A characterization of missingness at random in a generalized shared‐parameter joint modeling framework for longitudinal and time‐to‐event data,and sensitivity analysis

We consider a conceptual correspondence between the missing data setting, and joint modeling of longitudinal and time‐to‐event outcomes. Based on this, we formulate an extended shared random effects joint model. Based on this, we provide a characterization of missing at random, which is in line with that in the missing data setting. The ideas are illustrated using data from a study on liver cirrhosis, contrasting the new framework with conventional joint models.     

Bayesian estimation of two-part joint models for a longitudinal semicontinuous biomarker and a terminal event with INLA: Interests for cancer clinical trial evaluation

Two-part joint models for a longitudinal semicontinuous biomarker and a terminal event have been recently introduced based on frequentist estimation. The biomarker distribution is decomposed into a probability of positive value and the expected value among positive values. Shared random effects can represent the association structure between the biomarker and the terminal event. The computational burden increases compared to standard joint models with a single regression model for the biomarker. In this context, the frequentist estimation implemented in the R package frailtypack can be challenging for complex models (i.e., a large number of parameters and dimension of the random effects). As an alternative, we propose a Bayesian estimation of two-part joint models based on the Integrated Nested Laplace Approximation (INLA) algorithm to alleviate the computational burden and fit more complex models. Our simulation studies confirm that INLA provides accurate approximation of posterior estimates and to reduced computation time and variability of estimates compared to frailtypack in the situations considered. We contrast the Bayesian and frequentist approaches in the analysis of two randomized cancer clinical trials (GERCOR and PRIME studies), where INLA has a reduced variability for the association between the biomarker and the risk of event. Moreover, the Bayesian approach was able to characterize subgroups of patients associated with different responses to treatment in the PRIME study. Our study suggests that the Bayesian approach using the INLA algorithm enables to fit complex joint models that might be of interest in a wide range of clinical applications.     

Hierarchical Bayesian modeling of heterogeneous cluster‐ and subject‐level associations between continuous and binary outcomes in dairy production

The augmentation of categorical outcomes with underlying Gaussian variables in bivariate generalized mixed effects models has facilitated the joint modeling of continuous and binary response variables. These models typically assume that random effects and residual effects (co)variances are homogeneous across all clusters and subjects, respectively. Motivated by conflicting evidence about the association between performance outcomes in dairy production systems, we consider the situation where these (co)variance parameters may themselves be functions of systematic and/or random effects. We present a hierarchical Bayesian extension of bivariate generalized linear models whereby functions of the (co)variance matrices are specified as linear combinations of fixed and random effects following a square‐root‐free Cholesky reparameterization that ensures necessary positive semidefinite constraints. We test the proposed model by simulation and apply it to the analysis of a dairy cattle data set in which the random herd‐level and residual cow‐level effects (co)variances between a continuous production trait and binary reproduction trait are modeled as functions of fixed management effects and random cluster effects.     

A tutorial on dynamic risk prediction of a binary outcome based on a longitudinal biomarker

Dynamic risk predictions based on all available information are useful in timely identification of high-risk patients. However, in contrast with time to event outcomes, there is still a lack of studies that clearly demonstrate how to obtain and update predictions for a future binary outcome using a repeatedly measured biomarker. The aim of this study is to give an illustrative overview of four approaches to obtain such predictions: likelihood based two-stage method (2SMLE), likelihood based joint model (JMMLE), Bayesian two-stage method (2SB), and Bayesian joint model (JMB). We applied the approaches to provide weekly updated predictions of post–molar gestational trophoblastic neoplasia (GTN) based on age and repeated measurements of human chorionic gonadotropin (hCG). Discrimination and calibration measures were used to compare the accuracy of the weekly predictions. Internal validation of the models was conducted using bootstrapping. The four approaches resulted in the same predictive and discriminative performance in predicting GTN. A simulation study showed that the joint models outperform the two-stage methods when we increase the within- and the between-patients variability of the biomarker. The applicability of these models to produce dynamic predictions has been illustrated through a comprehensive explanation and accompanying syntax (R and SAS^®).     

Shared parameter models under random effects misspecification

A common objective in longitudinal studies is the investigationof the association structure between a longitudinal responseprocess and the time to an event of interest. An attractiveparadigm for the joint modelling of longitudinal and survivalprocesses is the shared parameter framework, where a set ofrandom effects is assumed to induce their interdependence. Inthis work, we propose an alternative parameterization for sharedparameter models and investigate the effect of misspecifyingthe random effects distribution in the parameter estimates andtheir standard errors.     

Score Test for Conditional Independence Between Longitudinal Outcome and Time to Event Given the Classes in the Joint Latent Class Model

Summary .  Latent class models have been recently developed for the joint analysis of a longitudinal quantitative outcome and a time to event. These models assume that the population is divided in  G  latent classes characterized by different risk functions for the event, and different profiles of evolution for the markers that are described by a mixed model for each class. However, the key assumption of conditional independence between the marker and the event given the latent classes is difficult to evaluate because the latent classes are not observed. Using a joint model with latent classes and shared random effects, we propose a score test for the null hypothesis of independence between the marker and the outcome given the latent classes versus the alternative hypothesis that the risk of event depends on one or several random effects from the mixed model in addition to the latent classes. A simulation study was performed to compare the behavior of the score test to other previously proposed tests, including situations where the alternative hypothesis or the baseline risk function are misspecified. In all the investigated situations, the score test was the most powerful. The methodology was applied to develop a prognostic model for recurrence of prostate cancer given the evolution of prostate-specific antigen in a cohort of patients treated by radiation therapy.     

