Joint modeling of longitudinal and survival data with missing and left-censored time-varying covariates

GPU Accelerated Estimation of a Shared Random Effect Joint Model for Dynamic Prediction

In longitudinal cohort studies, it is often of interest to predict the risk of a terminal clinical event using longitudinal predictor data among subjects at risk by the time of the prediction. The at-risk population changes over time; so does the association between predictors and the outcome, as well as the accumulating longitudinal predictor history. The dynamic nature of this prediction problem has received increasing interest in the literature, but computation often poses a challenge. The widely used joint model of longitudinal and survival data often comes with intensive computation and excessive model fitting time, due to numerical optimization and the analytically intractable high-dimensional integral in the likelihood function. This problem is exacerbated when the model is fit to a large dataset or the model involves multiple longitudinal predictors with nonlinear trajectories. This challenge can be addressed from an algorithmic perspective, by a novel two-stage estimation procedure, and from a computing perspective, by Graphics Processing Unit (GPU) programming. The latter is implemented through PyTorch, an emerging deep learning framework. The numerical studies demonstrate that the proposed algorithm and software can substantially speed up the estimation of the joint model, particularly with large datasets. The numerical studies also concluded that accounting for nonlinearity in longitudinal predictor trajectories can improve the prediction accuracy in comparison to joint modeling that ignore nonlinearity.

Analysis of Recurrent Times-to-Clinical Malaria Episodes and Plasmodium falciparum Parasitemia: A Joint Modeling Approach Applied to a Cohort Data

Background

Recurrent clinical malaria episodes due to Plasmodium falciparum parasite infection are common in endemic regions. With each infection, acquired immunity develops, making subsequent disease episodes less likely. To capture the effect of acquired immunity to malaria, it may be necessary to model recurrent clinical disease episodes jointly with P. falciparum parasitemia data. A joint model of longitudinal parasitemia and time-to-first clinical malaria episode (single-event joint model) may be inaccurate because acquired immunity is lost when subsequent episodes are excluded. This study's informativeness assessed whether joint modeling of recurrent clinical malaria episodes and parasitemia is more accurate than a single-event joint model where the subsequent episodes are ignored.

Methods

The single event joint model comprised Cox Proportional Hazards (PH) sub-model for time-to-first clinical malaria episode and Negative Binomial (NB) mixed-effects sub-model for the longitudinal parasitemia. The recurrent events joint model extends the survival sub-model to a Gamma shared frailty model to include all recurrent clinical episodes. The models were applied to cohort data from Malawi. Simulations were also conducted to assess the performance of the model under different conditions.

Results

The recurrent events joint model, which yielded higher hazard ratios of clinical malaria, was more precise and in most cases produced smaller standard errors than the single-event joint model; hazard ratio (HR) = 1.42, [95% confidence interval [CI]: 1.22, 2.03] vs. HR = 1.29, [95% CI:1.60, 2.45] among participants who reported not to use LLINs every night compared to those who used the nets every night; HR = 0.96, [ 95% CI: 0.94, 0.98] vs. HR = 0.81, [95% CI: 0.75, 0.88] for each 1-year increase in participants' age; and HR = 1.36, [95% CI: 1.05, 1.75] vs. HR = 1.10, [95% CI: 0.83, 4.11] for observations during the rainy season compared to the dry season.

Conclusion

The recurrent events joint model in this study provides a way of estimating the risk of recurrent clinical malaria in a cohort where the effect of immunity on malaria disease acquired due to P. falciparum parasitemia with aging is captured. The simulation study has shown that if correctly specified, the recurrent events joint model can give risk estimates with low bias.

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.

Congestive Heart Failure Patients’ Pulse Rate Progression and Time to Death at Debre Tabor Referral Hospital, Ethiopia

Background. Heart failure is a progressive condition marked by worsening symptoms such as shortness of breath, coughing, exhaustion and lethargy, fluid retention with swelling of the legs and abdomen, and a reduced ability to exercise. As a result, this study aims to use a joint model application to determine the joint risk factors of longitudinal change in pulse rate and time to death of congestive heart failure patients and their association admitted to a hospital. Methods. A retrospective study was undertaken on congestive heart failure patients admitted to the Debre Tabor Referral Hospital from January 2016 to December 2019. A statistical joint modeling strategy was employed to match the repeated biomarker pulse rate and a survival outcome at the same time. A total of 271 patients with congestive heart failure were chosen. Data were analyzed with R statistical software via joineRML. Results. According to the findings, the association between longitudinal changes in pulse rate and time to death in heart failure patients is statistically significant. Sex, residence, left ventricular injection fraction, New York Heart Association class, and diabetes mellitus were all found to be significant risk factors for congestive heart failure patients’ short survival time to death. Age, sex, residence, hypertension, left ventricular injection fraction, congestive heart failure, diabetes mellitus, tuberculosis, and etiology were all significant contributors in pulse rate progression. Conclusion. The computed association parameters revealed subject-specific values. The subject-specific linear time slope of PR measurement was positively related to the hazard rate of time to death of CHF patients in the study area. To reduce the risk level of CHF, health professionals, governmental organizations, and nongovernmental organizations must promote and allocate a suitable amount of budget for the treatment of CHF patients.

