Joint modeling of multivariate longitudinal measurements and survival data with applications to Parkinson's disease

Joint modeling of longitudinal CD4 count data and time to first occurrence of composite outcome

In this study, we jointly modeled longitudinal CD4 count data and survival outcome (time-to-first occurrence of composite outcome of death, cardiac tamponade or constriction) in other to investigate the effects of Mycobacterium indicus pranii immunotherapy and the CD4 count measurements on the hazard of the composite outcome among patients with HIV and tuberculous (TB) pericarditis. In this joint modeling framework, the models for longitudinal and the survival data are linked by an association structure. The association structure represents the hazard of the event for 1-unit increase in the longitudinal measurement. Models fitting and parameter estimation were carried out using R version 4.2.3. The association structure that represents the strength of the association between the hazard for an event at time point j and the area under the longitudinal trajectory up to the same time j provides the best fit. We found that 1-unit increase in CD4 count results in 2 % significant reduction in the hazard of the composite outcome. Among HIV and TB pericarditis individuals, the hazard of the composite outcome does not differ between of M.indicus pranii versus placebo. Application of joint models to investigate the effect of M.indicus pranii on the hazard of the composite outcome is limited. Hence, this study provides information on the effect of M.indicus pranii on the hazard of the composite outcome among HIV and TB pericarditis patients.

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.

A two‐level copula joint model for joint analysis of longitudinal and competing risks data

In this article, we propose a two-level copula joint model to analyze clinical data with multiple disparate continuous longitudinal outcomes and multiple event-times in the presence of competing risks. At the first level, we use a copula to model the dependence between competing latent event-times, in the process constructing the submodel for the observed event-time, and employ the Gaussian copula to construct the submodel for the longitudinal outcomes that accounts for their conditional dependence; these submodels are glued together at the second level via the Gaussian copula to construct a joint model that incorporates conditional dependence between the observed event-time and the longitudinal outcomes. To have the flexibility to accommodate skewed data and examine possibly different covariate effects on quantiles of a non-Gaussian outcome, we propose linear quantile mixed models for the continuous longitudinal data. We adopt a Bayesian framework for model estimation and inference via Markov Chain Monte Carlo sampling. We examine the performance of the copula joint model through a simulation study and show that our proposed method outperforms the conventional approach assuming conditional independence with smaller biases and better coverage probabilities of the Bayesian credible intervals. Finally, we carry out an analysis of clinical data on renal transplantation for illustration.

Prognostic Modeling of Parkinson's Disease Progression Using Early Longitudinal Patterns of Change

BACKGROUND

Predicting Parkinson's disease (PD) progression may enable better adaptive and targeted treatment planning.

OBJECTIVE

Develop a prognostic model using multiple, easily acquired longitudinal measures to predict temporal clinical progression from Hoehn and Yahr (H&Y) stage 1 or 2 to stage 3 in early PD.

METHODS

Predictive longitudinal measures of PD progression were identified by the joint modeling method. Measures were extracted by multivariate functional principal component analysis methods and used as covariates in Cox proportional hazards models. The optimal model was developed from the Parkinson's Progression Marker Initiative (PPMI) data set and confirmed with external validation from the Longitudinal and Biomarker Study in PD (LABS-PD) study.

RESULTS

The proposed prognostic model with longitudinal information of selected clinical measures showed significant advantages in predicting PD temporal progression in comparison to a model with only baseline information (iAUC = 0.812 vs. 0.743). The modeling results allowed the development of a prognostic index for categorizing PD patients into low, mid, and high risk of progression to HY 3 that is offered to facilitate physician-patient discussion on prognosis.

CONCLUSION

Incorporating longitudinal information of multiple clinical measures significantly enhances predictive performance of prognostic models. Furthermore, the proposed prognostic index enables clinicians to classify patients into different risk groups, which could be adaptively updated as new longitudinal information becomes available. Modeling of this type allows clinicians to utilize observational data sets that inform on disease natural history and specifically, for precision medicine, allows the insertion of a patient's clinical data to calculate prognostic estimates at the individual case level. © 2021 International Parkinson and Movement Disorder Society.

