Bayesian inference on joint models of HIV dynamics for time-to-event and longitudinal data with skewness and covariate measurement errors

Semiparametric multivariate joint model for skewed-longitudinal and survival data: A Bayesian approach

Joint models and statistical inference for longitudinal and survival data have been an active area of statistical research and have mostly coupled a longitudinal biomarker-based mixed-effects model with normal distribution and an event time-based survival model. In practice, however, the following issues may standout: (i) Normality of model error in longitudinal models is a routine assumption, but it may be unrealistically violating data features of subject variations. (ii) Data collected are often featured by the mixed types of multiple longitudinal outcomes which are significantly correlated, ignoring their correlation may lead to biased estimation. Additionally, a parametric model specification may be inflexible to capture the complicated patterns of longitudinal data. (iii) Missing observations in the longitudinal data are often encountered; the missing measures are likely to be informative (nonignorable) and ignoring this phenomenon may result in inaccurate inference. Multilevel item response theory (MLIRT) models have been increasingly used to analyze the multiple longitudinal data of mixed types (ie, continuous and categorical) in clinical studies. In this article, we develop an MLIRT-based semiparametric joint model with skew-t distribution that consists of an extended MLIRT model for the mixed types of multiple longitudinal data and a Cox proportional hazards model, linked through random-effects. A Bayesian approach is employed for joint modeling. Simulation studies are conducted to assess performance of the proposed models and method. A real example from primary biliary cirrhosis clinical study is analyzed to estimate parameters in the joint model and also evaluate sensitivity of parameter estimates for various plausible nonignorable missing data mechanisms.

Variable selection for joint models of multivariate skew-normal longitudinal and survival data

Many joint models of multivariate skew-normal longitudinal and survival data have been presented to accommodate for the non-normality of longitudinal outcomes in recent years. But existing work did not consider variable selection. This article investigates simultaneous parameter estimation and variable selection in joint modeling of longitudinal and survival data. The penalized splines technique is used to estimate unknown log baseline hazard function, the rectangle integral method is adopted to approximate conditional survival function. Monte Carlo expectation-maximization algorithm is developed to estimate model parameters. Based on local linear approximations to conditional expectation of likelihood function and penalty function, a one-step sparse estimation procedure is proposed to circumvent the computationally challenge in optimizing the penalized conditional expectation of likelihood function, which is utilized to select significant covariates and trajectory functions, and identify the departure from normality of longitudinal data. The conditional expectation of likelihood function-based Bayesian information criterion is developed to select the optimal tuning parameter. Simulation studies and a real example from the clinical trial are used to illustrate the proposed methodologies.

Flexible modeling of multiple nonlinear longitudinal trajectories with censored and non-ignorable missing outcomes

Multivariate nonlinear mixed-effects models (MNLMMs) have become a promising tool for analyzing multi-outcome longitudinal data following nonlinear trajectory patterns. However, such a classical analysis can be challenging due to censorship induced by detection limits of the quantification assay or non-response occurring when participants missed scheduled visits intermittently or discontinued participation. This article proposes an extension of the MNLMM approach, called the MNLMM-CM, by taking the censored and non-ignorable missing responses into account simultaneously. The non-ignorable missingness is described by the selection-modeling factorization to tackle the missing not at random mechanism. A Monte Carlo expectation conditional maximization algorithm coupled with the first-order Taylor approximation is developed for parameter estimation. The techniques for the calculation of standard errors of fixed effects, estimation of unobservable random effects, imputation of censored and missing responses and prediction of future values are also provided. The proposed methodology is motivated and illustrated by the analysis of a clinical HIV/AIDS dataset with censored RNA viral loads and the presence of missing CD4 and CD8 cell counts. The superiority of our method on the provision of more adequate estimation is validated by a simulation study.