A Comparative Study of Different Joint Modeling Approaches for HIV/AIDS Patients in Southern Iran

Background:

The prevalence of HIV/AIDS has been increasing in Iran, especially amongst the young population, recently. The joint model (JM) is a statistical method that represents an effective strategy to incorporate all information of repeated measurements and survival outcomes simultaneously. In many theoretical studies, the population under the study were heterogeneous. This study aimed at comparing three approaches by considering heterogeneity in the patients.

Methods:

This study was conducted on 750 archived files of patients infected with HIV in Fars Province, southern Iran, from 1994 to 2017. Proposed Approach (PA), Joint Latent Class Models (JLCM), and Separated Approach (SA) were compared to evaluate the influence covariates on the longitudinal and time-to-event outcomes in the heterogeneous HIV/AIDS patients.

Results:

Gender (P<0.001) and HCV (P<0.01) were two significant covariates in the classification of HIV/AIDS patients. Time had a significant effect on CD4 (P<0.001) in both classes in the three approaches. In PA and SA, females had higher CD4 than males (P<0.001) in the first class. In JLCM, females had higher CD4 than males (P<0.01) in both classes. The patients with higher Hgb had also higher CD4 (P<0.001) in both classes in the three approaches. HCV reduced the CD4 significantly in both classes in PA (P<0.05) and SA (P<0.001). Within the survival sub-model, HCV reduced survival rate significantly in the second class in PA (P<0.05), JLCM (P<0.01) and SA (P<0.001).

Conclusion:

PA was an appropriate approach for joint modeling longitudinal and survival outcomes for this heterogeneous population.

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.

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.

A latent class approach for joint modeling of a time-to-event outcome and multiple longitudinal biomarkers subject to limits of detection

Multiple biomarkers on different biological pathways are often measured over time to investigate the complex mechanism of disease development and progression. Identification of informative subpopulation patterns of longitudinal biomarkers and clinical endpoint may assist in risk stratification and provide insights into new therapeutic targets. Motivated by a multicenter study to assess the inflammatory markers of sepsis in patients with community-acquired pneumonia, we propose a joint latent class analysis of multiple biomarkers and a time-to-event outcome while accounting for censored biomarker measurements due to detection limits. The interrelationship between biomarker trajectories and clinical endpoint is fully captured by a latent class structure, which reveals the subpopulation profiles of biomarkers and clinical outcome. The estimation of joint latent class models becomes more complicated when biomarkers are subject to detection limits. Based on a Metropolis-Hastings method, we develop a Monte Carlo Expectation-Maximization (MCEM) algorithm to estimate model parameters. We demonstrate the satisfactory performance of our MCEM algorithm using simulation studies, and apply our method to the motivating study to examine the heterogeneous patterns of cytokine responses to pneumonia and associated mortality risks.

A joint model for mixed and truncated longitudinal data and survival data, with application to HIV vaccine studies

In HIV vaccine studies, a major research objective is to identify immune response biomarkers measured longitudinally that may be associated with risk of HIV infection. This objective can be assessed via joint modeling of longitudinal and survival data. Joint models for HIV vaccine data are complicated by the following issues: (i) left truncations of some longitudinal data due to lower limits of quantification; (ii) mixed types of longitudinal variables; (iii) measurement errors and missing values in longitudinal measurements; (iv) computational challenges associated with likelihood inference. In this article, we propose a joint model of complex longitudinal and survival data and a computationally efficient method for approximate likelihood inference to address the foregoing issues simultaneously. In particular, our model does not make unverifiable distributional assumptions for truncated values, which is different from methods commonly used in the literature. The parameters are estimated based on the h-likelihood method, which is computationally efficient and offers approximate likelihood inference. Moreover, we propose a new approach to estimate the standard errors of the h-likelihood based parameter estimates by using an adaptive Gauss-Hermite method. Simulation studies show that our methods perform well and are computationally efficient. A comprehensive data analysis is also presented.