A flexible joint model for multiple longitudinal biomarkers and a time-to-event outcome: With applications to dynamic prediction using highly correlated biomarkers

In biomedical studies it is common to collect data on multiple biomarkers during study follow-up for dynamic prediction of a time-to-event clinical outcome. The biomarkers are typically intermittently measured, missing at some event times, and may be subject to high biological variations, which cannot be readily used as time-dependent covariates in a standard time-to-event model. Moreover, they can be highly correlated if they are from in the same biological pathway. To address these issues, we propose a flexible joint model framework that models the multiple biomarkers with a shared latent reduced rank longitudinal principal component model and correlates the latent process to the event time by the Cox model for dynamic prediction of the event time. The proposed joint model for highly correlated biomarkers is more flexible than some existing methods since the latent trajectory shared by the multiple biomarkers does not require specification of a priori parametric time trend and is determined by data. We derive an expectation-maximization (EM) algorithm for parameter estimation, study large sample properties of the estimators, and adapt the developed method to make dynamic prediction of the time-to-event outcome. Bootstrap is used for standard error estimation and inference. The proposed method is evaluated using simulations and illustrated on a lung transplant data to predict chronic lung allograft dysfunction (CLAD) using chemokines measured in bronchoalveolar lavage fluid of the patients.

Pathway testing for longitudinal metabolomics

We propose a top‐down approach for pathway analysis of longitudinal metabolite data. We apply a score test based on a shared latent process mixed model which can identify pathways with differentially progressing metabolites. The strength of our approach is that it can handle unbalanced designs, deals with potential missing values in the longitudinal markers, and gives valid results even with small sample sizes. Contrary to bottom‐up approaches, correlations between metabolites are explicitly modeled leveraging power gains. For large pathway sizes, a computationally efficient solution is proposed based on pseudo‐likelihood methodology. We demonstrate the advantages of the proposed method in identification of differentially expressed pathways through simulation studies. Finally, longitudinal metabolite data from a mice experiment is analyzed to demonstrate our methodology.

Predicting Alzheimer's disease based on survival data and longitudinally measured performance on cognitive and functional scales

This study assessed how well longitudinally taken cognitive and functional scales from people with mild cognitive impairment (MCI) predict conversion to Alzheimer's disease (AD). Participants were individuals with baseline MCI from the Alzheimer's Disease Neuroimaging Initiative. Scales included the Alzheimer Disease Assessment Scale-Cognitive (ADAS-Cog) 11 and 13, the Mini Mental State Examination (MMSE), and the Functional Assessment Questionnaire (FAQ). A joint modelling approach compared performance on the four scales for dynamic prediction of risk for AD. The goodness of fit measures included log likelihood, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The area under the curve (AUC) of the receiver operating characteristic assessed predictive accuracy. The parameter α in the ADAS-Cog11, ADAS-Cog13, MMSE, and FAQ joint models was statistically significant. Joint MMSE and FAQ models had better goodness of fit. FAQ had the best predictive accuracy. Cognitive and functional impairment assessment scales are strong screening predictors when repeated measures are available. They could be useful for predicting risk for AD in primary healthcare.

Incorporating longitudinal biomarkers for dynamic risk prediction in the era of big data: A pseudo-observation approach

Longitudinal biomarker data are often collected in studies, providing important information regarding the probability of an outcome of interest occurring at a future time. With many new and evolving technologies for biomarker discovery, the number of biomarker measurements available for analysis of disease progression has increased dramatically. A large amount of data provides a more complete picture of a patient's disease progression, potentially allowing us to make more accurate and reliable predictions, but the magnitude of available data introduces challenges to most statistical analysts. Existing approaches suffer immensely from the curse of dimensionality. In this article, we propose methods for making dynamic risk predictions using repeatedly measured biomarkers of a large dimension, including cases when the number of biomarkers is close to the sample size. The proposed methods are computationally simple, yet sufficiently flexible to capture complex relationships between longitudinal biomarkers and potentially censored events times. The proposed approaches are evaluated by extensive simulation studies and are further illustrated by an application to a data set from the Nephrotic Syndrome Study Network.

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.

Dynamic prediction of Alzheimer's disease progression using features of multiple longitudinal outcomes and time-to-event data

This paper is motivated by combining serial neurocognitive assessments and other clinical variables for monitoring the progression of Alzheimer's disease (AD). We propose a novel framework for the use of multiple longitudinal neurocognitive markers to predict the progression of AD. The conventional joint modeling longitudinal and survival data approach is not applicable when there is a large number of longitudinal outcomes. We introduce various approaches based on functional principal component for dimension reduction and feature extraction from multiple longitudinal outcomes. We use these features to extrapolate the health outcome trajectories and use scores on these features as predictors in a Cox proportional hazards model to conduct predictions over time. We propose a personalized dynamic prediction framework that can be updated as new observations collected to reflect the patient's latest prognosis, and thus intervention could be initiated in a timely manner. Simulation studies and application to the Alzheimer's Disease Neuroimaging Initiative dataset demonstrate the robustness of the method for the prediction of future health outcomes and risks of target events under various scenarios.

