Exploring heterogeneity in tumour data using Markov chain Monte Carlo

We describe a Bayesian approach to incorporate between-individual heterogeneity associated with parameters of complicated biological models. We emphasize the use of the Markov chain Monte Carlo (MCMC) method in this context and demonstrate the implementation and use of MCMC by analysis of simulated overdispersed Poisson counts and by analysis of an experimental data set on preneoplastic liver lesions (their number and sizes) in the presence of heterogeneity. These examples show that MCMC-based estimates, derived from the posterior distribution with uniform priors, may agree well with maximum likelihood estimates (if available). However, with heterogeneous parameters, maximum likelihood estimates can be difficult to obtain, involving many integrations. In this case, the MCMC method offers substantial computational advantages.     

Use of Bayesian Markov Chain Monte Carlo methods to model cost-of-illness data.

It is well known that the modeling of cost data is often problematic due to the distribution of such data. Commonly observed problems include 1) a strongly right-skewed data distribution and 2) a significant percentage of zero-cost observations. This article demonstrates how a hurdle model can be implemented from a Bayesian perspective by means of Markov Chain Monte Carlo simulation methods using the freely available software WinBUGS. Assessment of model fit is addressed through the implementation of two cross-validation methods. The relative merits of this Bayesian approach compared to the classical equivalent are discussed in detail. To illustrate the methods described, patient-specific non-health-care resource-use data from a prospective longitudinal study and the Norfolk Arthritis Register (NOAR) are utilized for 218 individuals with early inflammatory polyarthritis (IP). The NOAR database also includes information on various patient-level covariates.     

Cluster modelling of disease incidence via RJMCMC methods: a comparative evaluation. Reversible jump Markov chain Monte Carlo

The spatial modelling of small area health data has, for some time, included spatial autocorrelation as a random effect. This effect is non-specific and global and does not address the location of clusters of disease (a specific task). This paper addresses the need for specific and non-specific random effects within spatial epidemiology. In addition, individual frailty is also considered important and a computational algorithm based on reversible jump Markov chain Monte Carlo (RJMCMC) methods is described.     

Identifying influential observations in Bayesian models by using Markov chain Monte Carlo

In statistical modelling, it is often important to know how much parameter estimates are influenced by particular observations. An attractive approach is to re-estimate the parameters with each observation deleted in turn, but this is computationally demanding when fitting models by using Markov chain Monte Carlo (MCMC), as obtaining complete sample estimates is often in itself a very time-consuming task. Here we propose two efficient ways to approximate the case-deleted estimates by using output from MCMC estimation. Our first proposal, which directly approximates the usual influence statistics in maximum likelihood analyses of generalised linear models (GLMs), is easy to implement and avoids any further evaluation of the likelihood. Hence, unlike the existing alternatives, it does not become more computationally intensive as the model complexity increases. Our second proposal, which utilises model perturbations, also has this advantage and does not require the form of the GLM to be specified. We show how our two proposed methods are related and evaluate them against the existing method of importance sampling and case deletion in a logistic regression analysis with missing covariates. We also provide practical advice for those implementing our procedures, so that they may be used in many situations where MCMC is used to fit statistical models.     

Determining joint carrier probabilities of cancer-causing genes using Markov chain Monte Carlo methods

In genetic counseling for cancer risk, the probability of carrying a mutation of a cancer-causing gene plays an important role. Family history of various cancers is important in calculating this probability. BRCAPRO is a widely used software for calculating the probability of carrying mutations in BRCA1 and BRCA2 genes given the family history of breast and ovarian cancer in first- and second-degree relatives. BRCAPRO uses an analytical (exact) calculational procedure. Using Markov chain Monte Carlo (MCMC) methods, we extend BRCAPRO to handle, in principle, any type of cancer, family history, any number of genes and alleles that each gene may have. When the information used in this MCMC approach is the same as for BRCAPRO (two genes: BRCA1 and BRCA2; two cancers: breast and ovarian; first- and second-degree relatives only), the two approaches give essentially the same answer. Extending the model to include (1) prostate cancer, (2) two mutated alleles of BRCA2, namely, mutations in Ovarian Cancer Cluster Region (OCCR) and non-OCCR region, and (3) relatives of degree greater than second-degree, leads to different carrier probabilities. The MCMC approach is a useful tool in building a comprehensive model to give accurate estimates of carrier probabilities. Such an approach will be even more important as additional information about the genetics of various cancers becomes available.     

