is considered. The problem is to estimate the reaction and transport fluxes, as well as the concentrations in venous blood when the stoichiometry and bound constraints for the fluxes and the concentrations are given. The problem has been addressed previously by a number of authors, and optimization based approaches as well as extreme pathway analysis have been proposed. These approaches are briefly discussed here. The main emphasis of this work is a Bayesian statistical approach to the flux balance analysis (FBA). We show how the bound constraints and optimality conditions such as maximizing the oxidative phosphorylation flux can be incorporated into the model in the Bayesian framework by proper construction of the prior densities. We propose an effective Markov Chain Monte Carlo (MCMC) scheme to explore the posterior densities, and compare the results with those obtained via the previously studied Linear Programming (LP) approach. The proposed methodology, which is applied here to a two-compartment model for skeletal muscle metabolism, can be extended to more complex models.
PMCID: PMC2065751  PMID: 17568615
Flux balance analysis; steady state; skeletal muscle metabolism; linear programming; Bayesian statistics; Markov Chain Monte Carlo; Gibbs sampler
18.  Impact of Population Stratification on Family-Based Association Tests with Longitudinal Measurements 
Several family-based approaches for testing genetic association with traits obtained from longitudinal or repeated measurement studies have been previously proposed. These approaches utilize the multivariate data more efficiently by using estimated optimal weights to combine univariate tests. We show that these FBAT approaches are still robust against hidden population stratification, but their power can be heavily affected since the estimated weights might provide poor approximation of the true theoretical optimal weights with the presence of population stratification. We introduce a permutation-based approach FBAT-MinP and an equal combination approach FBAT-EW, both of which do not involve the use of estimated weights. Through simulation studies, FBAT-MinP and FBAT-EW are shown to be powerful even in the presence of population stratification, when other approaches may substantially lose their power. An application of these approaches to the Childhood Asthma Management Program (CAMP) study data for testing an association between body mass index and a previously reported candidate SNP is given as an example.
PMCID: PMC2861319  PMID: 19222384
19.  Family-Based Bivariate Association Tests for Quantitative Traits 
PLoS ONE  2009;4(12):e8133.
The availability of a large number of dense SNPs, high-throughput genotyping and computation methods promotes the application of family-based association tests. While most of the current family-based analyses focus only on individual traits, joint analyses of correlated traits can extract more information and potentially improve the statistical power. However, current TDT-based methods are low-powered. Here, we develop a method for tests of association for bivariate quantitative traits in families. In particular, we correct for population stratification by the use of an integration of principal component analysis and TDT. A score test statistic in the variance-components model is proposed. Extensive simulation studies indicate that the proposed method not only outperforms approaches limited to individual traits when pleiotropic effect is present, but also surpasses the power of two popular bivariate association tests termed FBAT-GEE and FBAT-PC, respectively, while correcting for population stratification. When applied to the GAW16 datasets, the proposed method successfully identifies at the genome-wide level the two SNPs that present pleiotropic effects to HDL and TG traits.
PMCID: PMC2779861  PMID: 19956578
20.  Case-Control Studies of Gene-Environment Interaction: Bayesian Design and Analysis 
Biometrics  2010;66(3):934-948.
With increasing frequency, epidemiologic studies are addressing hypotheses regarding gene-environment interaction. In many well studied candidate genes and for standard dietary and behavioral epidemiologic exposures, there is often substantial prior information available which may be used to analyze current data as well as for designing a new study. In this paper, first, we propose a proper full Bayesian approach for analyzing studies of gene-environment interaction. The Bayesian approach provides a natural way to incorporate uncertainties around the assumption of gene-environment independence, often used in such analysis. We then consider Bayesian sample size determination criteria for both estimation and hypothesis testing regarding the multiplicative gene-environment interaction parameter. We illustrate our proposed methods using data from a large ongoing case-control study of colorectal cancer investigating the interaction of N-acetyl transferase type 2 (NAT2) with smoking and red meat consumption. We use the existing data to elicit a design prior and show how to use this information in allocating cases and controls in planning a future study which investigates the same interaction parameters. The Bayesian design and analysis strategies are compared with their corresponding frequentist counterparts.
PMCID: PMC3103064  PMID: 19930190
case-only design; gene-environment independence; highest posterior density interval; molecular epidemiology of colorectal cancer; multinomial-Dirichlet; posterior odds
21.  Rare Variant Analysis for Family-Based Design 
PLoS ONE  2013;8(1):e48495.
Genome-wide association studies have been able to identify disease associations with many common variants; however most of the estimated genetic contribution explained by these variants appears to be very modest. Rare variants are thought to have larger effect sizes compared to common SNPs but effects of rare variants cannot be tested in the GWAS setting. Here we propose a novel method to test for association of rare variants obtained by sequencing in family-based samples by collapsing the standard family-based association test (FBAT) statistic over a region of interest. We also propose a suitable weighting scheme so that low frequency SNPs that may be enriched in functional variants can be upweighted compared to common variants. Using simulations we show that the family-based methods perform at par with the population-based methods under no population stratification. By construction, family-based tests are completely robust to population stratification; we show that our proposed methods remain valid even when population stratification is present.