Joint modelling of time-to-clinical malaria and parasite count in a cohort in an endemic area

Background

In malaria endemic areas such as sub-Saharan Africa, repeated exposure to malaria results in acquired immunity to clinical disease but not infection. In prospective studies, time-to-clinical malaria and longitudinal parasite count trajectory are often analysed separately which may result in inefficient estimates since these two processes can be associated. Including parasite count as a time-dependent covariate in a model of time-to-clinical malaria episode may also be inaccurate because while clinical malaria disease frequently leads to treatment which may instantly affect the level of parasite count, standard time-to-event models require that time-dependent covariates be external to the event process. We investigated whether jointly modelling time-to-clinical malaria disease and longitudinal parasite count improves precision in risk factor estimates and assessed the strength of association between the hazard of clinical malaria and parasite count.

Methods

Using a cohort data of participants enrolled with uncomplicated malaria in Malawi, a conventional Cox Proportional Hazards (PH) model of time-to-first clinical malaria episode with time-dependent parasite count was compared with three competing joint models. The joint models had different association structures linking a quasi-Poisson mixed-effects of parasite count and event-time Cox PH sub-models.

Results

There were 120 participants of whom 115 (95.8%) had >1 follow-up visit and 100 (87.5%) experienced the episode. Adults >15 years being reference, log hazard ratio for children <5 years was 0.74 (95% CI: 0.17, 1.26) in the joint model with best fit vs. 0.62 (95% CI: 0.04, 1.18) from the conventional Cox PH model. The log hazard ratio for the 5-15 years was 0.72 (95% CI: 0.22, 1.22) in the joint model vs.0.63 (95% CI: 0.11, 1.17) in the Cox PH model. The area under parasite count trajectory was strongly associated with the risk of clinical malaria, with a unit increase corresponding to-0.0012 (95% CI: -0.0021, -0.0004) decrease in log hazard ratio.

Conclusion

Jointly modelling longitudinal parasite count and time-to-clinical malaria disease improves precision in log hazard ratio estimates compared to conventional time-dependent Cox PH model. The improved precision of joint modelling may improve study efficiency and allow for design of clinical trials with relatively lower sample sizes with increased power.

Cox model with interval-censored covariate in cohort studies

In cohort studies the outcome is often time to a particular event, and subjects are followed at regular intervals. Periodic visits may also monitor a secondary irreversible event influencing the event of primary interest, and a significant proportion of subjects develop the secondary event over the period of follow-up. The status of the secondary event serves as a time-varying covariate, but is recorded only at the times of the scheduled visits, generating incomplete time-varying covariates. While information on a typical time-varying covariate is missing for entire follow-up period except the visiting times, the status of the secondary event are unavailable only between visits where the status has changed, thus interval-censored. One may view interval-censored covariate of the secondary event status as missing time-varying covariates, yet missingness is partial since partial information is provided throughout the follow-up period. Current practice of using the latest observed status produces biased estimators, and the existing missing covariate techniques cannot accommodate the special feature of missingness due to interval censoring. To handle interval-censored covariates in the Cox proportional hazards model, we propose an available-data estimator, a doubly robust-type estimator as well as the maximum likelihood estimator via EM algorithm and present their asymptotic properties. We also present practical approaches that are valid. We demonstrate the proposed methods using our motivating example from the Northern Manhattan Study.

A Proposed Approach for Joint Modeling of the Longitudinal and Time-To-Event Data in Heterogeneous Populations: An Application to HIV/AIDS's Disease

In recent years, the joint models have been widely used for modeling the longitudinal and time-to-event data simultaneously. In this study, we proposed an approach (PA) to study the longitudinal and survival outcomes simultaneously in heterogeneous populations. PA relaxes the assumption of conditional independence (CI). We also compared PA with joint latent class model (JLCM) and separate approach (SA) for various sample sizes (150, 300, and 600) and different association parameters (0, 0.2, and 0.5). The average bias of parameters estimation (AB-PE), average SE of parameters estimation (ASE-PE), and coverage probability of the 95% confidence interval (CP) among the three approaches were compared. In most cases, when the sample sizes increased, AB-PE and ASE-PE decreased for the three approaches, and CP got closer to the nominal level of 0.95. When there was a considerable association, PA in comparison with SA and JLCM performed better in the sense that PA had the smallest AB-PE and ASE-PE for the longitudinal submodel among the three approaches for the small and moderate sample sizes. Moreover, JLCM was desirable for the none-association and the large sample size. Finally, the evaluated approaches were applied on a real HIV/AIDS dataset for validation, and the results were compared.

