eQuIPS: eQTL Analysis Using Informed Partitioning of SNPs – A Fully Bayesian Approach

We develop a Bayesian multi‐SNP Markov chain Monte Carlo approach that allows published functional significance scores to objectively inform single nucleotide polymorphism (SNP) prior effect sizes in expression quantitative trait locus (eQTL) studies. We developed the Normal Gamma prior to allow the inclusion of functional information. We partition SNPs into predefined functional groups and select prior distributions that fit the group‐specific observed functional significance scores. We test our method on two simulated datasets and previously analysed human eQTL data containing validated causal SNPs. In our simulations the modified Normal Gamma always performs at least as well, and generally outperforms, the other methods considered. When analysing the human eQTL data, we placed all SNPs into their actual functional group. The ranks of the four validated causal SNPs analysed using the modified Normal Gamma increase dramatically compared to those of the other methods considered. Using our new method, three of the four validated SNPs are ranked in the top 1% of SNPs and the other is in the top 2%. For the standard Normal Gamma, the best of the other methods, the four validated SNPs had ranks in the top 1%, 4%, 20% and 59%. Crucially these substantive improvements in the ranks make it highly likely that most, if not all, of these validated SNPs would have been flagged for follow‐up using our new method, whereas at least two of them would certainly not have been using the current approaches.     

Incorporating Functional Genomic Information in Genetic Association Studies Using an Empirical Bayes Approach

There is a large amount of functional genetic data available, which can be used to inform fine‐mapping association studies (in diseases with well‐characterised disease pathways). Single nucleotide polymorphism (SNP) prioritization via Bayes factors is attractive because prior information can inform the effect size or the prior probability of causal association. This approach requires the specification of the effect size. If the information needed to estimate a priori the probability density for the effect sizes for causal SNPs in a genomic region isn't consistent or isn't available, then specifying a prior variance for the effect sizes is challenging. We propose both an empirical method to estimate this prior variance, and a coherent approach to using SNP‐level functional data, to inform the prior probability of causal association. Through simulation we show that when ranking SNPs by our empirical Bayes factor in a fine‐mapping study, the causal SNP rank is generally as high or higher than the rank using Bayes factors with other plausible values of the prior variance. Importantly, we also show that assigning SNP‐specific prior probabilities of association based on expert prior functional knowledge of the disease mechanism can lead to improved causal SNPs ranks compared to ranking with identical prior probabilities of association. We demonstrate the use of our methods by applying the methods to the fine mapping of the CASP8 region of chromosome 2 using genotype data from the Collaborative Oncological Gene‐Environment Study (COGS) Consortium. The data we analysed included approximately 46,000 breast cancer case and 43,000 healthy control samples.     

Using GWAS top hits to inform priors in Bayesian fine-mapping association studies

The default causal single-nucleotide polymorphism (SNP) effect size prior in Bayesian fine-mapping studies is usually the Normal distribution. This choice is often based on computational convenience, rather than evidence that it is the most suitable prior distribution. The choice of prior is important because previous studies have shown considerable sensitivity of causal SNP Bayes factors to the form of the prior. In some well-studied diseases there are now considerable numbers of genome-wide association study (GWAS) top hits along with estimates of the number of yet-to-be-discovered causal SNPs. We show how the effect sizes of the top hits and estimates of the number of yet-to-be-discovered causal SNPs can be used to choose between the Laplace and Normal priors, to estimate the prior parameters and to quantify the uncertainty in this estimation. The methodology can readily be applied to other priors. We show that the top hits available from breast cancer GWAS provide overwhelming support for the Laplace over the Normal prior, which has important consequences for variant prioritisation. This work in this paper enables practitioners to derive more objective priors than are currently being used and could lead to prioritisation of different variants.     

Bayesian methods for highly correlated exposure data

Studies that include individuals with multiple highly correlated exposures are common in epidemiology. Because standard maximum likelihood techniques often fail to converge in such instances, hierarchical regression methods have seen increasing use. Bayesian hierarchical regression places prior distributions on exposure-specific regression coefficients to stabilize estimation and incorporate prior knowledge, if available. A common parametric approach in epidemiology is to treat the prior mean and variance as fixed constants. An alternative parametric approach is to place distributions on the prior mean and variance to allow the data to help inform their values. As a more flexible semiparametric option, one can place an unknown distribution on the coefficients that simultaneously clusters exposures into groups using a Dirichlet process prior. We also present a semiparametric model with a variable-selection prior to allow clustering of coefficients at 0. We compare these 4 hierarchical regression methods and demonstrate their application in an example estimating the association of herbicides with retinal degeneration among wives of pesticide applicators.     

