Bayesian mixed hidden Markov models: a multi‐level approach to modeling categorical outcomes with differential misclassification

Questionnaire‐based health status outcomes are often prone to misclassification. When studying the effect of risk factors on such outcomes, ignoring any potential misclassification may lead to biased effect estimates. Analytical challenges posed by these misclassified outcomes are further complicated when simultaneously exploring factors for both the misclassification and health processes in a multi‐level setting. To address these challenges, we propose a fully Bayesian mixed hidden Markov model (BMHMM) for handling differential misclassification in categorical outcomes in a multi‐level setting. The BMHMM generalizes the traditional hidden Markov model (HMM) by introducing random effects into three sets of HMM parameters for joint estimation of the prevalence, transition, and misclassification probabilities. This formulation not only allows joint estimation of all three sets of parameters but also accounts for cluster‐level heterogeneity based on a multi‐level model structure. Using this novel approach, both the true health status prevalence and the transition probabilities between the health states during follow‐up are modeled as functions of covariates. The observed, possibly misclassified, health states are related to the true, but unobserved, health states and covariates. Results from simulation studies are presented to validate the estimation procedure, to show the computational efficiency due to the Bayesian approach and also to illustrate the gains from the proposed method compared to existing methods that ignore outcome misclassification and cluster‐level heterogeneity. We apply the proposed method to examine the risk factors for both asthma transition and misclassification in the Southern California Children's Health Study. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimating recurrence and incidence of preterm birth subject to measurement error in gestational age: A hidden Markov modeling approach

Prediction of preterm birth as well as characterizing the etiological factors affecting both the recurrence and incidence of preterm birth (defined as gestational age at birth ≤ 37 wk) are important problems in obstetrics. The National Institute of Child Health and Human Development (NICHD) consecutive pregnancy study recently examined this question by collecting data on a cohort of women with at least 2 pregnancies over a fixed time interval. Unfortunately, measurement error due to the dating of conception may induce sizable error in computing gestational age at birth. This article proposes a flexible approach that accounts for measurement error in gestational age when making inference. The proposed approach is a hidden Markov model that accounts for measurement error in gestational age by exploiting the relationship between gestational age at birth and birth weight. We initially model the measurement error as being normally distributed, followed by a mixture of normals that has been proposed on the basis of biological considerations. We examine the asymptotic bias of the proposed approach when measurement error is ignored and also compare the efficiency of this approach to a simpler hidden Markov model formulation where only gestational age and not birth weight is incorporated. The proposed model is compared with alternative models for estimating important covariate effects on the risk of subsequent preterm birth using a unique set of data from the NICHD consecutive pregnancy study.     

Correlated gamma‐based hidden Markov model for the smart asthma management based on rescue inhaler usage

Asthma is a very common chronic disease that affects a large portion of population in many nations. Driven by the fast development in sensor and mobile communication technology, a smart asthma management system has become available to continuously monitor the key health indicators of asthma patients. Such data provides opportunities for healthcare practitioners to examine patients not only in the clinic (on‐site) but also outside of the clinic (off‐site) in their daily life. In this paper, taking advantage from this data availability, we propose a correlated gamma‐based hidden Markov model framework, which can reveal and highlight useful information from the rescue inhaler‐usage profiles of individual patients for practitioners. The proposed method can provide diagnostic information about the asthma control status of individual patients and can help practitioners to make more informed therapeutic decisions accordingly. The proposed method is validated through both numerical study and case study based on real world data. Copyright © 2017 John Wiley & Sons, Ltd.     

A hidden Markov model approach to analyze longitudinal ternary outcomes when some observed states are possibly misclassified

Understanding the dynamic disease process is vital in early detection, diagnosis, and measuring progression. Continuous‐time Markov chain (CTMC) methods have been used to estimate state‐change intensities but challenges arise when stages are potentially misclassified. We present an analytical likelihood approach where the hidden state is modeled as a three‐state CTMC model allowing for some observed states to be possibly misclassified. Covariate effects of the hidden process and misclassification probabilities of the hidden state are estimated without information from a ‘gold standard’ as comparison. Parameter estimates are obtained using a modified expectation‐maximization (EM) algorithm, and identifiability of CTMC estimation is addressed. Simulation studies and an application studying Alzheimer's disease caregiver stress‐levels are presented. The method was highly sensitive to detecting true misclassification and did not falsely identify error in the absence of misclassification. In conclusion, we have developed a robust longitudinal method for analyzing categorical outcome data when classification of disease severity stage is uncertain and the purpose is to study the process' transition behavior without a gold standard. Copyright © 2016 John Wiley & Sons, Ltd.     