Bayesian Joint Modeling of Multivariate Longitudinal and Survival Data With an Application to Diabetes Study

Joint models of longitudinal and time-to-event data have received a lot of attention in epidemiological and clinical research under a linear mixed-effects model with the normal assumption for a single longitudinal outcome and Cox proportional hazards model. However, those model-based analyses may not provide robust inference when longitudinal measurements exhibit skewness and/or heavy tails. In addition, the data collected are often featured by multivariate longitudinal outcomes which are significantly correlated, and ignoring their correlation may lead to biased estimation. Under the umbrella of Bayesian inference, this article introduces multivariate joint (MVJ) models with a skewed distribution for multiple longitudinal exposures in an attempt to cope with correlated multiple longitudinal outcomes, adjust departures from normality, and tailor linkage in specifying a time-to-event process. We develop a Bayesian joint modeling approach to MVJ models that couples a multivariate linear mixed-effects (MLME) model with the skew-normal (SN) distribution and a Cox proportional hazards model. Our proposed models and method are evaluated by simulation studies and are applied to a real example from a diabetes study.

A two-stage approach for joint modeling of longitudinal measurements and competing risks data

Joint modeling of longitudinal measurements and time-to-event data is used in many practical studies of medical sciences. Most of the time, particularly in clinical studies and health inquiry, there are more than one event and they compete for failing an individual. In this situation, assessing the competing risk failure time is important. In most cases, implementation of joint modeling involves complex calculations. Therefore, we propose a two-stage method for joint modeling of longitudinal measurements and competing risks (JMLC) data based on the full likelihood approach via the conditional EM (CEM) algorithm. In the first stage, a linear mixed effect model is used to estimate the parameters of the longitudinal sub-model. In the second stage, we consider a cause-specific sub-model to construct competing risks data and describe an approximation for the joint log-likelihood that uses the estimated parameters of the first stage. We express the results of a simulation study and perform this method on the "standard and new anti-epileptic drugs" trial to check the effect of drug assaying on the treatment effects of lamotrigine and carbamazepine through treatment failure.

Handling of uncertainty in medical data using machine learning and probability theory techniques: a review of 30 years (1991-2020)

Understanding the data and reaching accurate conclusions are of paramount importance in the present era of big data. Machine learning and probability theory methods have been widely used for this purpose in various fields. One critically important yet less explored aspect is capturing and analyzing uncertainties in the data and model. Proper quantification of uncertainty helps to provide valuable information to obtain accurate diagnosis. This paper reviewed related studies conducted in the last 30 years (from 1991 to 2020) in handling uncertainties in medical data using probability theory and machine learning techniques. Medical data is more prone to uncertainty due to the presence of noise in the data. So, it is very important to have clean medical data without any noise to get accurate diagnosis. The sources of noise in the medical data need to be known to address this issue. Based on the medical data obtained by the physician, diagnosis of disease, and treatment plan are prescribed. Hence, the uncertainty is growing in healthcare and there is limited knowledge to address these problems. Our findings indicate that there are few challenges to be addressed in handling the uncertainty in medical raw data and new models. In this work, we have summarized various methods employed to overcome this problem. Nowadays, various novel deep learning techniques have been proposed to deal with such uncertainties and improve the performance in decision making.

Bayesian joint modeling for partially linear mixed-effects quantile regression of longitudinal and time-to-event data with limit of detection, covariate measurement errors and skewness

Joint modeling analysis of longitudinal and time-to-event data has been an active area of statistical methodological study and biomedical research, but the majority of them are based on mean-regression. Quantile regression (QR) can characterize the entire conditional distribution of the outcome variable, and may be more robust to outliers/heavy tails and misspecification of error distribution. Additionally, a parametric specification may be insufficient and inflexible to capture the complicated longitudinal pattern of biomarkers. Thus, this study proposes novel QR-based partially linear mixed-effects joint models with three components (QR-based longitudinal response, longitudinal covariate, and time-to-event processes), and applies to Multicenter AIDS Cohort Study (MACS). Many common data features, including left-censoring due to a limit of detection, covariate measurement error, and asymmetric distribution, are considered to obtain reliable parameter estimates. Many interesting findings are discovered by the complicated joint models under Bayesian inference framework. Simulation studies are also implemented to assess the performance of the proposed joint models under different scenarios.