Markov chain Monte Carlo linkage analysis of a complex qualitative phenotype

We tested a new computer program, LOKI, that implements a reversible jump Markov chain Monte Carlo (MCMC) technique for segregation and linkage analysis. Our objective was to determine whether this software, designed for use with continuously distributed phenotypes, has any efficacy when applied to the discrete disease states of the simulated data from the Mordor data from GAW Problem 1. Although we were able to identify the genomic location for two of the three quantitative trait loci by repeated application of the software, the MCMC sampler experienced significant mixing problems indicating that the method, as currently formulated in LOKI, was not suitable for the discrete phenotypes in this data set.     

Dispersion of radionuclides released into a stable planetary boundary layer using a Monte Carlo model.

In this paper a Monte Carlo model for describing the atmospheric dispersion of radionuclides (represented by Lagrangian particles/neutral tracers) continuously released into a stable planetary boundary layer is presented. The effect of variation in release height and wind directional shear on plume dispersion is studied. The resultant plume concentration and dose rate at the ground is also calculated. The turbulent atmospheric parameters, like vertical profiles of fluctuating wind velocity components and eddy lifetime, were calculated using empirical relations for a stable atmosphere. The horizontal and vertical dispersion coefficients calculated by a numerical Lagrangian model are compared with the original and modified Pasquill-Gifford and Briggs empirical sigmas. The comparison shows that the Monte Carlo model can successfully predict dispersion in a stable atmosphere using the empirical turbulent parameters. The predicted ground concentration and dose rate contours indicate a significant increase in the affected area when wind shear is accounted for in the calculations.     

Markov chain Monte Carlo estimation of a multiparameter decision model: consistency of evidence and the accurate assessment of uncertainty.

Decision models are usually populated 1 parameter at a time, with 1 item of information informing each parameter. Often, however, data may not be available on the parameters themselves but on several functions of parameters, and there may be more items of information than there are parameters to be estimated. The authors show how in these circumstances all the model parameters can be estimated simultaneously using Bayesian Markov chain Monte Carlo methods. Consistency of the information and/or the adequacy of the model can also be assessed within this framework. Statistical evidence synthesis using all available data should result in more precise estimates of parameters and functions of parameters, and is compatible with the emphasis currently placed on systematic use of evidence. To illustrate this, WinBUGS software is used to estimate a simple 9-parameter model of the epidemiology of HIV in women attending prenatal clinics, using information on 12 functions of parameters, and to thereby compute the expected net benefit of 2 alternative prenatal testing strategies, universal testing and targeted testing of high-risk groups. The authors demonstrate improved precision of estimates, and lower estimates of the expected value of perfect information, resulting from the use of all available data.     

Changes in surrogate outcomes can be translated into clinical outcomes using a Monte Carlo model