Latent time‐varying factors in longitudinal analysis: a linear mixed hidden Markov model for heart rates

Longitudinal data are often segmented by unobserved time‐varying factors, which introduce latent heterogeneity at the observation level, in addition to heterogeneity across subjects. We account for this latent structure by a linear mixed hidden Markov model. It integrates subject‐specific random effects and Markovian sequences of time‐varying effects in the linear predictor. We propose an expectation?‐maximization algorithm for maximum likelihood estimation, based on data augmentation. It reduces to the iterative maximization of the expected value of a complete likelihood function, derived from an augmented dataset with case weights, alternated with weights updating. In a case study of the Survey on Stress Aging and Health in Russia, the model is exploited to estimate the influence of the observed covariates under unobserved time‐varying factors, which affect the cardiovascular activity of each subject during the observation period. Copyright © 2014 John Wiley & Sons, Ltd.     

Reducing Monte Carlo error in the Bayesian estimation of risk ratios using log‐binomial regression models

In cohort studies, binary outcomes are very often analyzed by logistic regression. However, it is well known that when the goal is to estimate a risk ratio, the logistic regression is inappropriate if the outcome is common. In these cases, a log‐binomial regression model is preferable. On the other hand, the estimation of the regression coefficients of the log‐binomial model is difficult owing to the constraints that must be imposed on these coefficients. Bayesian methods allow a straightforward approach for log‐binomial regression models and produce smaller mean squared errors in the estimation of risk ratios than the frequentist methods, and the posterior inferences can be obtained using the software WinBUGS. However, Markov chain Monte Carlo methods implemented in WinBUGS can lead to large Monte Carlo errors in the approximations to the posterior inferences because they produce correlated simulations, and the accuracy of the approximations are inversely related to this correlation. To reduce correlation and to improve accuracy, we propose a reparameterization based on a Poisson model and a sampling algorithm coded in R. Copyright © 2015 John Wiley & Sons, Ltd.     

Applications of continuous time hidden Markov models to the study of misclassified disease outcomes

Disease progression in prospective clinical and epidemiological studies is often conceptualized in terms of transitions between disease states. Analysis of data from such studies can be complicated by a number of factors, including the presence of individuals in various prevalent disease states and with unknown prior disease history, interval censored observations of state transitions and misclassified measurements of disease states. We present an approach where the disease states are modelled as the hidden states of a continuous time hidden Markov model using the imperfect measurements of the disease state as observations. Covariate effects on transitions between disease states are incorporated using a generalized regression framework. Parameter estimation and inference are based on maximum likelihood methods and rely on an EM algorithm. In addition, techniques for model assessment are proposed. Applications to two binary disease outcomes are presented: the oral lesion hairy leukoplakia in a cohort of HIV infected men and cervical human papillomavirus (HPV) infection in a cohort of young women. Estimated transition rates and misclassification probabilities for the hairy leukoplakia data agree well with clinical observations on the persistence and diagnosis of this lesion, lending credibility to the interpretation of hidden states as representing the actual disease states. By contrast, interpretation of the results for the HPV data are more problematic, illustrating that successful application of the hidden Markov model may be highly dependent on the degree to which the assumptions of the model are satisfied.     

A joint logistic regression and covariate‐adjusted continuous‐time Markov chain model

The use of longitudinal measurements to predict a categorical outcome is an increasingly common goal in research studies. Joint models are commonly used to describe two or more models simultaneously by considering the correlated nature of their outcomes and the random error present in the longitudinal measurements. However, there is limited research on joint models with longitudinal predictors and categorical cross‐sectional outcomes. Perhaps the most challenging task is how to model the longitudinal predictor process such that it represents the true biological mechanism that dictates the association with the categorical response. We propose a joint logistic regression and Markov chain model to describe a binary cross‐sectional response, where the unobserved transition rates of a two‐state continuous‐time Markov chain are included as covariates. We use the method of maximum likelihood to estimate the parameters of our model. In a simulation study, coverage probabilities of about 95%, standard deviations close to standard errors, and low biases for the parameter values show that our estimation method is adequate. We apply the proposed joint model to a dataset of patients with traumatic brain injury to describe and predict a 6‐month outcome based on physiological data collected post‐injury and admission characteristics. Our analysis indicates that the information provided by physiological changes over time may help improve prediction of long‐term functional status of these severely ill subjects. Copyright © 2017 John Wiley & Sons, Ltd.     

