Bayesian latent variable models for hierarchical clustered count outcomes with repeated measures in microbiome studies

Motivated by the multivariate nature of microbiome data with hierarchical taxonomic clusters, counts that are often skewed and zero inflated, and repeated measures, we propose a Bayesian latent variable methodology to jointly model multiple operational taxonomic units within a single taxonomic cluster. This novel method can incorporate both negative binomial and zero‐inflated negative binomial responses, and can account for serial and familial correlations. We develop a Markov chain Monte Carlo algorithm that is built on a data augmentation scheme using Pólya‐Gamma random variables. Hierarchical centering and parameter expansion techniques are also used to improve the convergence of the Markov chain. We evaluate the performance of our proposed method through extensive simulations. We also apply our method to a human microbiome study.     

Proportional hazards model with random effects

We propose a general proportional hazards model with random effects for handling clustered survival data. This generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk, in a way similar to the modelling of random effects in linear, generalized linear and non-linear mixed models. The distribution of the random effects is generally assumed to be multivariate normal, but other (preferably symmetrical) distributions are also possible. Maximum likelihood estimates of the regression parameters, the variance components and the baseline hazard function are obtained via the EM algorithm. The E-step of the algorithm involves computation of the conditional expectations of functions of the random effects, for which we use Markov chain Monte Carlo (MCMC) methods. Approximate variances of the estimates are computed by Louis' formula, and posterior expectations and variances of the individual random effects can be obtained as a by-product of the estimation. The inference procedure is exemplified on two data sets.     

Joint analysis of stochastic processes with application to smoking patterns and insomnia

This article proposes a joint modeling framework for longitudinal insomnia measurements and a stochastic smoking cessation process in the presence of a latent permanent quitting state (i.e., ‘cure’). We use a generalized linear mixed‐effects model and a stochastic mixed‐effects model for the longitudinal measurements of insomnia symptom and for the smoking cessation process, respectively. We link these two models together via the latent random effects. We develop a Bayesian framework and Markov Chain Monte Carlo algorithm to obtain the parameter estimates. We formulate and compute the likelihood functions involving time‐dependent covariates. We explore the within‐subject correlation between insomnia and smoking processes. We apply the proposed methodology to simulation studies and the motivating dataset, that is, the Alpha‐Tocopherol, Beta‐Carotene Lung Cancer Prevention study, a large longitudinal cohort study of smokers from Finland. Copyright © 2013 John Wiley & Sons, Ltd.     

A mixture model for the joint analysis of latent developmental trajectories and survival

A general joint modeling framework is proposed that includes a parametric stratified survival component for continuous time survival data, and a mixture multilevel item response component to model latent developmental trajectories given mixed discrete response data. The joint model is illustrated in a real data setting, where the utility of longitudinally measured cognitive function as a predictor for survival is investigated in a group of elderly persons. The object is partly to determine whether cognitive impairment is accompanied by a higher mortality rate. Time-dependent cognitive function is measured using the generalized partial credit model given occasion-specific mini-mental state examination response data. A parametric survival model is applied for the survival information, and cognitive function as a continuous latent variable is included as a time-dependent explanatory variable along with other explanatory information. A mixture model is defined, which incorporates the latent developmental trajectory and the survival component. The mixture model captures the heterogeneity in the developmental trajectories that could not be fully explained by the multilevel item response model and other explanatory variables. A Bayesian modeling approach is pursued, where a Markov chain Monte Carlo algorithm is developed for simultaneous estimation of the joint model parameters. Practical issues as model building and assessment are addressed using the DIC and various posterior predictive tests.     

Semiparametric estimation of major gene effects for age of onset

Analysis of age of onset is a key factor in the segregation and linkage analysis of some complex genetic traits. Previous work in the genetics literature has used parametric distributional assumptions on age of onset. In this paper, a Cox model with latent major gene effects is used: a semiparametric model with unspecified baseline hazard. A Monte Carlo EM procedure is used to obtain maximum likelihood estimates. Markov chain Monte Carlo is used to realize genotypic configurations from the posterior distribution given the current model and the observed data, and these genotypic configurations are used to estimate the expectations in the EM algorithm. Simulated data sets indicate that the parameters can be estimated well, and one real data set shows the practical applicability of the proposed method. Genet. Epidemiol. 15:279–298, 1998. © 1998 Wiley-Liss, Inc.     

