Bayesian shrinkage approach for a joint model of longitudinal and survival outcomes assuming different association structures

The joint modeling of longitudinal and survival data has recently received much attention. Several extensions of the standard joint model that consists of one longitudinal and one survival outcome have been proposed including the use of different association structures between the longitudinal and the survival outcomes. However, in general, relatively little attention has been given to the selection of the most appropriate functional form to link the two outcomes. In common practice, it is assumed that the underlying value of the longitudinal outcome is associated with the survival outcome. However, it could be that different characteristics of the patients' longitudinal profiles influence the hazard. For example, not only the current value but also the slope or the area under the curve of the longitudinal outcome. The choice of which functional form to use is an important decision that needs to be investigated because it could influence the results. In this paper, we use a Bayesian shrinkage approach in order to determine the most appropriate functional forms. We propose a joint model that includes different association structures of different biomarkers and assume informative priors for the regression coefficients that correspond to the terms of the longitudinal process. Specifically, we assume Bayesian lasso, Bayesian ridge, Bayesian elastic net, and horseshoe. These methods are applied to a dataset consisting of patients with a chronic liver disease, where it is important to investigate which characteristics of the biomarkers have an influence on survival. Copyright © 2016 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.     

SNP Prioritization Using a Bayesian Probability of Association

Prioritization is the process whereby a set of possible candidate genes or SNPs is ranked so that the most promising can be taken forward into further studies. In a genome‐wide association study, prioritization is usually based on the P‐values alone, but researchers sometimes take account of external annotation information about the SNPs such as whether the SNP lies close to a good candidate gene. Using external information in this way is inherently subjective and is often not formalized, making the analysis difficult to reproduce. Building on previous work that has identified 14 important types of external information, we present an approximate Bayesian analysis that produces an estimate of the probability of association. The calculation combines four sources of information: the genome‐wide data, SNP information derived from bioinformatics databases, empirical SNP weights, and the researchers’ subjective prior opinions. The calculation is fast enough that it can be applied to millions of SNPS and although it does rely on subjective judgments, those judgments are made explicit so that the final SNP selection can be reproduced. We show that the resulting probability of association is intuitively more appealing than the P‐value because it is easier to interpret and it makes allowance for the power of the study. We illustrate the use of the probability of association for SNP prioritization by applying it to a meta‐analysis of kidney function genome‐wide association studies and demonstrate that SNP selection performs better using the probability of association compared with P‐values alone.     

Integrative variable selection via Bayesian model uncertainty

We are interested in developing integrative approaches for variable selection problems that incorporate external knowledge on a set of predictors of interest. In particular, we have developed an integrative Bayesian model uncertainty (iBMU) method, which formally incorporates multiple sources of data via a second‐stage probit model on the probability that any predictor is associated with the outcome of interest. Using simulations, we demonstrate that iBMU leads to an increase in power to detect true marginal associations over more commonly used variable selection techniques, such as least absolute shrinkage and selection operator and elastic net. In addition, iBMU leads to a more efficient model search algorithm over the basic BMU method even when the predictor‐level covariates are only modestly informative. The increase in power and efficiency of our method becomes more substantial as the predictor‐level covariates become more informative. Finally, we demonstrate the power and flexibility of iBMU for integrating both gene structure and functional biomarker information into a candidate gene study investigating over 50 genes in the brain reward system and their role with smoking cessation from the Pharmacogenetics of Nicotine Addiction and Treatment Consortium. Copyright © 2013 John Wiley & Sons, Ltd.     

Country specific cost comparisons from multinational clinical trials using empirical Bayesian shrinkage estimation: the Canadian ASSENT-3 economic analysis

The growing number of multinational clinical trials in which patient-level health care resource data are collected have raised the issue of which is the best approach for making inference for individual countries with respect to the between-treatment difference in mean cost. We describe and discuss the relative merits of three approaches. The first uses the random effects pooled estimate from all countries to estimate the difference for any particular country. The second approach estimates the difference using only the data from the specific country in question. Using empirical Bayes estimation a third approach estimates the country-specific difference using a variance-weighted linear sum of the estimates provided by the other two approaches. The approaches are illustrated and compared using the data from the ASSENT-3 trial.     