Adult myeloid leukaemia and radon exposure: a Bayesian model for a case-control study with error in covariates

The possible association between radon exposure in dwellings and adult myeloid leukaemia had been explored in an Italian province by a case-control study. A total of 44 cases and 211 controls were selected from death certificates file. No association had been found in the original study (OR = 0.58 for > 185 vs 80 < or = Bq/cm). Here we reanalyse the data taking into account the measurement error of radon concentration and the presence of missing data. A Bayesian hierarchical model with error in covariates is proposed which allows appropriate imputation of missing values.The general conclusion of no evidence of association with radon does not change, but a negative association is not observed anymore (OR = 0.99 for > 185 vs 80 < or = Bq/cm). After adjusting for residential house radon and gamma radiation, and for the multilevel data structure, geological features of the soil is associated with adult myeloid leukaemia risk (OR = 2.14, 95 per cent Cr.I. 1.0-5.5).     

Bayesian monotonic errors‐in‐variables models with applications to pathogen susceptibility testing

Drug dilution (MIC) and disk diffusion (DIA) are the 2 most common antimicrobial susceptibility assays used by hospitals and clinics to determine an unknown pathogen's susceptibility to various antibiotics. Since only one assay is commonly used, it is important that the 2 assays give similar results. Calibration of the DIA assay to the MIC assay is typically done using the error‐rate bounded method, which selects DIA breakpoints that minimize the observed discrepancies between the 2 assays. In 2000, Craig proposed a model‐based approach that specifically models the measurement error and rounding processes of each assay, the underlying pathogen distribution, and the true monotonic relationship between the 2 assays. The 2 assays are then calibrated by focusing on matching the probabilities of correct classification (susceptible, indeterminant, and resistant). This approach results in greater precision and accuracy for estimating DIA breakpoints. In this paper, we expand the flexibility of the model‐based method by introducing a Bayesian 4‐parameter logistic model (extending Craig's original 3‐parameter model) as well as a Bayesian nonparametric spline model to describe the relationship between the 2 assays. We propose 2 ways to handle spline knot selection, considering many equally spaced knots but restricting overfitting via a random walk prior and treating the number and location of knots as additional unknown parameters. We demonstrate the 2 approaches via a series of simulation studies and apply the methods to 2 real data sets.     

Sensitivity of regression calibration to non‐perfect validation data with application to the Norwegian Women and Cancer Study

Measurement error occurs when we observe error‐prone surrogates, rather than true values. It is common in observational studies and especially so in epidemiology, in nutritional epidemiology in particular. Correcting for measurement error has become common, and regression calibration is the most popular way to account for measurement error in continuous covariates. We consider its use in the context where there are validation data, which are used to calibrate the true values given the observed covariates. We allow for the case that the true value itself may not be observed in the validation data, but instead, a so‐called reference measure is observed. The regression calibration method relies on certain assumptions.This paper examines possible biases in regression calibration estimators when some of these assumptions are violated. More specifically, we allow for the fact that (i) the reference measure may not necessarily be an ‘alloyed gold standard’ (i.e., unbiased) for the true value; (ii) there may be correlated random subject effects contributing to the surrogate and reference measures in the validation data; and (iii) the calibration model itself may not be the same in the validation study as in the main study; that is, it is not transportable. We expand on previous work to provide a general result, which characterizes potential bias in the regression calibration estimators as a result of any combination of the violations aforementioned. We then illustrate some of the general results with data from the Norwegian Women and Cancer Study. Copyright © 2015 John Wiley & Sons, Ltd.     

Methods to adjust for misclassification in the quantiles for the generalized linear model with measurement error in continuous exposures

In many biomedical studies, covariates of interest may be measured with errors. However, frequently in a regression analysis, the quantiles of the exposure variable are often used as the covariates in the regression analysis. Because of measurement errors in the continuous exposure variable, there could be misclassification in the quantiles for the exposure variable. Misclassification in the quantiles could lead to bias estimation in the association between the exposure variable and the outcome variable. Adjustment for misclassification will be challenging when the gold standard variables are not available. In this paper, we develop two regression calibration estimators to reduce bias in effect estimation. The first estimator is normal likelihood‐based. The second estimator is linearization‐based, and it provides a simple and practical correction. Finite sample performance is examined via a simulation study. We apply the methods to a four‐arm randomized clinical trial that tested exercise and weight loss interventions in women aged 50–75years. Copyright © 2015 John Wiley & Sons, Ltd.     