A joint model for multiple dynamic processes and clinical endpoints: Application to Alzheimer's disease

As other neurodegenerative diseases, Alzheimer's disease, the most frequent dementia in the elderly, is characterized by multiple progressive impairments in the brain structure and in clinical functions such as cognitive functioning and functional disability. Until recently, these components were mostly studied independently because no joint model for multivariate longitudinal data and time to event was available in the statistical community. Yet, these components are fundamentally interrelated in the degradation process toward dementia and should be analyzed together. We thus propose a joint model to simultaneously describe the dynamics of multiple correlated components. Each component, defined as a latent process, is measured by one or several continuous markers (not necessarily Gaussian). Rather than considering the associated time to diagnosis as in standard joint models, we assume diagnosis corresponds to the passing above a covariate-specific threshold (to be estimated) of a pathological process that is modeled as a combination of the component-specific latent processes. This definition captures the clinical complexity of diagnoses such as dementia diagnosis but also benefits from simplifications for the computation of maximum likelihood estimates. We show that the model and estimation procedure can also handle competing clinical endpoints. The estimation procedure, implemented in an R package, is validated by simulations and the method is illustrated on a large French population-based cohort of cerebral aging in which we focused on the dynamics of three clinical manifestations and the associated risk of dementia and death before dementia.

Joint modeling of multiple repeated measures and survival data using multidimensional latent trait linear mixed model

Impairment caused by Amyotrophic lateral sclerosis (ALS) is multidimensional (e.g. bulbar, fine motor, gross motor) and progressive. Its multidimensional nature precludes a single outcome to measure disease progression. Clinical trials of ALS use multiple longitudinal outcomes to assess the treatment effects on overall improvement. A terminal event such as death or dropout can stop the follow-up process. Moreover, the time to the terminal event may be dependent on the multivariate longitudinal measurements. In this article, we develop a joint model consisting of a multidimensional latent trait linear mixed model (MLTLMM) for the multiple longitudinal outcomes, and a proportional hazards model with piecewise constant baseline hazard for the event time data. Shared random effects are used to link together two models. The model inference is conducted using a Bayesian framework via Markov chain Monte Carlo simulation implemented in Stan language. Our proposed model is evaluated by simulation studies and is applied to the Ceftriaxone study, a motivating clinical trial assessing the effect of ceftriaxone on ALS patients.

Estimating the Evolution of Disease in the Parkinson's Progression Markers Initiative

Parkinson's disease is the second most common neurological disease and affects about 1% of persons over the age of 60 years. Due to the lack of approved surrogate markers, confirmation of the disease still requires postmortem examination. Identifying and validating biomarkers are essential steps toward improving clinical diagnosis and accelerating the search for therapeutic drugs to ameliorate disease symptoms. Until recently, statistical analysis of multicohort longitudinal studies of neurodegenerative diseases has usually been restricted to a single analysis per outcome with simple comparisons between diagnostic groups. However, an important methodological consideration is to allow the modeling framework to handle multiple outcomes simultaneously and consider the transitions between diagnostic groups. This enables researchers to monitor multiple trajectories, correctly account for the correlation among biomarkers, and assess how these associations may jointly change over the long-term course of disease. In this study, we apply a latent time joint mixed-effects model to study biomarker progression and disease dynamics in the Parkinson's Progression Markers Initiative (PPMI) and examine which markers might be most informative in the earliest phases of disease. The results reveal that, even though diagnostic category was not included in the model, it seems to accurately reflect the temporal ordering of the disease state consistent with diagnosis categorization at baseline. In addition, results indicated that the specific binding ratio on striatum and the total Unified Parkinson's Disease Rating Scale (UPDRS) show high discriminability between disease stages. An extended latent time joint mixed-effects model with heterogeneous latent time variance also showed improvement in model fit in a simulation study and when applied to real data.