A variance components factor model for genetic association studies: A Bayesian analysis

Studies of gene‐trait associations for complex diseases often involve multiple traits that may vary by genotype groups or patterns. Such traits are usually manifestations of lower‐dimensional latent factors or disease syndromes. We illustrate the use of a variance components factor (VCF) model to model the association between multiple traits and genotype groups as well as any other existing patient‐level covariates. This model characterizes the correlations between traits as underlying latent factors that can be used in clinical decision‐making. We apply it within the Bayesian framework and provide a straightforward implementation using the WinBUGS software. The VCF model is illustrated with simulated data and an example that comprises changes in plasma lipid measurements of patients who were treated with statins to lower low‐density lipoprotein cholesterol, and polymorphisms from the apolipoprotein‐E gene. The simulation shows that this model clearly characterizes existing multiple trait manifestations across genotype groups where individuals' group assignments are fully observed or can be deduced from the observed data. It also allows one to investigate covariate by genotype group interactions that may explain the variability in the traits. The flexibility to characterize such multiple trait manifestations makes the VCF model more desirable than the univariate variance components model, which is applied to each trait separately. The Bayesian framework offers a flexible approach that allows one to incorporate prior information. Genet. Epidemiol. 34: 529–536, 2010. © 2010 Wiley‐Liss, Inc.     

Incorporating model uncertainty in detecting rare variants: the Bayesian risk index

We are interested in investigating the involvement of multiple rare variants within a given region by conducting analyses of individual regions with two goals: (1) to determine if regional rare variation in aggregate is associated with risk; and (2) conditional upon the region being associated, to identify specific genetic variants within the region that are driving the association. In particular, we seek a formal integrated analysis that achieves both of our goals. For rare variants with low minor allele frequencies, there is very little power to statistically test the null hypothesis of equal allele or genotype counts for each variant. Thus, genetic association studies are often limited to detecting association within a subset of the common genetic markers. However, it is very likely that associations exist for the rare variants that may not be captured by the set of common markers. Our framework aims at constructing a risk index based on multiple rare variants within a region. Our analytical strategy is novel in that we use a Bayesian approach to incorporate model uncertainty in the selection of variants to include in the index as well as the direction of the associated effects. Additionally, the approach allows for inference at both the group and variant-specific levels. Using a set of simulations, we show that our methodology has added power over other popular rare variant methods to detect global associations. In addition, we apply the approach to sequence data from the WECARE Study of second primary breast cancers.     

A spatial probit model for fine-scale mapping of disease genes

We present a novel statistical method for linkage disequilibrium (LD) mapping of disease susceptibility loci in case-control studies. Such studies exploit the statistical correlation or LD that exist between variants physically close along the genome to identify those that correlate with disease status and might thus be close to a causative mutation, generally assumed unobserved. LD structure, however, varies markedly over short distances because of variation in local recombination rates, mutation and genetic drift among other factors. We propose a Bayesian multivariate probit model that flexibly accounts for the local spatial correlation between markers. In a case-control setting, we use a retrospective model that properly reflects the sampling scheme and identify regions where single- or multi-locus marker frequencies differ across cases and controls. We formally quantify these differences using information-theoretic distance measures while the fully Bayesian approach naturally accommodates unphased or missing genotype data. We demonstrate our approach on simulated data and on real data from the CYP2D6 region that has a confirmed role in drug metabolism.     

A Bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time-to-event

Motivated by a real data example on renal graft failure, we propose a new semiparametric multivariate joint model that relates multiple longitudinal outcomes to a time-to-event. To allow for greater flexibility, key components of the model are modelled nonparametrically. In particular, for the subject-specific longitudinal evolutions we use a spline-based approach, the baseline risk function is assumed piecewise constant, and the distribution of the latent terms is modelled using a Dirichlet Process prior formulation. Additionally, we discuss the choice of a suitable parameterization, from a practitioner's point of view, to relate the longitudinal process to the survival outcome. Specifically, we present three main families of parameterizations, discuss their features, and present tools to choose between them.     