Adverse subpopulation regression for multivariate outcomes with high‐dimensional predictors

Biomedical studies have a common interest in assessing relationships between multiple related health outcomes and high‐dimensional predictors. For example, in reproductive epidemiology, one may collect pregnancy outcomes such as length of gestation and birth weight and predictors such as single nucleotide polymorphisms in multiple candidate genes and environmental exposures. In such settings, there is a need for simple yet flexible methods for selecting true predictors of adverse health responses from a high‐dimensional set of candidate predictors. To address this problem, one may either consider linear regression models for the continuous outcomes or convert these outcomes into binary indicators of adverse responses using predefined cutoffs. The former strategy has the disadvantage of often leading to a poorly fitting model that does not predict risk well, whereas the latter approach can be very sensitive to the cutoff choice. As a simple yet flexible alternative, we propose a method for adverse subpopulation regression, which relies on a two‐component latent class model, with the dominant component corresponding to (presumed) healthy individuals and the risk of falling in the minority component characterized via a logistic regression. The logistic regression model is designed to accommodate high‐dimensional predictors, as occur in studies with a large number of gene by environment interactions, through the use of a flexible nonparametric multiple shrinkage approach. The Gibbs sampler is developed for posterior computation. We evaluate the methods with the use of simulation studies and apply these to a genetic epidemiology study of pregnancy outcomes. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of infection and recovery rates for highly polymorphic parasites when detectability is imperfect,using hidden Markov models

A Bayesian hierarchical model is proposed for estimating parasitic infection dynamics for highly polymorphic parasites when detectability of the parasite using standard tests is imperfect. The parasite dynamics are modelled as a non-homogeneous hidden two-state Markov process, where the observed process is the detection or failure to detect a parasitic genotype. This is assumed to be conditionally independent given the hidden process, that is, the underlying true presence of the parasite, which evolves according to a first-order Markov chain. The model allows the transition probabilities of the hidden states as well as the detectability parameter of the test to depend on a number of covariates. Full Bayesian inference is implemented using Markov chain Monte Carlo simulation. The model is applied to a panel data set of malaria genotype data from a randomized controlled trial of bed nets in Tanzanian children aged 6-30 months, with the age of the host and bed net use as covariates. This analysis confirmed that the duration of infections with parasites belonging to the MSP-2 FC27 allelic family increased with age.     

Comparison of Bayesian model averaging and stepwise methods for model selection in logistic regression

Logistic regression is the standard method for assessing predictors of diseases. In logistic regression analyses, a stepwise strategy is often adopted to choose a subset of variables. Inference about the predictors is then made based on the chosen model constructed of only those variables retained in that model. This method subsequently ignores both the variables not selected by the procedure, and the uncertainty due to the variable selection procedure. This limitation may be addressed by adopting a Bayesian model averaging approach, which selects a number of all possible such models, and uses the posterior probabilities of these models to perform all inferences and predictions. This study compares the Bayesian model averaging approach with the stepwise procedures for selection of predictor variables in logistic regression using simulated data sets and the Framingham Heart Study data. The results show that in most cases Bayesian model averaging selects the correct model and out-performs stepwise approaches at predicting an event of interest.     

Application of the Bayesian dynamic survival model in medicine

The Bayesian dynamic survival model (BDSM), a time‐varying coefficient survival model from the Bayesian prospective, was proposed in early 1990s but has not been widely used or discussed. In this paper, we describe the model structure of the BDSM and introduce two estimation approaches for BDSMs: the Markov Chain Monte Carlo (MCMC) approach and the linear Bayesian (LB) method. The MCMC approach estimates model parameters through sampling and is computationally intensive. With the newly developed geoadditive survival models and software BayesX, the BDSM is available for general applications. The LB approach is easier in terms of computations but it requires the prespecification of some unknown smoothing parameters. In a simulation study, we use the LB approach to show the effects of smoothing parameters on the performance of the BDSM and propose an ad hoc method for identifying appropriate values for those parameters. We also demonstrate the performance of the MCMC approach compared with the LB approach and a penalized partial likelihood method available in software R packages. A gastric cancer trial is utilized to illustrate the application of the BDSM. Copyright © 2009 John Wiley & Sons, Ltd.     