Multivariate-t linear mixed models with censored responses, intermittent missing values and heavy tails

Multivariate longitudinal data arisen in medical studies often exhibit complex features such as censored responses, intermittent missing values, and atypical or outlying observations. The multivariate-t linear mixed model (MtLMM) has been recognized as a powerful tool for robust modeling of multivariate longitudinal data in the presence of potential outliers or fat-tailed noises. This paper presents a generalization of MtLMM, called the MtLMM-CM, to properly adjust for censorship due to detection limits of the assay and missingness embodied within multiple outcome variables recorded at irregular occasions. An expectation conditional maximization either (ECME) algorithm is developed to compute parameter estimates using the maximum likelihood (ML) approach. The asymptotic standard errors of the ML estimators of fixed effects are obtained by inverting the empirical information matrix according to Louis' method. The techniques for the estimation of random effects and imputation of missing responses are also investigated. The proposed methodology is illustrated on two real-world examples from HIV-AIDS studies and a simulation study under a variety of scenarios.

Bayesian joint modelling of longitudinal and time to event data: a methodological review

BACKGROUND

In clinical research, there is an increasing interest in joint modelling of longitudinal and time-to-event data, since it reduces bias in parameter estimation and increases the efficiency of statistical inference. Inference and prediction from frequentist approaches of joint models have been extensively reviewed, and due to the recent popularity of data-driven Bayesian approaches, a review on current Bayesian estimation of joint model is useful to draw recommendations for future researches.

METHODS

We have undertaken a comprehensive review on Bayesian univariate and multivariate joint models. We focused on type of outcomes, model assumptions, association structure, estimation algorithm, dynamic prediction and software implementation.

RESULTS

A total of 89 articles have been identified, consisting of 75 methodological and 14 applied articles. The most common approach to model the longitudinal and time-to-event outcomes jointly included linear mixed effect models with proportional hazards. A random effect association structure was generally used for linking the two sub-models. Markov Chain Monte Carlo (MCMC) algorithms were commonly used (93% articles) to estimate the model parameters. Only six articles were primarily focused on dynamic predictions for longitudinal or event-time outcomes.

CONCLUSION

Methodologies for a wide variety of data types have been proposed; however the research is limited if the association between the two outcomes changes over time, and there is also lack of methods to determine the association structure in the absence of clinical background knowledge. Joint modelling has been proved to be beneficial in producing more accurate dynamic prediction; however, there is a lack of sufficient tools to validate the prediction.

Quantile regression-based Bayesian joint modeling analysis of longitudinal-survival data, with application to an AIDS cohort study

In longitudinal studies, it is of interest to investigate how repeatedly measured markers are associated with time to an event. Joint models have received increasing attention on analyzing such complex longitudinal-survival data with multiple data features, but most of them are mean regression-based models. This paper formulates a quantile regression (QR) based joint models in general forms that consider left-censoring due to the limit of detection, covariates with measurement errors and skewness. The joint models consist of three components: (i) QR-based nonlinear mixed-effects Tobit model using asymmetric Laplace distribution for response dynamic process; (ii) nonparametric linear mixed-effects model with skew-normal distribution for mismeasured covariate; and (iii) Cox proportional hazard model for event time. For the purpose of simultaneously estimating model parameters, we propose a Bayesian method to jointly model the three components which are linked through the random effects. We apply the proposed modeling procedure to analyze the Multicenter AIDS Cohort Study data, and assess the performance of the proposed models and method through simulation studies. The findings suggest that our QR-based joint models may provide comprehensive understanding of heterogeneous outcome trajectories at different quantiles, and more reliable and robust results if the data exhibits these features.

Review and Comparison of Computational Approaches for Joint Longitudinal and Time-to-Event Models

Joint models for longitudinal and time-to-event data are useful in situations where an association exists between a longitudinal marker and an event time. These models are typically complicated due to the presence of shared random effects and multiple submodels. As a consequence, software implementation is warranted that is not prohibitively time consuming. While methodological research in this area continues, several statistical software procedures exist to assist in the fitting of some joint models. We review the available implementation for frequentist and Bayesian models in the statistical programming languages R, SAS, and Stata. A description of each procedure is given including estimation techniques, input and data requirements, available options for customization, and some available extensions, such as competing risks models. The software implementations are compared and contrasted through extensive simulation, highlighting their strengths and weaknesses. Data from an ongoing trial on adrenal cancer patients is used to study different nuances of software fitting on a practical example.