Analysis of crossover designs with nonignorable dropout

This article addresses the analysis of crossover designs with nonignorable dropout. We study nonreplicated crossover designs and replicated designs separately. With a primary objective of comparing the treatment mean effects, we jointly model the longitudinal measures and discrete time to dropout. We propose shared‐parameter models and mixed‐effects selection models. We adapt a linear‐mixed effects model as the conditional model for the longitudinal outcomes. We invoke a discrete‐time hazards model with a complementary log‐log link function for the conditional distribution of time to dropout. We apply maximum likelihood for parameter estimation. We perform simulation studies to investigate the robustness of our proposed approaches under various missing data mechanisms. We then apply the approaches to two examples with a continuous outcome and one example with a binary outcome using existing software. We also implement the controlled multiple imputation methods as a sensitivity analysis of the missing data assumption.     

类别归因比例的Bayes估计与极大似然估计比较

目的：比较类别归因比例的Ｂａｙｅｓ图模型估计与极大似然估计方法。方法：应用Ｇｉｂｂｓ抽样和迭代蒙特卡罗方法得到参数后验分布,得到类别归因比例的模型法估计,应用婴儿低出生体重资料分析孕妇吸烟状况的类别归因比例。结果：类别归因比例的Ｂａｙｅｓ估计与极大似然估计一致性较好。结论：Ｂａｙｅｓ方法避免了复杂伯解析和高维积分运算,对于复杂模型估计,比极大似然估计有更实际的可行性。     

Finite mixture varying coefficient models for analyzing longitudinal heterogenous data

This paper aims to develop a mixture model to study heterogeneous longitudinal data on the treatment effect of heroin use from a California Civil Addict Program. Each component of the mixture is characterized by a varying coefficient mixed effect model. We use the Bayesian P-splines approach to approximate the varying coefficient functions. We develop Markov chain Monte Carlo algorithms to estimate the smooth functions, unknown parameters, and latent variables in the model. We use modified deviance information criterion to determine the number of components in the mixture. A simulation study demonstrates that the modified deviance information criterion selects the correct number of components and the estimation of unknown quantities is accurate. We apply the proposed model to the heroin treatment study. Furthermore, we identify heterogeneous longitudinal patterns.     

Estimation and inference on correlations between biomarkers with repeated measures and left-censoring due to minimum detection levels

Statistical approaches for estimating and drawing inference on the correlation between two biomarkers that are repeatedly assessed over time and subject to left‐censoring because minimum detection levels are lacking. We propose a linear mixed‐effects model and estimate the parameters with the Monte Carlo expectation maximization (MCEM) method. Inferences regarding the model parameters and the correlation between the biomarkers are performed by applying Louis's method and the delta method. Simulation studies were conducted to compare the proposed MCEM method with existing methods including the maximum likelihood estimation method, the multiple imputation method, and two widely used ad hoc approaches: replacing the censored values with the detection limit or with half of the detection limit. The results show that the performance of the MCEM with respect to relative bias and coverage probability for the 95% confidence interval is superior to the detection limit and half of the detection limit approaches and exceeds that of the multiple imputation method at medium to high levels of censoring, and the standard error estimates from the MCEM method are close to ideal. The maximum likelihood estimation method can estimate the parameters accurately; however, a nonpositive definite information matrix can occur so that the variances are not estimable. These five methods are illustrated with data from a longitudinal human immunodeficiency virus study to estimate and draw inference on the correlation between human immunodeficiency virus RNA levels measured in plasma and in cervical secretions at multiple time points. Copyright © 2012 John Wiley & Sons, Ltd.     

Joint analysis of left‐censored longitudinal biomarker and binary outcome via latent class modeling

Joint latent class modeling is an appealing approach for evaluating the association between a longitudinal biomarker and clinical outcome when the study population is heterogeneous. The link between the biomarker trajectory and the risk of event is reflected by the latent classes, which accommodate the underlying population heterogeneity. The estimation of joint latent class models may be complicated by the censored data in the biomarker measurements due to detection limits. We propose a modified likelihood function under the parametric assumption of biomarker distribution and develop a Monte Carlo expectation‐maximization algorithm for joint analysis of a biomarker and a binary outcome. We conduct simulation studies to demonstrate the satisfactory performance of our Monte Carlo expectation‐maximization algorithm and the superiority of our method to the naive imputation method for handling censored biomarker data. In addition, we apply our method to the Genetic and Inflammatory Markers of Sepsis study to investigate the role of inflammatory biomarker profile in predicting 90‐day mortality for patients hospitalized with community‐acquired pneumonia.     