PMCID: PMC3546113  PMID: 23341868
22.  Hierarchical likelihood opens a new way of estimating genetic values using genome-wide dense marker maps 
BMC Proceedings  2011;5(Suppl 3):S14.
Genome-wide dense markers have been used to detect genes and estimate relative genetic values. Among many methods, Bayesian techniques have been widely used and shown to be powerful in genome-wide breeding value estimation and association studies. However, computation is known to be intensive under the Bayesian framework, and specifying a prior distribution for each parameter is always required for Bayesian computation. We propose the use of hierarchical likelihood to solve such problems.
Using double hierarchical generalized linear models, we analyzed the simulated dataset provided by the QTLMAS 2010 workshop. Marker-specific variances estimated by double hierarchical generalized linear models identified the QTL with large effects for both the quantitative and binary traits. The QTL positions were detected with very high accuracy. For young individuals without phenotypic records, the true and estimated breeding values had Pearson correlation of 0.60 for the quantitative trait and 0.72 for the binary trait, where the quantitative trait had a more complicated genetic architecture involving imprinting and epistatic QTL.
Hierarchical likelihood enables estimation of marker-specific variances under the likelihoodist framework. Double hierarchical generalized linear models are powerful in localizing major QTL and computationally fast.
PMCID: PMC3103199  PMID: 21624170
23.  A Bayesian Hierarchical Model for Relating Multiple SNPs within Multiple Genes to Disease Risk 
A variety of methods have been proposed for studying the association of multiple genes thought to be involved in a common pathway for a particular disease. Here, we present an extension of a Bayesian hierarchical modeling strategy that allows for multiple SNPs within each gene, with external prior information at either the SNP or gene level. The model involves variable selection at the SNP level through latent indicator variables and Bayesian shrinkage at the gene level towards a prior mean vector and covariance matrix that depend on external information. The entire model is fitted using Markov chain Monte Carlo methods. Simulation studies show that the approach is capable of recovering many of the truly causal SNPs and genes, depending upon their frequency and size of their effects. The method is applied to data on 504 SNPs in 38 candidate genes involved in DNA damage response in the WECARE study of second breast cancers in relation to radiotherapy exposure.
PMCID: PMC3892936  PMID: 24490143
24.  Cyber-T web server: differential analysis of high-throughput data 
Nucleic Acids Research  2012;40(Web Server issue):W553-W559.
The Bayesian regularization method for high-throughput differential analysis, described in Baldi and Long (A Bayesian framework for the analysis of microarray expression data: regularized t-test and statistical inferences of gene changes. Bioinformatics 2001: 17: 509-519) and implemented in the Cyber-T web server, is one of the most widely validated. Cyber-T implements a t-test using a Bayesian framework to compute a regularized variance of the measurements associated with each probe under each condition. This regularized estimate is derived by flexibly combining the empirical measurements with a prior, or background, derived from pooling measurements associated with probes in the same neighborhood. This approach flexibly addresses problems associated with low replication levels and technology biases, not only for DNA microarrays, but also for other technologies, such as protein arrays, quantitative mass spectrometry and next-generation sequencing (RNA-seq). Here we present an update to the Cyber-T web server, incorporating several useful new additions and improvements. Several preprocessing data normalization options including logarithmic and (Variance Stabilizing Normalization) VSN transforms are included. To augment two-sample t-tests, a one-way analysis of variance is implemented. Several methods for multiple tests correction, including standard frequentist methods and a probabilistic mixture model treatment, are available. Diagnostic plots allow visual assessment of the results. The web server provides comprehensive documentation and example data sets. The Cyber-T web server, with R source code and data sets, is publicly available at
PMCID: PMC3394347  PMID: 22600740
25.  Identification of Pharmacogenetic Markers in Smoking Cessation Therapy 
Pharmacogenetic clinical trials seek to identify genetic modifiers of treatment effects. When a trial has collected data on many potential genetic markers, a first step in analysis is to screen for evidence of pharmacogenetic effects by testing for treatment-by-marker interactions in a statistical model for the outcome of interest. This approach is potentially problematic because i) individual significance tests can be overly sensitive, particularly when sample sizes are large; and ii) standard significance tests fail to distinguish between markers that are likely, on biological grounds, to have an effect, and those that are not. One way to address these concerns is to perform Bayesian hypothesis tests (Berger 1985; Kass and Raftery 1995), which are typically more conservative than standard uncorrected frequentist tests, less conservative than multiplicity-corrected tests, and make explicit use of relevant biological information through specification of the prior distribution. In this article we use a Bayesian testing approach to screen a panel of genetic markers recorded in a randomized clinical trial of bupropion versus placebo for smoking cessation. From a panel of 59 single-nucleotide polymorphisms (SNPs) located on 11 candidate genes, we identify four SNPs (one each on CHRNA5 and CHRNA2 and two on CHAT) that appear to have pharmacogenetic relevance. Of these, the SNP on CHRNA5 is most robust to specification of the prior. An unadjusted frequentist test identifies seven SNPs, including these four, none of which remains significant upon correction for multiplicity. In a panel of 43 randomly selected control SNPs, none is significant by either the Bayesian or the corrected frequentist test.
PMCID: PMC2655206  PMID: 18165968
Bayes factor; Bayesian hypothesis test; bupropion; importance sampling; pharmacogenomics; single-nucleotide polymorphism

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