ObjectiveTo meaningfully interpret trials using surrogate outcomes, one must translate changes in the surrogate outcome to changes in the clinical outcome. Formulae to do this are uncommon because they require primary data from multiple randomized trials that measure both the surrogate and clinical outcome.Study Design and SettingWe developed a model to translate changes in anticoagulation control (the surrogate outcome) into hemorrhagic and thromboembolic event rates (the clinical outcome). The model used Monte Carlo simulation and association measures between the surrogate and the clinical outcome from a meta-analysis. In randomized trials having interventions that improved anticoagulation control, we used the model to predict and statistically compare event rates between the study groups.ResultsSeven randomized trials found significantly improved anticoagulation control (mean increase in proportion of time in therapeutic range: 8.4%; range: 1.8–18%). These improvements in anticoagulation control translated to small decreases in hemorrhagic and thromboembolic events (mean: 0.66%/yr; range: 0.13–1.42%). These changes were never statistically significant.ConclusionMonte Carlo modeling can be used to translate surrogate outcomes into clinical outcomes. Statistically significant changes in anticoagulation control did not translate to significant differences in clinical outcomes. This methodology could be applied to other areas in medicine to assess surrogate outcomes.     

Effect of voxel size when calculating patient specific radionuclide dosimetry estimates using direct Monte Carlo simulation

The scalable XCAT voxelised phantom was used with the GATE Monte Carlo toolkit to investigate the effect of voxel size on dosimetry estimates of internally distributed radionuclide calculated using direct Monte Carlo simulation. A uniformly distributed Fluorine-18 source was simulated in the Kidneys of the XCAT phantom with the organ self dose (kidney ← kidney) and organ cross dose (liver ← kidney) being calculated for a number of organ and voxel sizes. Patient specific dose factors (DF) from a clinically acquired FDG PET/CT study have also been calculated for kidney self dose and liver ← kidney cross dose. Using the XCAT phantom it was found that significantly small voxel sizes are required to achieve accurate calculation of organ self dose. It has also been used to show that a voxel size of 2 mm or less is suitable for accurate calculations of organ cross dose. To compensate for insufficient voxel sampling a correction factor is proposed. This correction factor is applied to the patient specific dose factors calculated with the native voxel size of the PET/CT study.     

Ascertainment correction for Markov chain Monte Carlo segregation and linkage analysis of a quantitative trait

Although extended pedigrees are often sampled through probands with extreme levels of a quantitative trait, Markov chain Monte Carlo (MCMC) methods for segregation and linkage analysis have not been able to perform ascertainment corrections. Further, the extent to which ascertainment of pedigrees leads to biases in the estimation of segregation and linkage parameters has not been previously studied for MCMC procedures. In this paper, we studied these issues with a Bayesian MCMC approach for joint segregation and linkage analysis, as implemented in the package Loki. We first simulated pedigrees ascertained through individuals with extreme values of a quantitative trait in spirit of the sequential sampling theory of Cannings and Thompson [Cannings and Thompson [1977] Clin. Genet. 12:208-212]. Using our simulated data, we detected no bias in estimates of the trait locus location. However, in addition to allele frequencies, when the ascertainment threshold was higher than or close to the true value of the highest genotypic mean, bias was also found in the estimation of this parameter. When there were multiple trait loci, this bias destroyed the additivity of the effects of the trait loci, and caused biases in the estimation all genotypic means when a purely additive model was used for analyzing the data. To account for pedigree ascertainment with sequential sampling, we developed a Bayesian ascertainment approach and implemented Metropolis-Hastings updates in the MCMC samplers used in Loki. Ascertainment correction greatly reduced biases in parameter estimates. Our method is designed for multiple, but a fixed number of trait loci.     

Identifiability and convergence issues for Markov chain Monte Carlo fitting of spatial models

The marked increase in popularity of Bayesian methods in statistical practice over the last decade owes much to the simultaneous development of Markov chain Monte Carlo (MCMC) methods for the evaluation of requisite posterior distributions. However, along with this increase in computing power has come the temptation to fit models larger than the data can readily support, meaning that often the propriety of the posterior distributions for certain parameters depends on the propriety of the associated prior distributions. An important example arises in spatial modelling, wherein separate random effects for capturing unstructured heterogeneity and spatial clustering are of substantive interest, even though only their sum is well identified by the data. Increasing the informative content of the associated prior distributions offers an obvious remedy, but one that hampers parameter interpretability and may also significantly slow the convergence of the MCMC algorithm. In this paper we investigate the relationship among identifiability, Bayesian learning and MCMC convergence rates for a common class of spatial models, in order to provide guidance for prior selection and algorithm tuning. We are able to elucidate the key issues with relatively simple examples, and also illustrate the varying impacts of covariates, outliers and algorithm starting values on the resulting algorithms and posterior distributions.     