The use of Gaussian quadrature for estimation in frailty proportional hazards models

In this paper, we propose a novel Gaussian quadrature estimation method in various frailty proportional hazards models. We approximate the unspecified baseline hazard by a piecewise constant one, resulting in a parametric model that can be fitted conveniently by Gaussian quadrature tools in standard software such as SAS Proc NLMIXED. We first apply our method to simple frailty models for correlated survival data (e.g. recurrent or clustered failure times), then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death. Simulation studies show that our method compares favorably with the well-received penalized partial likelihood method and the Monte Carlo EM (MCEM) method, for both normal and Gamma frailty models. We apply our method to three real data examples: (1) the time to blindness of both eyes in a diabetic retinopathy study, (2) the joint analysis of recurrent opportunistic diseases in the presence of death for HIV-infected patients, and (3) the joint modeling of local, distant tumor recurrences and patients survival in a soft tissue sarcoma study. The proposed method greatly simplifies the implementation of the (joint) frailty models and makes them much more accessible to general statistical practitioners.     

Inference on cancer screening exam accuracy using population‐level administrative data

This paper develops a model for cancer screening and cancer incidence data, accommodating the partially unobserved disease status, clustered data structures, general covariate effects, and dependence between exams. The true unobserved cancer and detection status of screening participants are treated as latent variables, and a Markov Chain Monte Carlo algorithm is used to estimate the Bayesian posterior distributions of the diagnostic error rates and disease prevalence. We show how the Bayesian approach can be used to draw inferences about screening exam properties and disease prevalence while allowing for the possibility of conditional dependence between two exams. The techniques are applied to the estimation of the diagnostic accuracy of mammography and clinical breast examination using data from the Ontario Breast Screening Program in Canada. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimating overall exposure effects for the clustered and censored outcome using random effect Tobit regression models

The random effect Tobit model is a regression model that accommodates both left‐ and/or right‐censoring and within‐cluster dependence of the outcome variable. Regression coefficients of random effect Tobit models have conditional interpretations on a constructed latent dependent variable and do not provide inference of overall exposure effects on the original outcome scale. Marginalized random effects model (MREM) permits likelihood‐based estimation of marginal mean parameters for the clustered data. For random effect Tobit models, we extend the MREM to marginalize over both the random effects and the normal space and boundary components of the censored response to estimate overall exposure effects at population level. We also extend the ‘Average Predicted Value’ method to estimate the model‐predicted marginal means for each person under different exposure status in a designated reference group by integrating over the random effects and then use the calculated difference to assess the overall exposure effect. The maximum likelihood estimation is proposed utilizing a quasi‐Newton optimization algorithm with Gauss–Hermite quadrature to approximate the integration of the random effects. We use these methods to carefully analyze two real datasets. Copyright © 2016 John Wiley & Sons, Ltd.     

Identification of treatment responders based on multiple longitudinal outcomes with applications to multiple sclerosis patients

Identification of treatment responders is a challenge in comparative studies where treatment efficacy is measured by multiple longitudinally collected continuous and count outcomes. Existing procedures often identify responders on the basis of only a single outcome. We propose a novel multiple longitudinal outcome mixture model that assumes that, conditionally on a cluster label, each longitudinal outcome is from a generalized linear mixed effect model. We utilize a Monte Carlo expectation‐maximization algorithm to obtain the maximum likelihood estimates of our high‐dimensional model and classify patients according to their estimated posterior probability of being a responder. We demonstrate the flexibility of our novel procedure on two multiple sclerosis clinical trial datasets with distinct data structures. Our simulation study shows that incorporating multiple outcomes improves the responder identification performance; this can occur even if some of the outcomes are ineffective. Our general procedure facilitates the identification of responders who are comprehensively defined by multiple outcomes from various distributions. Copyright © 2017 John Wiley & Sons, Ltd.     

Latent transition analysis: inference and estimation

Parameters for latent transition analysis (LTA) are easily estimated by maximum likelihood (ML) or Bayesian method via Markov chain Monte Carlo (MCMC). However, unusual features in the likelihood can cause difficulties in ML and Bayesian inference and estimation, especially with small samples. In this study we explore several problems in drawing inference for LTA in the context of a simulation study and a substance use example. We argue that when conventional ML and Bayesian estimates behave erratically, problems often may be alleviated with a small amount of prior input for LTA with small samples. This paper proposes a dynamic data-dependent prior for LTA with small samples and compares the performance of the estimation methods with the proposed prior in drawing inference.     