Bayesian proportional hazards model with latent variables

We consider a joint modeling approach that incorporates latent variables into a proportional hazards model to examine the observed and latent risk factors of the failure time of interest. An exploratory factor analysis model is used to characterize the latent risk factors through multiple observed variables. In commonly used confirmatory factor analysis, the number of latent variables and their observed indicators are specified prior to analysis. By contrast, the exploratory factor analysis model allows such information to be fully determined by the data. A Bayesian approach coupled with efficient sampling methods is developed to conduct statistical inference, and the performance of the proposed methodology is confirmed through simulations. The model is applied to a study on the risk factors of chronic kidney disease for patients with type 2 diabetes.

Bivariate longitudinal data analysis: a case of hypertensive patients at Felege Hiwot Referral Hospital, Bahir Dar, Ethiopia

Objective

Longitudinal data are often collected to study the evolution of biomedical markers. The study of the joint evolution of response variables concerning hypertension over time was the aim of this paper. A hospital based retrospective data were collected from September 2014 to August 2015 to identify factors that affect hypertensive. The joint mixed effect model with unstructured covariance was fitted. A total of 172 patients screened for antihypertensive drugs treated were longitudinally considered from Felege Hiwot referral.

Results

The joint mixed effect model with unstructured covariance (AIC: 12,236.9 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi_{12}^{2} $$\end{document}χ122 = 1007.8, P < 10⁻⁴) was significantly best fit to the data. The correlation between the evolutions of DBP and SBP was 0.429 and the evolution of the association between responses over-time was found 0.257. Among all covariates included in joint-mixed-effect-models, sex, residence, related disease and time were statistically significant on evolution of systolic and diastolic blood pressure. The joint modeling of longitudinal bivariate responses is necessary to explore the association between paired response variables like systolic and diastolic blood pressure. Fitting joint model with modern computing method is recommended to address questions for association of the evolutions with better accuracy.

DYNAMIC PREDICTION FOR MULTIPLE REPEATED MEASURES AND EVENT TIME DATA: AN APPLICATION TO PARKINSON'S DISEASE

In many clinical trials studying neurodegenerative diseases such as Parkinson's disease (PD), multiple longitudinal outcomes are collected to fully explore the multidimensional impairment caused by this disease. If the outcomes deteriorate rapidly, patients may reach a level of functional disability sufficient to initiate levodopa therapy for ameliorating disease symptoms. An accurate prediction of the time to functional disability is helpful for clinicians to monitor patients' disease progression and make informative medical decisions. In this article, we first propose a joint model that consists of a semiparametric multilevel latent trait model (MLLTM) for the multiple longitudinal outcomes, and a survival model for event time. The two submodels are linked together by an underlying latent variable. We develop a Bayesian approach for parameter estimation and a dynamic prediction framework for predicting target patients' future outcome trajectories and risk of a survival event, based on their multivariate longitudinal measurements. Our proposed model is evaluated by simulation studies and is applied to the DATATOP study, a motivating clinical trial assessing the effect of deprenyl among patients with early PD.

Bayesian multivariate augmented Beta rectangular regression models for patient-reported outcomes and survival data

Many longitudinal studies (e.g. observational studies and randomized clinical trials) have collected multiple rating scales at each visit in the form of patient-reported outcomes (PROs) in the close unit interval [0 ,1]. We propose a joint modeling framework to address the issues from the following data features: (1) multiple correlated PROs; (2) the presence of the boundary values of zeros and ones; (3) extreme outliers and heavy tails; (4) the PRO-dependent terminal events such as death and dropout. Our modeling framework consists of a multivariate augmented mixed-effects sub-model based on Beta rectangular distributions for the multiple longitudinal outcomes and a Cox model for the terminal events. The simulation studies suggest that in the presence of outliers, heavy tails, and dependent terminal event, our proposed models provide more accurate parameter estimates than the joint model based on Beta distributions. The proposed models are applied to the motivating Long-term Study-1 (LS-1 study, n = 1741) of Parkinson's disease patients.

Bayesian Hierarchical Joint Modeling Using Skew-Normal/Independent Distributions

The multiple longitudinal outcomes collected in many clinical trials are often analyzed by multilevel item response theory (MLIRT) models. The normality assumption for the continuous outcomes in the MLIRT models can be violated due to skewness and/or outliers. Moreover, patients' follow-up may be stopped by some terminal events (e.g., death or dropout) which are dependent on the multiple longitudinal outcomes. We proposed a joint modeling framework based on the MLIRT model to account for three data features: skewness, outliers, and dependent censoring. Our method development was motivated by a clinical study for Parkinson's disease.