Bayesian Latent Variable Collapsing Model for Detecting Rare Variant Interaction Effect in Twin Study

By analyzing more next‐generation sequencing data, researchers have affirmed that rare genetic variants are widespread among populations and likely play an important role in complex phenotypes. Recently, a handful of statistical models have been developed to analyze rare variant (RV) association in different study designs. However, due to the scarce occurrence of minor alleles in data, appropriate statistical methods for detecting RV interaction effects are still difficult to develop. We propose a hierarchical Bayesian latent variable collapsing method (BLVCM), which circumvents the obstacles by parameterizing the signals of RVs with latent variables in a Bayesian framework and is parameterized for twin data. The BLVCM can tackle nonassociated variants, allow both protective and deleterious effects, capture SNP‐SNP synergistic effect, provide estimates for the gene level and individual SNP contributions, and can be applied to both independent and various twin designs. We assessed the statistical properties of the BLVCM using simulated data, and found that it achieved better performance in terms of power for interaction effect detection compared to the Granvil and the SKAT. As proof of practical application, the BLVCM was then applied to a twin study analysis of more than 20,000 gene regions to identify significant RVs associated with low‐density lipoprotein cholesterol level. The results show that some of the findings are consistent with previous studies, and we identified some novel gene regions with significant SNP–SNP synergistic effects.     

Bayesian methods for the design and interpretation of clinical trials in very rare diseases

This paper considers the design and interpretation of clinical trials comparing treatments for conditions so rare that worldwide recruitment efforts are likely to yield total sample sizes of 50 or fewer, even when patients are recruited over several years. For such studies, the sample size needed to meet a conventional frequentist power requirement is clearly infeasible. Rather, the expectation of any such trial has to be limited to the generation of an improved understanding of treatment options. We propose a Bayesian approach for the conduct of rare‐disease trials comparing an experimental treatment with a control where patient responses are classified as a success or failure. A systematic elicitation from clinicians of their beliefs concerning treatment efficacy is used to establish Bayesian priors for unknown model parameters. The process of determining the prior is described, including the possibility of formally considering results from related trials. As sample sizes are small, it is possible to compute all possible posterior distributions of the two success rates. A number of allocation ratios between the two treatment groups can be considered with a view to maximising the prior probability that the trial concludes recommending the new treatment when in fact it is non‐inferior to control. Consideration of the extent to which opinion can be changed, even by data from the best feasible design, can help to determine whether such a trial is worthwhile. © 2014 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Estimating the effects of copy-number variants on intelligence using hierarchical Bayesian models

It is challenging to estimate the phenotypic impact of the structural genome changes known as copy-number variations (CNVs), since there are many unique CNVs which are nonrecurrent, and most are too rare to be studied individually. In recent work, we found that CNV-aggregated genomic annotations, that is, specifically the intolerance to mutation as measured by the pLI score (probability of being loss-of-function intolerant), can be strong predictors of intellectual quotient (IQ) loss. However, this aggregation method only estimates the individual CNV effects indirectly. Here, we propose the use of hierarchical Bayesian models to directly estimate individual effects of rare CNVs on measures of intelligence. Annotation information on the impact of major mutations in genomic regions is extracted from genomic databases and used to define prior information for the approach we call HBIQ. We applied HBIQ to the analysis of CNV deletions and duplications from three datasets and identified several genomic regions containing CNVs demonstrating significant deleterious effects on IQ, some of which validate previously known associations. We also show that several CNVs were identified as deleterious by HBIQ even if they have a zero pLI score, and the converse is also true. Furthermore, we show that our new model yields higher out-of-sample concordance (78%) for predicting the consequences of carrying known recurrent CNVs compared with our previous approach.     