A Bayesian approach for individual-level drug benefit-risk assessment

In existing benefit-risk assessment (BRA) methods, benefit and risk criteria are usually identified and defined separately based on aggregated clinical data and therefore ignore the individual-level differences as well as the association among the criteria. We proposed a Bayesian multicriteria decision-making method for BRA of drugs using individual-level data. We used a multidimensional latent trait model to account for the heterogeneity of treatment effects with latent variables introducing the dependencies among outcomes. We then applied the stochastic multicriteria acceptability analysis approach for BRA incorporating imprecise and heterogeneous patient preference information. We adopted an efficient Markov chain Monte Carlo algorithm when implementing the proposed method. We applied our method to a case study to illustrate how individual-level benefit-risk profiles could inform decision-making.     

Estimation for zero‐inflated over‐dispersed count data model with missing response

In this paper, we develop estimation procedure for the parameters of a zero‐inflated over‐dispersed/under‐dispersed count model in the presence of missing responses. In particular, we deal with a zero‐inflated extended negative binomial model in the presence of missing responses. A weighted expectation maximization algorithm is used for the maximum likelihood estimation of the parameters involved. Some simulations are conducted to study the properties of the estimators. Robustness of the procedure is shown when count data follow other over‐dispersed models, such as the log‐normal mixture of the Poisson distribution or even from a zero‐inflated Poisson model. An illustrative example and a discussion leading to some conclusions are given. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian adaptive group lasso with semiparametric hidden Markov models

This paper presents a Bayesian adaptive group least absolute shrinkage and selection operator method to conduct simultaneous model selection and estimation under semiparametric hidden Markov models. We specify the conditional regression model and the transition probability model in the hidden Markov model into additive nonparametric functions of covariates. A basis expansion is adopted to approximate the nonparametric functions. We introduce multivariate conditional Laplace priors to impose adaptive penalties on regression coefficients and different groups of basis expansions under the Bayesian framework. An efficient Markov chain Monte Carlo algorithm is then proposed to identify the nonexistent, constant, linear, and nonlinear forms of covariate effects in both conditional and transition models. The empirical performance of the proposed methodology is evaluated via simulation studies. We apply the proposed model to analyze a real data set that was collected from the Alzheimer's Disease Neuroimaging Initiative study. The analysis identifies important risk factors on cognitive decline and the transition from cognitive normal to Alzheimer's disease.     

A mechanistic nonlinear model for censored and mismeasured covariates in longitudinal models,with application in AIDS studies

When modeling longitudinal data, the true values of time‐varying covariates may be unknown because of detection‐limit censoring or measurement error. A common approach in the literature is to empirically model the covariate process based on observed data and then predict the censored values or mismeasured values based on this empirical model. Such an empirical model can be misleading, especially for censored values since the (unobserved) censored values may behave very differently than observed values due to the underlying data‐generation mechanisms or disease status. In this paper, we propose a mechanistic nonlinear covariate model based on the underlying data‐generation mechanisms to address censored values and mismeasured values. Such a mechanistic model is based on solid scientific or biological arguments, so the predicted censored or mismeasured values are more reasonable. We use a Monte Carlo EM algorithm for likelihood inference and apply the methods to an AIDS dataset, where viral load is censored by a lower detection limit. Simulation results confirm that the proposed models and methods offer substantial advantages over existing empirical covariate models for censored and mismeasured covariates.     

Evaluating latent class models with conditional dependence in record linkage

Record linkage methods commonly use a traditional latent class model to classify record pairs from different sources as true matches or non‐matches. This approach was first formally described by Fellegi and Sunter and assumes that the agreement in fields is independent conditional on the latent class. Consequences of violating the conditional independence assumption include bias in parameter estimates from the model. We sought to further characterize the impact of conditional dependence on the overall misclassification rate, sensitivity, and positive predictive value in the record linkage problem when the conditional independence assumption is violated. Additionally, we evaluate various methods to account for the conditional dependence. These methods include loglinear models with appropriate interaction terms identified through the correlation residual plot as well as Gaussian random effects models. The proposed models are used to link newborn screening data obtained from a health information exchange. On the basis of simulations, loglinear models with interaction terms demonstrated the best misclassification rate, although this type of model cannot accommodate other data features such as continuous measures for agreement. Results indicate that Gaussian random effects models, which can handle additional data features, perform better than assuming conditional independence and in some situations perform as well as the loglinear model with interaction terms. Copyright © 2014 John Wiley & Sons, Ltd.