Prediction of Conversion to Alzheimer's Disease with Longitudinal Measures and Time-To-Event Data

BACKGROUND

Identifying predictors of conversion to Alzheimer's disease (AD) is critically important for AD prevention and targeted treatment.

OBJECTIVE

To compare various clinical and biomarker trajectories for tracking progression and predicting conversion from amnestic mild cognitive impairment to probable AD.

METHODS

Participants were from the ADNI-1 study. We assessed the ability of 33 longitudinal biomarkers to predict time to AD conversion, accounting for demographic and genetic factors. We used joint modelling of longitudinal and survival data to examine the association between changes of measures and disease progression. We also employed time-dependent receiver operating characteristic method to assess the discriminating capability of the measures.

RESULTS

23 of 33 longitudinal clinical and imaging measures are significant predictors of AD conversion beyond demographic and genetic factors. The strong phenotypic and biological predictors are in the cognitive domain (ADAS-Cog; RAVLT), functional domain (FAQ), and neuroimaging domain (middle temporal gyrus and hippocampal volume). The strongest predictor is ADAS-Cog 13 with an increase of one SD in ADAS-Cog 13 increased the risk of AD conversion by 2.92 times.

CONCLUSION

Prediction of AD conversion can be improved by incorporating longitudinal change information, in addition to baseline characteristics. Cognitive measures are consistently significant and generally stronger predictors than imaging measures.

Dynamic Prediction of Motor Diagnosis in Huntington's Disease Using a Joint Modeling Approach

BACKGROUND

Prediction of motor diagnosis in Huntington's disease (HD) can be improved by incorporating other phenotypic and biological clinical measures in addition to cytosine-adenine-guanine (CAG) repeat length and age.

OBJECTIVE

The objective was to compare various clinical and biomarker trajectories for tracking HD progression and predicting motor conversion.

METHODS

Participants were from the PREDICT-HD study. We constructed a mixed-effect model to describe the change of measures while jointly modeling the process with time to HD diagnosis. The model was then used for subject-specific prediction. We employed the time-dependent receiver operating characteristic (ROC) method to assess the discriminating capability of the measures to identify high and low risk patients. The strongest predictor was used to illustrate the dynamic prediction of the disease risk and future trajectories of biomarkers for three hypothetical patients.

RESULTS

1078 individuals were included in this analysis. Five longitudinal clinical and imaging measures were compared. The putamen volume had the best discrimination performance with area under the curve (AUC) ranging from 0.74 to 0.82 over time. The total motor score showed a comparable discriminative ability with AUC ranging from 0.69 to 0.78 over time. The model showed that decreasing putamen volume was a significant predictor of motor conversion. A web-based calculator was developed for implementing the methods.

CONCLUSIONS

By jointly modeling longitudinal data with time-to-event outcomes, it is possible to construct an individualized dynamic event prediction model that renews over time with accumulating evidence. If validated, this could be a valuable tool to guide the clinician in predicting age of onset and potentially rate of progression.

Medical informatics research trend analysis: A text mining approach

The objective of this research is to identify major subject areas of medical informatics and explore the time-variant changes therein. As such it can inform the field about where medical informatics research has been and where it is heading. Furthermore, by identifying subject areas, this study identifies the development trends and the boundaries of medical informatics as an academic field. To conduct the study, first we identified 26,307 articles in PubMed archives which were published in the top medical informatics journals within the timeframe of 2002 to 2013. And then, employing a text mining -based semi-automated analytic approach, we clustered major research topics by analyzing the most frequently appearing subject terms extracted from the abstracts of these articles. The results indicated that some subject areas, such as biomedical, are declining, while other research areas such as health information technology (HIT), Internet-enabled research, and electronic medical/health records (EMR/EHR), are growing. The changes within the research subject areas can largely be attributed to the increasing capabilities and use of HIT. The Internet, for example, has changed the way medical research is conducted in the health care field. While discovering new medical knowledge through clinical and biological experiments is important, the utilization of EMR/EHR enabled the researchers to discover novel medical insight buried deep inside massive data sets, and hence, data analytics research has become a common complement in the medical field, rapidly growing in popularity.

Joint modelling of time-to-event and multivariate longitudinal outcomes: recent developments and issues

Background

Available methods for the joint modelling of longitudinal and time-to-event outcomes have typically only allowed for a single longitudinal outcome and a solitary event time. In practice, clinical studies are likely to record multiple longitudinal outcomes. Incorporating all sources of data will improve the predictive capability of any model and lead to more informative inferences for the purpose of medical decision-making.