Direct analysis of unphased SNP genotype data in population-based association studies via Bayesian partition modelling of haplotypes

We describe a novel method for assessing the strength of disease association with single nucleotide polymorphisms (SNPs) in a candidate gene or small candidate region, and for estimating the corresponding haplotype relative risks of disease, using unphased genotype data directly. We begin by estimating the relative frequencies of haplotypes consistent with observed SNP genotypes. Under the Bayesian partition model, we specify cluster centres from this set of consistent SNP haplotypes. The remaining haplotypes are then assigned to the cluster with the "nearest" centre, where distance is defined in terms of SNP allele matches. Within a logistic regression modelling framework, each haplotype within a cluster is assigned the same disease risk, reducing the number of parameters required. Uncertainty in phase assignment is addressed by considering all possible haplotype configurations consistent with each unphased genotype, weighted in the logistic regression likelihood by their probabilities, calculated according to the estimated relative haplotype frequencies. We develop a Markov chain Monte Carlo algorithm to sample over the space of haplotype clusters and corresponding disease risks, allowing for covariates that might include environmental risk factors or polygenic effects. Application of the algorithm to SNP genotype data in an 890-kb region flanking the CYP2D6 gene illustrates that we can identify clusters of haplotypes with similar risk of poor drug metaboliser (PDM) phenotype, and can distinguish PDM cases carrying different high-risk variants. Further, the results of a detailed simulation study suggest that we can identify positive evidence of association for moderate relative disease risks with a sample of 1,000 cases and 1,000 controls.     

Objective Bayesian model selection for Cox regression

There is now a large literature on objective Bayesian model selection in the linear model based on the g‐prior. The methodology has been recently extended to generalized linear models using test‐based Bayes factors. In this paper, we show that test‐based Bayes factors can also be applied to the Cox proportional hazards model. If the goal is to select a single model, then both the maximum a posteriori and the median probability model can be calculated. For clinical prediction of survival, we shrink the model‐specific log hazard ratio estimates with subsequent calculation of the Breslow estimate of the cumulative baseline hazard function. A Bayesian model average can also be employed. We illustrate the proposed methodology with the analysis of survival data on primary biliary cirrhosis patients and the development of a clinical prediction model for future cardiovascular events based on data from the Second Manifestations of ARTerial disease (SMART) cohort study. Cross‐validation is applied to compare the predictive performance with alternative model selection approaches based on Harrell's c‐Index, the calibration slope and the integrated Brier score. Finally, a novel application of Bayesian variable selection to optimal conditional prediction via landmarking is described. Copyright © 2016 John Wiley & Sons, Ltd.     

Semiparametric Bayesian modeling of random genetic effects in family-based association studies

We consider the inference problem of estimating covariate and genetic effects in a family-based case-control study where families are ascertained on the basis of the number of cases within the family. However, our interest lies not only in estimating the fixed covariate effects but also in estimating the random effects parameters that account for varying correlations among family members. These random effects parameters, though weakly identifiable in a strict theoretical sense, are often hard to estimate due to the small number of observations per family. A hierarchical Bayesian paradigm is a very natural route in this context with multiple advantages compared with a classical mixed effects estimation strategy based on the integrated likelihood. We propose a fully flexible Bayesian approach allowing nonparametric modeling of the random effects distribution using a Dirichlet process prior and provide estimation of both fixed effect and random effects parameters using a Markov chain Monte Carlo numerical integration scheme. The nonparametric Bayesian approach not only provides inference that is less sensitive to parametric specification of the random effects distribution but also allows possible uncertainty around a specific genetic correlation structure. The Bayesian approach has certain computational advantages over its mixed-model counterparts. Data from the Prostate Cancer Genetics Project, a family-based study at the University of Michigan Comprehensive Cancer Center including families having one or more members with prostate cancer, are used to illustrate the proposed methods. A small-scale simulation study is carried out to compare the proposed nonparametric Bayes methodology with a parametric Bayesian alternative.     

Incorporating model uncertainty in cost-effectiveness analysis: a Bayesian model averaging approach

Recently, several authors have proposed the use of linear regression models in cost-effectiveness analysis. In this paper, by modelling costs and outcomes using patient and Health Centre covariates, we seek to identify the part of the cost or outcome difference that is not attributable to the treatment itself, but to the patients' condition or to characteristics of the Centres. Selection of the covariates to be included as predictors of effectiveness and cost is usually assumed by the researcher. This behaviour ignores the uncertainty associated with model selection and leads to underestimation of the uncertainty about quantities of interest. We propose the use of Bayesian model averaging as a mechanism to account for such uncertainty about the model. Data from a clinical trial are used to analyze the effect of incorporating model uncertainty, by comparing two highly active antiretroviral treatments applied to asymptomatic HIV patients. The joint posterior density of incremental effectiveness and cost and cost-effectiveness acceptability curves are proposed as decision-making measures.     