Partially linear mixed-effects joint models for skewed and missing longitudinal competing risks outcomes

Longitudinal competing risks data frequently arise in clinical studies. Skewness and missingness are commonly observed for these data in practice. However, most joint models do not account for these data features. In this article, we propose partially linear mixed-effects joint models to analyze skew longitudinal competing risks data with missingness. In particular, to account for skewness, we replace the commonly assumed symmetric distributions by asymmetric distribution for model errors. To deal with missingness, we employ an informative missing data model. The joint models that couple the partially linear mixed-effects model for the longitudinal process, the cause-specific proportional hazard model for competing risks process and missing data process are developed. To estimate the parameters in the joint models, we propose a fully Bayesian approach based on the joint likelihood. To illustrate the proposed model and method, we implement them to an AIDS clinical study. Some interesting findings are reported. We also conduct simulation studies to validate the proposed method.

Bayesian quantile regression-based partially linear mixed-effects joint models for longitudinal data with multiple features

In longitudinal AIDS studies, it is of interest to investigate the relationship between HIV viral load and CD4 cell counts, as well as the complicated time effect. Most of common models to analyze such complex longitudinal data are based on mean-regression, which fails to provide efficient estimates due to outliers and/or heavy tails. Quantile regression-based partially linear mixed-effects models, a special case of semiparametric models enjoying benefits of both parametric and nonparametric models, have the flexibility to monitor the viral dynamics nonparametrically and detect the varying CD4 effects parametrically at different quantiles of viral load. Meanwhile, it is critical to consider various data features of repeated measurements, including left-censoring due to a limit of detection, covariate measurement error, and asymmetric distribution. In this research, we first establish a Bayesian joint models that accounts for all these data features simultaneously in the framework of quantile regression-based partially linear mixed-effects models. The proposed models are applied to analyze the Multicenter AIDS Cohort Study (MACS) data. Simulation studies are also conducted to assess the performance of the proposed methods under different scenarios.

Joint Models for Multiple Longitudinal Processes and Time-to-event Outcome

Joint models are statistical tools for estimating the association between time-to-event and longitudinal outcomes. One challenge to the application of joint models is its computational complexity. Common estimation methods for joint models include a two-stage method, Bayesian and maximum-likelihood methods. In this work, we consider joint models of a time-to-event outcome and multiple longitudinal processes and develop a maximum-likelihood estimation method using the expectation-maximization (EM) algorithm. We assess the performance of the proposed method via simulations and apply the methodology to a data set to determine the association between longitudinal systolic and diastolic blood pressure (BP) measures and time to coronary artery disease (CAD).

Prediction of coronary artery disease risk based on multiple longitudinal biomarkers

In the last decade, few topics in the area of cardiovascular disease (CVD) research have received as much attention as risk prediction. One of the well-documented risk factors for CVD is high blood pressure (BP). Traditional CVD risk prediction models consider BP levels measured at a single time and such models form the basis for current clinical guidelines for CVD prevention. However, in clinical practice, BP levels are often observed and recorded in a longitudinal fashion. Information on BP trajectories can be powerful predictors for CVD events. We consider joint modeling of time to coronary artery disease and individual longitudinal measures of systolic and diastolic BPs in a primary care cohort with up to 20 years of follow-up. We applied novel prediction metrics to assess the predictive performance of joint models. Predictive performances of proposed joint models and other models were assessed via simulations and illustrated using the primary care cohort.