Bayesian multivariant fine mapping using the Laplace prior

Currently, the only effect size prior that is routinely implemented in a Bayesian fine-mapping multi-single-nucleotide polymorphism (SNP) analysis is the Gaussian prior. Here, we show how the Laplace prior can be deployed in Bayesian multi-SNP fine mapping studies. We compare the ranking performance of the posterior inclusion probability (PIP) using a Laplace prior with the ranking performance of the corresponding Gaussian prior and FINEMAP. Our results indicate that, for the simulation scenarios we consider here, the Laplace prior can lead to higher PIPs than either the Gaussian prior or FINEMAP, particularly for moderately sized fine-mapping studies. The Laplace prior also appears to have better worst-case scenario properties. We reanalyse the iCOGS case–control data from the CASP8 region on Chromosome 2. Even though this study has a total sample size of nearly 90,000 individuals, there are still some differences in the top few ranked SNPs if the Laplace prior is used rather than the Gaussian prior. R code to implement the Laplace (and Gaussian) prior is available at https://github.com/Kevin-walters/lapmapr .     

Bayesian perspectives for epidemiological research. II. Regression analysis

This article describes extensions of the basic Bayesian methods using data priors to regression modelling, including hierarchical (multilevel) models. These methods provide an alternative to the parsimony-oriented approach of frequentist regression analysis. In particular, they replace arbitrary variable-selection criteria by prior distributions, and by doing so facilitate realistic use of imprecise but important prior information. They also allow Bayesian analyses to be conducted using standard regression packages; one need only be able to add variables and records to the data set. The methods thus facilitate the use of Bayesian solutions to problems of sparse data, multiple comparisons, subgroup analyses and study bias. Because these solutions have a frequentist interpretation as "shrinkage" (penalized) estimators, the methods can also be viewed as a means of implementing shrinkage approaches to multiparameter problems.     

Robust bayesian methods for monitoring clinical trials

Bayesian methods for the analysis of clinical trials data have received increasing attention recently as they offer an approach for dealing with difficult problems that arise in practice. A major criticism of the Bayesian approach, however, has focused on the need to specify a single, often subjective, prior distribution for the parameters of interest. In an attempt to address this critism, we describe methods for assessing the robustness of the posterior distribution to the specification of the prior. The robust Bayesian approach to data analysis replaces the prior distribution with a class of prior distributions and investigations and investigates how the inferences might change as the prior varies over this class. The purpose of this paper is to illustrate the application of robust Bayesian methods to the analysis of clinical trials data. Using two examples of clinical trials taken from the literature, we illustrate how to use these methods to help a data monitoring committee decide whether or not to stop a trial early.     

Bayesian methods of analysis for cluster randomized trials with binary outcome data

We explore the potential of Bayesian hierarchical modelling for the analysis of cluster randomized trials with binary outcome data, and apply the methods to a trial randomized by general practice. An approximate relationship is derived between the intracluster correlation coefficient (ICC) and the between-cluster variance used in a hierarchical logistic regression model. By constructing an informative prior for the ICC on the basis of available information, we are thus able implicitly to specify an informative prior for the between-cluster variance. The approach also provides us with a credible interval for the ICC for binary outcome data. Several approaches to constructing informative priors from empirical ICC values are described. We investigate the sensitivity of results to the prior specified and find that the estimate of intervention effect changes very little in this data set, while its interval estimate is more sensitive. The Bayesian approach allows us to assume distributions other than normality for the random effects used to model the clustering. This enables us to gain insight into the robustness of our parameter estimates to the classical normality assumption. In a model with a more complex variance structure, Bayesian methods can provide credible intervals for a difference between two variance components, in order for example to investigate whether the effect of intervention varies across clusters. We compare our results with those obtained from classical estimation, discuss the relative merits of the Bayesian framework, and conclude that the flexibility of the Bayesian approach offers some substantial advantages, although selection of prior distributions is not straightforward.     

Predictive distributions for between‐study heterogeneity and simple methods for their application in Bayesian meta‐analysis

Numerous meta‐analyses in healthcare research combine results from only a small number of studies, for which the variance representing between‐study heterogeneity is estimated imprecisely. A Bayesian approach to estimation allows external evidence on the expected magnitude of heterogeneity to be incorporated. The aim of this paper is to provide tools that improve the accessibility of Bayesian meta‐analysis. We present two methods for implementing Bayesian meta‐analysis, using numerical integration and importance sampling techniques. Based on 14 886 binary outcome meta‐analyses in the Cochrane Database of Systematic Reviews, we derive a novel set of predictive distributions for the degree of heterogeneity expected in 80 settings depending on the outcomes assessed and comparisons made. These can be used as prior distributions for heterogeneity in future meta‐analyses. The two methods are implemented in R, for which code is provided. Both methods produce equivalent results to standard but more complex Markov chain Monte Carlo approaches. The priors are derived as log‐normal distributions for the between‐study variance, applicable to meta‐analyses of binary outcomes on the log odds‐ratio scale. The methods are applied to two example meta‐analyses, incorporating the relevant predictive distributions as prior distributions for between‐study heterogeneity. We have provided resources to facilitate Bayesian meta‐analysis, in a form accessible to applied researchers, which allow relevant prior information on the degree of heterogeneity to be incorporated. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Bayesian mixture models for the incorporation of prior knowledge to inform genetic association studies