A Bayesian fixed effects analysis of the Mantel-Haenszel model applied to meta-analysis

When performing a meta analysis, it is often necessary to combine results from several 2 x 2 contingency tables. The Mantel-Haenszel model assumes a common measure of association between the treatment and outcome variables across the tables. A Bayesian method is described for drawing inferences regarding the measure of association, for checking the plausibility of the Mantel-Haenszel model, and for drawing inferences regarding the success rates for the individual studies. While the methodology is readily extendable to random effects models, a fixed effects approach avoids the complex statistical modelling of a mixture distribution which is required for the good application of random effects models.     

Bayesian random threshold estimation in a Cox proportional hazards cure model

In this paper, we develop a Bayesian approach to estimate a Cox proportional hazards model that allows a threshold in the regression coefficient, when some fraction of subjects are not susceptible to the event of interest. A data augmentation scheme with latent binary cure indicators is adopted to simplify the Markov chain Monte Carlo implementation. Given the binary cure indicators, the Cox cure model reduces to a standard Cox model and a logistic regression model. Furthermore, the threshold detection problem reverts to a threshold problem in a regular Cox model. The baseline cumulative hazard for the Cox model is formulated non‐parametrically using counting processes with a gamma process prior. Simulation studies demonstrate that the method provides accurate point and interval estimates. Application to a data set of oropharynx cancer patients suggests a significant threshold in age at diagnosis such that the effect of gender on disease‐specific survival changes after the threshold. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian analysis of multilocus association in quantitative and qualitative traits

A Bayesian model-based method for multilocus association analysis of quantitative and qualitative (binary) traits is presented. The method selects a trait-associated subset of markers among candidates, and is equally applicable for analyzing wide chromosomal segments (genome scans) and small candidate regions. The method can be applied in situations involving missing genotype data. The number of trait loci, their marker positions, and the magnitudes of their gene effects (strengths of association) are all estimated simultaneously. The inference of parameters is based on their posterior distributions, which are obtained through Markov chain Monte Carlo simulations. The strengths of the approach are: 1) flexible use of oligogenic models with unknown number of loci, 2) performing the estimation of association jointly with model selection, and 3) avoidance of the multiple testing problem, which typically complicates the approaches based on association testing. The performance of the method was tested and compared to the multilocus conditional search procedure by analyzing two simulated data sets. We also applied the method to cystic fibrosis haplotype data (two-locus haplotypes), where gene position has already been identified. The method is implemented as a software package, which is freely available for research purposes under the name BAMA.     

层次贝叶斯模型在校正低估以估计慢性病患病率中的应用

目的 探讨层次贝叶斯模型中在慢性病患病率估计研究中的应用,以校正低估的计数数据从而获得潜在的真实患病率。 方法 基于真实的患者人数服从Poisson分布,建档登记的患者人数服从二项分布的假设,考虑与疾病患病水平及建档过程相关的影响因素,建立层次贝叶斯模型对某地17个县(市)2018—2020年建档登记的高血压患者人数进行校正,并对影响患病及建档过程的相关因素的效应进行了估计。 结果 校正之后该地总的高血压患病率为25.62%(95%CI: 22.66%～28.58%),而建档登记的患病率为10.85%。在该地的17个县(市)中,建档率最高为17.70%(95%CI: 14.65%～20.74%),最低为12.96%(95%CI: 8.09%～17.82%)。老龄化率与高血压患病率呈现正相关关系。城镇化率、当地老龄人口的规范化管理率、已建档高血压患者的规范化管理率与高血压患者建档率呈正相关关系。 结论 层次贝叶斯模型在校正低估数据,估计真实的患病率中效果优良,在校正低估从而获得真实的患病人数方面具有潜在的应用价值。