Extending multivariate- t linear mixed models for multiple longitudinal data with censored responses and heavy tails

The analysis of complex longitudinal data is challenging due to several inherent features: (i) more than one series of responses are repeatedly collected on each subject at irregularly occasions over a period of time; (ii) censorship due to limits of quantification of responses arises left- and/or right- censoring effects; (iii) outliers or heavy-tailed noises are possibly embodied within multiple response variables. This article formulates the multivariate- t linear mixed model with censored responses (MtLMMC), which allows the analysts to model such data in the presence of the above described features simultaneously. An efficient expectation conditional maximization either (ECME) algorithm is developed to carry out maximum likelihood estimation of model parameters. The implementation of the E-step relies on the mean and covariance matrix of truncated multivariate- t distributions. To enhance the computational efficiency, two auxiliary permutation matrices are incorporated into the procedure to determine the observed and censored parts of each subject. The proposed methodology is demonstrated via a simulation study and a real application on HIV/AIDS data.

Joint modeling of survival and longitudinal non-survival data: current methods and issues. Report of the DIA Bayesian joint modeling working group

Explicitly modeling underlying relationships between a survival endpoint and processes that generate longitudinal measured or reported outcomes potentially could improve the efficiency of clinical trials and provide greater insight into the various dimensions of the clinical effect of interventions included in the trials. Various strategies have been proposed for using longitudinal findings to elucidate intervention effects on clinical outcomes such as survival. The application of specifically Bayesian approaches for constructing models that address longitudinal and survival outcomes explicitly has been recently addressed in the literature. We review currently available methods for carrying out joint analyses, including issues of implementation and interpretation, identify software tools that can be used to carry out the necessary calculations, and review applications of the methodology.

Mixed-effects varying-coefficient model with skewed distribution coupled with cause-specific varying-coefficient hazard model with random-effects for longitudinal-competing risks data analysis

It is well known that there is strong relationship between HIV viral load and CD4 cell counts in AIDS studies. However, the relationship between them changes during the course of treatment and may vary among individuals. During treatments, some individuals may experience terminal events such as death. Because the terminal event may be related to the individual's viral load measurements, the terminal mechanism is non-ignorable. Furthermore, there exists competing risks from multiple types of events, such as AIDS-related death and other death. Most joint models for the analysis of longitudinal-survival data developed in literatures have focused on constant coefficients and assume symmetric distribution for the endpoints, which does not meet the needs for investigating the nature of varying relationship between HIV viral load and CD4 cell counts in practice. We develop a mixed-effects varying-coefficient model with skewed distribution coupled with cause-specific varying-coefficient hazard model with random-effects to deal with varying relationship between the two endpoints for longitudinal-competing risks survival data. A fully Bayesian inference procedure is established to estimate parameters in the joint model. The proposed method is applied to a multicenter AIDS cohort study. Various scenarios-based potential models that account for partial data features are compared. Some interesting findings are presented.

A Bayesian mixture of semiparametric mixed-effects joint models for skewed-longitudinal and time-to-event data

In longitudinal studies, it is of interest to investigate how repeatedly measured markers in time are associated with a time to an event of interest, and in the mean time, the repeated measurements are often observed with the features of a heterogeneous population, non-normality, and covariate measured with error because of longitudinal nature. Statistical analysis may complicate dramatically when one analyzes longitudinal-survival data with these features together. Recently, a mixture of skewed distributions has received increasing attention in the treatment of heterogeneous data involving asymmetric behaviors across subclasses, but there are relatively few studies accommodating heterogeneity, non-normality, and measurement error in covariate simultaneously arose in longitudinal-survival data setting. Under the umbrella of Bayesian inference, this article explores a finite mixture of semiparametric mixed-effects joint models with skewed distributions for longitudinal measures with an attempt to mediate homogeneous characteristics, adjust departures from normality, and tailor accuracy from measurement error in covariate as well as overcome shortages of confidence in specifying a time-to-event model. The Bayesian mixture of joint modeling offers an appropriate avenue to estimate not only all parameters of mixture joint models but also probabilities of class membership. Simulation studies are conducted to assess the performance of the proposed method, and a real example is analyzed to demonstrate the methodology. The results are reported by comparing potential models with various scenarios.