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 untangle 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 the use of this method through a genome‐wide linkage study of colorectal cancer, and a genome‐wide association study of colorectal polyps. Genet. Epidemiol. 34:418–426, 2010. © 2010 Wiley‐Liss, Inc.     

COMPARING METHODS FOR CALCULATING CONFIDENCE INTERVALS FOR VACCINE EFFICACY

A method is introduced for computing a Bayesian 95 per cent posterior probability region for vaccine efficacy. This method assumes independent vague gamma prior distributions for the incidence rates on each arm of the trial, and a Poisson likelihood for the counts of incident cases of infection. The approach is similar in spirit to the Bayesian analysis of the binomial risk ratio described by Aitchison and Bacon-Shone. However, the focus of our interest is not on incorporating prior information into the design of trials for efficacy, but rather on evaluating whether or not the Bayesian approach with vague prior information produces comparable results to a frequentist approach. A review of methods for constructing exact and large sample intervals for vaccine efficacy is provided as a framework for comparison. The confidence interval methods are assessed by comparing the size and power of tests of vaccine efficacy in proposed intermediate sized randomized double blinded placebo controlled trials.     

Incorporation of genuine prior information in cost-effectiveness analysis of clinical trial data

The Bayesian approach to statistics has been growing rapidly in popularity as an alternative to the frequentist approach in the appraisal of healthcare technologies in clinical trials. Bayesian methods have significant advantages over classical frequentist statistical methods and the presentation of evidence to decision makers. A fundamental feature of a Bayesian analysis is the use of prior information as well as the clinical trial data in the final analysis. However, the incorporation of prior information remains a controversial subject that provides a potential barrier to the acceptance of practical uses of Bayesian methods. The purpose of this paper is to stimulate a debate on the use of prior information in evidence submitted to decision makers. We discuss the advantages of incorporating genuine prior information in cost-effectiveness analyses of clinical trial data and explore mechanisms to safeguard scientific rigor in the use of such prior information.     

Commentary: practical advantages of Bayesian analysis of epidemiologic data.

In the past decade, there have been enormous advances in the use of Bayesian methodology for analysis of epidemiologic data, and there are now many practical advantages to the Bayesian approach. Bayesian models can easily accommodate unobserved variables such as an individual's true disease status in the presence of diagnostic error. The use of prior probability distributions represents a powerful mechanism for incorporating information from previous studies and for controlling confounding. Posterior probabilities can be used as easily interpretable alternatives to p values. Recent developments in Markov chain Monte Carlo methodology facilitate the implementation of Bayesian analyses of complex data sets containing missing observations and multidimensional outcomes. Tools are now available that allow epidemiologists to take advantage of this powerful approach to assessment of exposure-disease relations.     

On the choice of parameterisation and priors for the Bayesian analyses of Mendelian randomisation studies

Mendelian randomisation is a form of instrumental variable analysis that estimates the causal effect of an intermediate phenotype or exposure on an outcome or disease in the presence of unobserved confounding, using a genetic variant as the instrument. A Bayesian approach allows current knowledge to be incorporated into the analysis in the form of informative prior distributions, and the unobserved confounder can be modelled explicitly. We consider Bayesian methods for Mendelian randomisation in the case where all relationships are linear and there are no interactions. A 'full' model in which the unobserved confounder is included explicitly is not completely identifiable, although the causal parameter can be estimated. We compare inferences from this general but non-identified model with a reduced parameter model that is identifiable. We show that, theoretically, additional information about the causal parameter can be obtained by using the non-identifiable full model, rather than the identifiable reduced model, but that this is advantageous only when realistically informative priors are used and when the instrument is weak or the sample size is small. Furthermore, we consider the impact of using 'vague' versus 'informative' priors.     