Two‐Locus Modeling of Asthma in a Hutterite Pedigree via Markov Chain Monte Carlo

Bayesian Markov chain Monte Carlo (MCMC) segregation analysis for asthma was performed on the whole 1,544‐member Hutterite pedigree. Heterogeneous and epistatic two‐locus models and complex one‐locus models were investigated, with trait loci postulated to be linked to markers in regions previously found to be possibly linked to asthma or atopy. The epistatic two‐locus dominant‐dominant model provided the best estimates, among the models investigated, in terms of prediction of population prevalence and relative risk for sibs of the affecteds. © 2001 Wiley‐Liss, Inc.     

A novel Bayesian continuous piecewise linear log-hazard model,with estimation and inference via reversible jump Markov chain Monte Carlo

We present a reversible jump Bayesian piecewise log-linear hazard model that extends the Bayesian piecewise exponential hazard to a continuous function of piecewise linear log hazards. A simulation study encompassing several different hazard shapes, accrual rates, censoring proportion, and sample sizes showed that the Bayesian piecewise linear log-hazard model estimated the true mean survival time and survival distributions better than the piecewsie exponential hazard. Survival data from Wake Forest Baptist Medical Center is analyzed by both methods and the posterior results are compared.     

Computational toxicology of chloroform: reverse dosimetry using Bayesian inference, Markov chain Monte Carlo simulation, and human biomonitoring data

BACKGROUND: One problem of interpreting population-based biomonitoring data is the reconstruction of corresponding external exposure in cases where no such data are available. OBJECTIVES: We demonstrate the use of a computational framework that integrates physiologically based pharmacokinetic (PBPK) modeling, Bayesian inference, and Markov chain Monte Carlo simulation to obtain a population estimate of environmental chloroform source concentrations consistent with human biomonitoring data. The biomonitoring data consist of chloroform blood concentrations measured as part of the Third National Health and Nutrition Examination Survey (NHANES III), and for which no corresponding exposure data were collected. METHODS: We used a combined PBPK and shower exposure model to consider several routes and sources of exposure: ingestion of tap water, inhalation of ambient household air, and inhalation and dermal absorption while showering. We determined posterior distributions for chloroform concentration in tap water and ambient household air using U.S. Environmental Protection Agency Total Exposure Assessment Methodology (TEAM) data as prior distributions for the Bayesian analysis. RESULTS: Posterior distributions for exposure indicate that 95% of the population represented by the NHANES III data had likely chloroform exposures < or = 67 microg/L [corrected] in tap water and < or = 0.02 microg/L in ambient household air. CONCLUSIONS: Our results demonstrate the application of computer simulation to aid in the interpretation of human biomonitoring data in the context of the exposure-health evaluation-risk assessment continuum. These results should be considered as a demonstration of the method and can be improved with the addition of more detailed data.     

A note on estimating the posterior density of a quantitative trait locus from a Markov chain Monte Carlo sample

We provide an overview of the use of kernel smoothing to summarize the quantitative trait locus posterior distribution from a Markov chain Monte Carlo sample. More traditional distributional summary statistics based on the histogram depend both on the bin width and on the sideway shift of the bin grid used. These factors influence both the overall mapping accuracy and the estimated location of the mode of the distribution. Replacing the histogram by kernel smoothing helps to alleviate these problems. Using simulated data, we performed numerical comparisons between the two approaches. The results clearly illustrate the superiority of the kernel method. The kernel approach is particularly efficient when one needs to point out the best putative quantitative trait locus position on the marker map. In such situations, the smoothness of the posterior estimate is especially important because rough posterior estimates easily produce biased mode estimates. Different kernel implementations are available from Rolf Nevanlinna Institute's web page (http://www.rni.helsinki.fi/;fjh).     