Methods

We reviewed current methodologies of joint modelling for time-to-event data and multivariate longitudinal data including the distributional and modelling assumptions, the association structures, estimation approaches, software tools for implementation and clinical applications of the methodologies.

Results

We found that a large number of different models have recently been proposed. Most considered jointly modelling linear mixed models with proportional hazard models, with correlation between multiple longitudinal outcomes accounted for through multivariate normally distributed random effects. So-called current value and random effects parameterisations are commonly used to link the models. Despite developments, software is still lacking, which has translated into limited uptake by medical researchers.

Conclusion

Although, in an era of personalized medicine, the value of multivariate joint modelling has been established, researchers are currently limited in their ability to fit these models routinely. We make a series of recommendations for future research needs.

Electronic supplementary material

The online version of this article (doi:10.1186/s12874-016-0212-5) contains supplementary material, which is available to authorized users.

Bayesian joint ordinal and survival modeling for breast cancer risk assessment

We propose a joint model to analyze the structure and intensity of the association between longitudinal measurements of an ordinal marker and time to a relevant event. The longitudinal process is defined in terms of a proportional‐odds cumulative logit model. Time‐to‐event is modeled through a left‐truncated proportional‐hazards model, which incorporates information of the longitudinal marker as well as baseline covariates. Both longitudinal and survival processes are connected by means of a common vector of random effects.

General inferences are discussed under the Bayesian approach and include the posterior distribution of the probabilities associated to each longitudinal category and the assessment of the impact of the baseline covariates and the longitudinal marker on the hazard function. The flexibility provided by the joint model makes possible to dynamically estimate individual event‐free probabilities and predict future longitudinal marker values.

The model is applied to the assessment of breast cancer risk in women attending a population‐based screening program. The longitudinal ordinal marker is mammographic breast density measured with the Breast Imaging Reporting and Data System (BI‐RADS) scale in biennial screening exams. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

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.

Robust Bayesian hierarchical model using normal/independent distributions

The multilevel item response theory (MLIRT) models have been increasingly used in longitudinal clinical studies that collect multiple outcomes. The MLIRT models account for all the information from multiple longitudinal outcomes of mixed types (e.g., continuous, binary, and ordinal) and can provide valid inference for the overall treatment effects. However, the continuous outcomes and the random effects in the MLIRT models are often assumed to be normally distributed. The normality assumption can sometimes be unrealistic and thus may produce misleading results. The normal/independent (NI) distributions have been increasingly used to handle the outlier and heavy tail problems in order to produce robust inference. In this article, we developed a Bayesian approach that implemented the NI distributions on both continuous outcomes and random effects in the MLIRT models and discussed different strategies of implementing the NI distributions. Extensive simulation studies were conducted to demonstrate the advantage of our proposed models, which provided parameter estimates with smaller bias and more reasonable coverage probabilities. Our proposed models were applied to a motivating Parkinson's disease study, the DATATOP study, to investigate the effect of deprenyl in slowing down the disease progression.

Analysis of multivariate longitudinal kidney function outcomes using generalized linear mixed models

Background

Renal transplant patients are mandated to have continuous assessment of their kidney function over time to monitor disease progression determined by changes in blood urea nitrogen (BUN), serum creatinine (Cr), and estimated glomerular filtration rate (eGFR). Multivariate analysis of these outcomes that aims at identifying the differential factors that affect disease progression is of great clinical significance. Thus our study aims at demonstrating the application of different joint modeling approaches with random coefficients on a cohort of renal transplant patients and presenting a comparison of their performance through a pseudo-simulation study. The objective of this comparison is to identify the model with best performance and to determine whether accuracy compensates for complexity in the different multivariate joint models.

Methods and results

We propose a novel application of multivariate Generalized Linear Mixed Models (mGLMM) to analyze multiple longitudinal kidney function outcomes collected over 3 years on a cohort of 110 renal transplantation patients. The correlated outcomes BUN, Cr, and eGFR and the effect of various covariates such patient’s gender, age and race on these markers was determined holistically using different mGLMMs. The performance of the various mGLMMs that encompass shared random intercept (SHRI), shared random intercept and slope (SHRIS), separate random intercept (SPRI) and separate random intercept and slope (SPRIS) was assessed to identify the one that has the best fit and most accurate estimates. A bootstrap pseudo-simulation study was conducted to gauge the tradeoff between the complexity and accuracy of the models. Accuracy was determined using two measures; the mean of the differences between the estimates of the bootstrapped datasets and the true beta obtained from the application of each model on the renal dataset, and the mean of the square of these differences. The results showed that SPRI provided most accurate estimates and did not exhibit any computational or convergence problem.