Bayesian inference on mixed-effects varying-coefficient joint models with skew- t distribution for longitudinal data with multiple features

In AIDS clinical study, two biomarkers, HIV viral load and CD4 cell counts, play important roles. It is well known that there is inverse relationship between the two. Nevertheless, the relationship is not constant but time varying. The mixed-effects varying-coefficient model is capable of capturing the time varying nature of such relationship from both population and individual perspective. In practice, the nucleic acid sequence-based amplification assay is used to measure plasma HIV-1 RNA with a limit of detection (LOD) and the CD4 cell counts are usually measured with much noise and missing data often occur during the treatment. Furthermore, most of the statistical models assume symmetric distribution, such as normal, for the response variables. Often time, normality assumption does not hold in practice. Therefore, it is important to explore all these factors when modeling the real data. In this article, we establish a joint model that accounts for asymmetric and LOD data for the response variable, and covariate measurement error and missingness simultaneously in the mixed-effects varying-coefficient modeling framework. A Bayesian inference procedure is developed to estimate the parameters in the joint model. The proposed model and method are applied to a real AIDS clinical study and various comparisons of a few models are performed.

Joint Analyses of Longitudinal and Time-to-Event Data in Research on Aging: Implications for Predicting Health and Survival

Longitudinal data on aging, health, and longevity provide a wealth of information to investigate different aspects of the processes of aging and development of diseases leading to death. Statistical methods aimed at analyses of time-to-event data jointly with longitudinal measurements became known as the "joint models" (JM). An important point to consider in analyses of such data in the context of studies on aging, health, and longevity is how to incorporate knowledge and theories about mechanisms and regularities of aging-related changes that accumulate in the research field into respective analytic approaches. In the absence of specific observations of longitudinal dynamics of relevant biomarkers manifesting such mechanisms and regularities, traditional approaches have a rather limited utility to estimate respective parameters that can be meaningfully interpreted from the biological point of view. A conceptual analytic framework for these purposes, the stochastic process model of aging (SPM), has been recently developed in the biodemographic literature. It incorporates available knowledge about mechanisms of aging-related changes, which may be hidden in the individual longitudinal trajectories of physiological variables and this allows for analyzing their indirect impact on risks of diseases and death. Despite, essentially, serving similar purposes, JM and SPM developed in parallel in different disciplines with very limited cross-referencing. Although there were several publications separately reviewing these two approaches, there were no publications presenting both these approaches in some detail. Here, we overview both approaches jointly and provide some new modifications of SPM. We discuss the use of stochastic processes to capture biological variation and heterogeneity in longitudinal patterns and important and promising (but still largely underused) applications of JM and SPM to predictions of individual and population mortality and health-related outcomes.

Jointly modeling time-to-event and longitudinal data: A Bayesian approach

This article explores Bayesian joint models of event times and longitudinal measures with an attempt to overcome departures from normality of the longitudinal response, measurement errors, and shortages of confidence in specifying a parametric time-to-event model. We allow the longitudinal response to have a skew distribution in the presence of measurement errors, and assume the time-to-event variable to have a nonparametric prior distribution. Posterior distributions of the parameters are attained simultaneously for inference based on Bayesian approach. An example from a recent AIDS clinical trial illustrates the methodology by jointly modeling the viral dynamics and the time to decrease in CD4/CD8 ratio in the presence of CD4 counts with measurement errors and to compare potential models with various scenarios and different distribution specifications. The analysis outcome indicates that the time-varying CD4 covariate is closely related to the first-phase viral decay rate, but the time to CD4/CD8 decrease is not highly associated with either the two viral decay rates or the CD4 changing rate over time. These findings may provide some quantitative guidance to better understand the relationship of the virological and immunological responses to antiretroviral treatments.

Analysis of Longitudinal and Survival Data: Joint Modeling, Inference Methods, and Issues

In the past two decades, joint models of longitudinal and survival data have received much attention in the literature. These models are often desirable in the following situations: (i) survival models with measurement errors or missing data in time-dependent covariates, (ii) longitudinal models with informative dropouts, and (iii) a survival process and a longitudinal process are associated via latent variables. In these cases, separate inferences based on the longitudinal model and the survival model may lead to biased or inefficient results. In this paper, we provide a brief overview of joint models for longitudinal and survival data and commonly used methods, including the likelihood method and two-stage methods.