The hazard of starting smoking: estimates from a split population duration model

Understanding the determinants of the hazard of starting smoking is of great importance in developing policy to reduce the number of smokers. This paper develops a split population duration model of the decision to start smoking. Using data from the 1978 and 1979 Smoking Supplements to the National Health Interview Survey, we find some evidence that lifetime educational attainment, gender, and race are important determinants of both whether and when the smoking habit is initiated. The study finds no evidence that higher cigarette prices would have a significant impact on teenage decisions to pick up the smoking habit.     

GATE Monte Carlo simulation of dose distribution using MapReduce in a cloud computing environment

The GATE Monte Carlo simulation platform has good application prospects of treatment planning and quality assurance. However, accurate dose calculation using GATE is time consuming. The purpose of this study is to implement a novel cloud computing method for accurate GATE Monte Carlo simulation of dose distribution using MapReduce. An Amazon Machine Image installed with Hadoop and GATE is created to set up Hadoop clusters on Amazon Elastic Compute Cloud (EC2). Macros, the input files for GATE, are split into a number of self-contained sub-macros. Through Hadoop Streaming, the sub-macros are executed by GATE in Map tasks and the sub-results are aggregated into final outputs in Reduce tasks. As an evaluation, GATE simulations were performed in a cubical water phantom for X-ray photons of 6 and 18 MeV. The parallel simulation on the cloud computing platform is as accurate as the single-threaded simulation on a local server and the simulation correctness is not affected by the failure of some worker nodes. The cloud-based simulation time is approximately inversely proportional to the number of worker nodes. For the simulation of 10 million photons on a cluster with 64 worker nodes, time decreases of 41× and 32× were achieved compared to the single worker node case and the single-threaded case, respectively. The test of Hadoop’s fault tolerance showed that the simulation correctness was not affected by the failure of some worker nodes. The results verify that the proposed method provides a feasible cloud computing solution for GATE.     

Adaptive Markov chain Monte Carlo forward projection for statistical analysis in epidemic modelling of human papillomavirus

A Bayesian statistical model and estimation methodology based on forward projection adaptive Markov chain Monte Carlo is developed in order to perform the calibration of a high‐dimensional nonlinear system of ordinary differential equations representing an epidemic model for human papillomavirus types 6 and 11 (HPV‐6, HPV‐11). The model is compartmental and involves stratification by age, gender and sexual‐activity group. Developing this model and a means to calibrate it efficiently is relevant because HPV is a very multi‐typed and common sexually transmitted infection with more than 100 types currently known. The two types studied in this paper, types 6 and 11, are causing about 90% of anogenital warts. We extend the development of a sexual mixing matrix on the basis of a formulation first suggested by Garnett and Anderson, frequently used to model sexually transmitted infections. In particular, we consider a stochastic mixing matrix framework that allows us to jointly estimate unknown attributes and parameters of the mixing matrix along with the parameters involved in the calibration of the HPV epidemic model. This matrix describes the sexual interactions between members of the population under study and relies on several quantities that are a priori unknown. The Bayesian model developed allows one to estimate jointly the HPV‐6 and HPV‐11 epidemic model parameters as well as unknown sexual mixing matrix parameters related to assortativity. Finally, we explore the ability of an extension to the class of adaptive Markov chain Monte Carlo algorithms to incorporate a forward projection strategy for the ordinary differential equation state trajectories. Efficient exploration of the Bayesian posterior distribution developed for the ordinary differential equation parameters provides a challenge for any Markov chain sampling methodology, hence the interest in adaptive Markov chain methods. We conclude with simulation studies on synthetic and recent actual data. Copyright © 2012 John Wiley & Sons, Ltd.