Efficient computation of partial expected value of sample information using Bayesian approximation

We describe a novel process for transforming the efficiency of partial expected value of sample information (EVSI) computation in decision models. Traditional EVSI computation begins with Monte Carlo sampling to produce new simulated data-sets with a specified sample size. Each data-set is synthesised with prior information to give posterior distributions for model parameters, either via analytic formulae or a further Markov Chain Monte Carlo (MCMC) simulation. A further 'inner level' Monte Carlo sampling then quantifies the effect of the simulated data on the decision. This paper describes a novel form of Bayesian Laplace approximation, which can be replace both the Bayesian updating and the inner Monte Carlo sampling to compute the posterior expectation of a function. We compare the accuracy of EVSI estimates in two case study cost-effectiveness models using 1st and 2nd order versions of our approximation formula, the approximation of Tierney and Kadane, and traditional Monte Carlo. Computational efficiency gains depend on the complexity of the net benefit functions, the number of inner level Monte Carlo samples used, and the requirement or otherwise for MCMC methods to produce the posterior distributions. This methodology provides a new and valuable approach for EVSI computation in health economic decision models and potential wider benefits in many fields requiring Bayesian approximation.     

Case-control studies and Bayesian inference

We outline the methods of Bayesian inference for applications to case-control studies. These methods appear as the natural way of making inferences, since much of the controversy that surrounds a specific case-control study is subjective. We derive conjugate prior distributions of exposure, posterior distributions of the ratio of the odds of being incident with a disease both with and without exposure to a potential causal agent, and convenient approximations. In particular, we show how one may carry out 'case-control studies' without necessarily having a control group. We illustrate these ideas with the data that first showed the relationship between in utero exposure to diethylstilbestrol and cancer of the vagina in young girls.     

Bayesian estimation of the average treatment effect on the treated using inverse weighting

We develop a Bayesian approach to estimate the average treatment effect on the treated in the presence of confounding. The approach builds on developments proposed by Saarela et al in the context of marginal structural models, using importance sampling weights to adjust for confounding and estimate a causal effect. The Bayesian bootstrap is adopted to approximate posterior distributions of interest and avoid the issue of feedback that arises in Bayesian causal estimation relying on a joint likelihood. We present results from simulation studies to estimate the average treatment effect on the treated, evaluating the impact of sample size and the strength of confounding on estimation. We illustrate our approach using the classic Right Heart Catheterization data set and find a negative causal effect of the exposure on 30-day survival, in accordance with previous analyses of these data. We also apply our approach to the data set of the National Center for Health Statistics Birth Data and obtain a negative effect of maternal smoking during pregnancy on birth weight.     

A comparison of bayesian and frequentist approaches to incorporating external information for the prediction of prostate cancer risk

We present the most comprehensive comparison to date of the predictive benefit of genetics in addition to currently used clinical variables, using genotype data for 33 single‐nucleotide polymorphisms (SNPs) in 1,547 Caucasian men from the placebo arm of the REduction by DUtasteride of prostate Cancer Events (REDUCE®) trial. Moreover, we conducted a detailed comparison of three techniques for incorporating genetics into clinical risk prediction. The first method was a standard logistic regression model, which included separate terms for the clinical covariates and for each of the genetic markers. This approach ignores a substantial amount of external information concerning effect sizes for these Genome Wide Association Study (GWAS)‐replicated SNPs. The second and third methods investigated two possible approaches to incorporating meta‐analysed external SNP effect estimates – one via a weighted PCa 'risk' score based solely on the meta analysis estimates, and the other incorporating both the current and prior data via informative priors in a Bayesian logistic regression model. All methods demonstrated a slight improvement in predictive performance upon incorporation of genetics. The two methods that incorporated external information showed the greatest receiver‐operating‐characteristic AUCs increase from 0.61 to 0.64. The value of our methods comparison is likely to lie in observations of performance similarities, rather than difference, between three approaches of very different resource requirements. The two methods that included external information performed best, but only marginally despite substantial differences in complexity.     

Finding factors influencing risk: comparing Bayesian stochastic search and standard variable selection methods applied to logistic regression models of cases and controls

When modeling the risk of a disease, the very act of selecting the factors to be included can heavily impact the results. This study compares the performance of several variable selection techniques applied to logistic regression. We performed realistic simulation studies to compare five methods of variable selection: (1) a confidence interval (CI) approach for significant coefficients, (2) backward selection, (3) forward selection, (4) stepwise selection, and (5) Bayesian stochastic search variable selection (SSVS) using both informed and uniformed priors. We defined our simulated diseases mimicking odds ratios for cancer risk found in the literature for environmental factors, such as smoking; dietary risk factors, such as fiber; genetic risk factors, such as XPD; and interactions. We modeled the distribution of our covariates, including correlation, after the reported empirical distributions of these risk factors. We also used a null data set to calibrate the priors of the Bayesian method and evaluate its sensitivity. Of the standard methods (95 per cent CI, backward, forward, and stepwise selection) the CI approach resulted in the highest average per cent of correct associations and the lowest average per cent of incorrect associations. SSVS with an informed prior had a higher average per cent of correct associations and a lower average per cent of incorrect associations than the CI approach. This study shows that the Bayesian methods offer a way to use prior information to both increase power and decrease false-positive results when selecting factors to model complex disease risk.