Conclusion

Higher accuracy was demonstrated when the level of complexity increased from shared random coefficient models to the separate random coefficient alternatives with SPRI showing to have the best fit and most accurate estimates.

Bayesian hierarchical model for multiple repeated measures and survival data: an application to Parkinson's disease

Multilevel item response theory models have been increasingly used to analyze the multivariate longitudinal data of mixed types (e.g., continuous and categorical) in clinical studies. To address the possible correlation between multivariate longitudinal measures and time to terminal events (e.g., death and dropout), joint models that consist of a multilevel item response theory submodel and a survival submodel have been previously developed. However, in multisite studies, multiple patients are recruited and treated by the same clinical site. There can be a significant site correlation because of common environmental and socioeconomic status, and similar quality of care within site. In this article, we develop and study several hierarchical joint models with the hazard of terminal events dependent on shared random effects from various levels. We conduct extensive simulation study to evaluate the performance of various models under different scenarios. The proposed hierarchical joint models are applied to the motivating deprenyl and tocopherol antioxidative therapy of Parkinsonism study to investigate the effect of tocopherol in slowing Parkinson's disease progression.

A Bayesian approach to joint analysis of multivariate longitudinal data and parametric accelerated failure time

Impairment caused by Parkinson's disease (PD) is multidimensional (e.g., sensoria, functions, and cognition) and progressive. Its multidimensional nature precludes a single outcome to measure disease progression. Clinical trials of PD use multiple categorical and continuous longitudinal outcomes to assess the treatment effects on overall improvement. A terminal event such as death or dropout can stop the follow-up process. Moreover, the time to the terminal event may be dependent on the multivariate longitudinal measurements. In this article, we consider a joint random-effects model for the correlated outcomes. A multilevel item response theory model is used for the multivariate longitudinal outcomes and a parametric accelerated failure time model is used for the failure time because of the violation of proportional hazard assumption. These two models are linked via random effects. The Bayesian inference via MCMC is implemented in 'BUGS' language. Our proposed method is evaluated by a simulation study and is applied to DATATOP study, a motivating clinical trial to determine if deprenyl slows the progression of PD.

Joint model for a diagnostic test without a gold standard in the presence of a dependent terminal event

Breast cancer patients after breast conservation therapy often develop ipsilateral breast tumor relapse (IBTR), whose classification (true local recurrence versus new ipsilateral primary tumor) is subject to error, and there is no available gold standard. Some patients may die because of breast cancer before IBTR develops. Because this terminal event may be related to the individual patient's unobserved disease status and time to IBTR, the terminal mechanism is non-ignorable. This article presents a joint analysis framework to model the binomial regression with misclassified binary outcome and the correlated time to IBTR, subject to a dependent terminal event and in the absence of a gold standard. Shared random effects are used to link together two survival times. The proposed approach is evaluated by a simulation study and is applied to a breast cancer data set consisting of 4477 breast cancer patients. The proposed joint model can be conveniently fit using adaptive Gaussian quadrature tools implemented in SAS 9.3 (SAS Institute Inc., Cary, NC, USA) procedure NLMIXED.

A Joint Modeling Approach for Right Censored High Dimensional Multivariate Longitudinal Data

Analysis of multivariate longitudinal data becomes complicated when the outcomes are of high dimension and informative right censoring is prevailing. Here, we propose a likelihood based approach for high dimensional outcomes wherein we jointly model the censoring process along with the slopes of the multivariate outcomes in the same likelihood function. We utilized pseudo likelihood function to generate parameter estimates for the population slopes and Empirical Bayes estimates for the individual slopes. The proposed approach was applied to jointly model longitudinal measures of blood urea nitrogen, plasma creatinine, and estimated glomerular filtration rate which are key markers of kidney function in a cohort of renal transplant patients followed from kidney transplant to kidney failure. Feasibility of the proposed joint model for high dimensional multivariate outcomes was successfully demonstrated and its performance was compared to that of a pairwise bivariate model. Our simulation study results suggested that there was a significant reduction in bias and mean squared errors associated with the joint model compared to the pairwise bivariate model.