A Bayesian model for misclassified binary outcomes and correlated survival data with applications to breast cancer

Breast cancer patients may experience ipsilateral breast tumor relapse (IBTR) after breast conservation therapy. IBTR is classified as either true local recurrence or new ipsilateral primary tumor. The correct classification of IBTR status has significant implications in therapeutic decision‐making and patient management. However, the diagnostic tests to classify IBTR are imperfect and prone to misclassification. In addition, some observed survival data (e.g., time to relapse, time from relapse to death) are strongly correlated with IBTR status. We present a Bayesian approach to model the potentially misclassified IBTR status and the correlated survival information. We conduct the inference using a Bayesian framework via Markov chain Monte Carlo simulation implemented in WinBUGS . Extensive simulation shows that the proposed method corrects biases and provides more efficient estimates for the covariate effects on the probability of IBTR and the diagnostic test accuracy. Moreover, our method provides useful subject‐specific patient prognostic information. Our method is motivated by, and applied to, a dataset of 397 breast cancer patients. Copyright © 2012 John Wiley & Sons, Ltd.     

Correcting for multivariate measurement error by regression calibration in meta‐analyses of epidemiological studies

Within‐person variability in measured values of multiple risk factors can bias their associations with disease. The multivariate regression calibration (RC) approach can correct for such measurement error and has been applied to studies in which true values or independent repeat measurements of the risk factors are observed on a subsample. We extend the multivariate RC techniques to a meta‐analysis framework where multiple studies provide independent repeat measurements and information on disease outcome. We consider the cases where some or all studies have repeat measurements, and compare study‐specific, averaged and empirical Bayes estimates of RC parameters. Additionally, we allow for binary covariates (e.g. smoking status) and for uncertainty and time trends in the measurement error corrections. Our methods are illustrated using a subset of individual participant data from prospective long‐term studies in the Fibrinogen Studies Collaboration to assess the relationship between usual levels of plasma fibrinogen and the risk of coronary heart disease, allowing for measurement error in plasma fibrinogen and several confounders. Copyright © 2009 John Wiley & Sons, Ltd.     

贝叶斯累加回归树模型在高血压药物个性化疗效评价中的应用

目的  以高血压合并糖尿病人群为例,应用贝叶斯累加回归树(Bayesian additive regression tree, BART)模型评价卡托普利及其联合尼群地平用药对血压控制的个性化疗效,并识别高收益患者亚组特征。 方法  纳入2011年1月至2015年7月山东省胶南市“全人群高血压、糖尿病综合防治项目”中的高血压合并糖尿病患者,按不同用药方式分为三组,采用倾向性得分随机化匹配形成可比样本后,建立BART模型探索用药的个性化疗效。 结果  在单用卡托普利与不用药、联合用药与不用药,联合用药与单用卡托普利三组对比中,三组模型曲线下面积(area under curre, AUC)及其95% CI分别为0.710(0.686~0.734)、0.796(0.754~0.838)、0.768(0.725~0.812)。对于大多数患者,联合用药效果优于单用卡托普利和不用药,其中,相比不用药者,SBP较低且有良好运动习惯是单用卡托普利和联合用药疗效更佳者的共同特征。 结论  构建的BART模型用于预测高血压合并糖尿病患者的高血压药物个性化疗效效果良好,并且能够用于总结高收益亚组特征,为精准医疗在高血压治疗中的实践提供依据。     

Bayesian inference for the stereotype regression model: Application to a case–control study of prostate cancer

The stereotype regression model for categorical outcomes, proposed by Anderson (J. Roy. Statist. Soc. B. 1984; 46 :1–30) is nested between the baseline‐category logits and adjacent category logits model with proportional odds structure. The stereotype model is more parsimonious than the ordinary baseline‐category (or multinomial logistic) model due to a product representation of the log‐odds‐ratios in terms of a common parameter corresponding to each predictor and category‐specific scores. The model could be used for both ordered and unordered outcomes. For ordered outcomes, the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable, but are inherently multi‐dimensional in nature. As pointed out by Greenland (Statist. Med. 1994; 13 :1665–1677), an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome‐stratified sampling as in case–control studies. In addition, for matched case–control studies, the stereotype model is amenable to classical conditional likelihood principle, whereas there is no reduction due to sufficiency under the proportional odds model. In spite of these attractive features, the model has been applied less, as there are issues with maximum likelihood estimation and likelihood‐based testing approaches due to non‐linearity and lack of identifiability of the parameters. We present comprehensive Bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach. We illustrate our methodology by analyzing data from The Flint Men's Health Study, a case–control study of prostate cancer in African‐American men aged 40–79 years. We use clinical staging of prostate cancer in terms of Tumors, Nodes and Metastasis as the categorical response of interest. Copyright © 2009 John Wiley & Sons, Ltd.