Clustering high-dimensional mixed data to uncover sub-phenotypes: joint analysis of phenotypic and genotypic data

The LIPGENE-SU.VI.MAX study, like many others, recorded high-dimensional continuous phenotypic data and categorical genotypic data. LIPGENE-SU.VI.MAX focuses on the need to account for both phenotypic and genetic factors when studying the metabolic syndrome (MetS), a complex disorder that can lead to higher risk of type 2 diabetes and cardiovascular disease. Interest lies in clustering the LIPGENE-SU.VI.MAX participants into homogeneous groups or sub-phenotypes, by jointly considering their phenotypic and genotypic data, and in determining which variables are discriminatory. A novel latent variable model that elegantly accommodates high dimensional, mixed data is developed to cluster LIPGENE-SU.VI.MAX participants using a Bayesian finite mixture model. A computationally efficient variable selection algorithm is incorporated, estimation is via a Gibbs sampling algorithm and an approximate BIC-MCMC criterion is developed to select the optimal model. Two clusters or sub-phenotypes ('healthy' and 'at risk') are uncovered. A small subset of variables is deemed discriminatory, which notably includes phenotypic and genotypic variables, highlighting the need to jointly consider both factors. Further, 7 years after the LIPGENE-SU.VI.MAX data were collected, participants underwent further analysis to diagnose presence or absence of the MetS. The two uncovered sub-phenotypes strongly correspond to the 7-year follow-up disease classification, highlighting the role of phenotypic and genotypic factors in the MetS and emphasising the potential utility of the clustering approach in early screening. Additionally, the ability of the proposed approach to define the uncertainty in sub-phenotype membership at the participant level is synonymous with the concepts of precision medicine and nutrition. Copyright © 2017 John Wiley & Sons, Ltd.

Joint modeling of longitudinal and survival data with a covariate subject to a limit of detection

We develop and study an innovative method for jointly modeling longitudinal response and time-to-event data with a covariate subject to a limit of detection. The joint model assumes a latent process based on random effects to describe the association between longitudinal and time-to-event data. We study the role of the association parameter on the regression parameters estimators. We model the longitudinal and survival outcomes using linear mixed-effects and Weibull frailty models, respectively. Because of the limit of detection, missing covariate (explanatory variable, x) values may lead to the non-ignorable missing, resulting in biased parameter estimates with poor coverage probabilities of the confidence interval. We define and estimate the probability of missing due to the limit of detection. Then we develop a novel joint density and hence the likelihood function that incorporates the effect of left-censored covariate. Monte Carlo simulations show that the estimators of the proposed method are approximately unbiased and provide expected coverage probabilities for both longitudinal and survival submodels parameters. We also present an application of the proposed method using a large clinical dataset of pneumonia patients obtained from the Genetic and Inflammatory Markers of Sepsis study.

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 longitudinal and time-to-event data: a review of reporting quality with a view to meta-analysis

BACKGROUND

Joint models for longitudinal and time-to-event data are commonly used to simultaneously analyse correlated data in single study cases. Synthesis of evidence from multiple studies using meta-analysis is a natural next step but its feasibility depends heavily on the standard of reporting of joint models in the medical literature. During this review we aim to assess the current standard of reporting of joint models applied in the literature, and to determine whether current reporting standards would allow or hinder future aggregate data meta-analyses of model results.

METHODS

We undertook a literature review of non-methodological studies that involved joint modelling of longitudinal and time-to-event medical data. Study characteristics were extracted and an assessment of whether separate meta-analyses for longitudinal, time-to-event and association parameters were possible was made.

RESULTS

The 65 studies identified used a wide range of joint modelling methods in a selection of software. Identified studies concerned a variety of disease areas. The majority of studies reported adequate information to conduct a meta-analysis (67.7% for longitudinal parameter aggregate data meta-analysis, 69.2% for time-to-event parameter aggregate data meta-analysis, 76.9% for association parameter aggregate data meta-analysis). In some cases model structure was difficult to ascertain from the published reports.

CONCLUSIONS

Whilst extraction of sufficient information to permit meta-analyses was possible in a majority of cases, the standard of reporting of joint models should be maintained and improved. Recommendations for future practice include clear statement of model structure, of values of estimated parameters, of software used and of statistical methods applied.