A joint model for repeated events of different types and multiple longitudinal outcomes with application to a follow‐up study of patients after kidney transplant

This paper presents an extension of the joint modeling strategy for the case of multiple longitudinal outcomes and repeated infections of different types over time, motivated by postkidney transplantation data. Our model comprises two parts linked by shared latent terms. On the one hand is a multivariate mixed linear model with random effects, where a low‐rank thin‐plate spline function is incorporated to collect the nonlinear behavior of the different profiles over time. On the other hand is an infection‐specific Cox model, where the dependence between different types of infections and the related times of infection is through a random effect associated with each infection type to catch the within dependence and a shared frailty parameter to capture the dependence between infection types. We implemented the parameterization used in joint models which uses the fitted longitudinal measurements as time‐dependent covariates in a relative risk model. Our proposed model was implemented in OpenBUGS using the MCMC approach.     

Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

Properties of cell death models calibrated and compared using Bayesian approaches

Using models to simulate and analyze biological networks requires principled approaches to parameter estimation and model discrimination. We use Bayesian and Monte Carlo methods to recover the full probability distributions of free parameters (initial protein concentrations and rate constants) for mass‐action models of receptor‐mediated cell death. The width of the individual parameter distributions is largely determined by non‐identifiability but covariation among parameters, even those that are poorly determined, encodes essential information. Knowledge of joint parameter distributions makes it possible to compute the uncertainty of model‐based predictions whereas ignoring it (e.g., by treating parameters as a simple list of values and variances) yields nonsensical predictions. Computing the Bayes factor from joint distributions yields the odds ratio (～20‐fold) for competing ‘direct’ and ‘indirect’ apoptosis models having different numbers of parameters. Our results illustrate how Bayesian approaches to model calibration and discrimination combined with single‐cell data represent a generally useful and rigorous approach to discriminate between competing hypotheses in the face of parametric and topological uncertainty.     

Marginal hazard ratio estimates in joint frailty models for heart failure trials

This work is motivated by clinical trials in chronic heart failure disease, where treatment has effects both on morbidity (assessed as recurrent non‐fatal hospitalisations) and on mortality (assessed as cardiovascular death, CV death). Recently, a joint frailty proportional hazards model has been proposed for these kind of efficacy outcomes to account for a potential association between the risk rates for hospital admissions and CV death. However, more often clinical trial results are presented by treatment effect estimates that have been derived from marginal proportional hazards models, that is, a Cox model for mortality and an Andersen–Gill model for recurrent hospitalisations. We show how these marginal hazard ratios and their estimates depend on the association between the risk processes, when these are actually linked by shared or dependent frailty terms. First we derive the marginal hazard ratios as a function of time. Then, applying least false parameter theory, we show that the marginal hazard ratio estimate for the hospitalisation rate depends on study duration and on parameters of the underlying joint frailty model. In particular, we identify parameters, for example the treatment effect on mortality, that determine if the marginal hazard ratio estimate for hospitalisations is smaller, equal or larger than the conditional one. How this affects rejection probabilities is further investigated in simulation studies. Our findings can be used to interpret marginal hazard ratio estimates in heart failure trials and are illustrated by the results of the CHARM‐Preserved trial (where CHARM is the ‘Candesartan in Heart failure Assessment of Reduction in Mortality and morbidity’ programme).     

Logistic regression when covariates are random effects from a non‐linear mixed model

In many studies, the association of longitudinal measurements of a continuous response and a binary outcome are often of interest. A convenient framework for this type of problems is the joint model, which is formulated to investigate the association between a binary outcome and features of longitudinal measurements through a common set of latent random effects. The joint model, which is the focus of this article, is a logistic regression model with covariates defined as the individual‐specific random effects in a non‐linear mixed‐effects model (NLMEM) for the longitudinal measurements. We discuss different estimation procedures, which include two‐stage, best linear unbiased predictors, and various numerical integration techniques. The proposed methods are illustrated using a real data set where the objective is to study the association between longitudinal hormone levels and the pregnancy outcome in a group of young women. The numerical performance of the estimating methods is also evaluated by means of simulation.     

A class of Bayesian shared gamma frailty models with multivariate failure time data

For multivariate failure time data, we propose a new class of shared gamma frailty models by imposing the Box-Cox transformation on the hazard function, and the product of the baseline hazard and the frailty. This novel class of models allows for a very broad range of shapes and relationships between the hazard and baseline hazard functions. It includes the well-known Cox gamma frailty model and a new additive gamma frailty model as two special cases. Due to the nonnegative hazard constraint, this shared gamma frailty model is computationally challenging in the Bayesian paradigm. The joint priors are constructed through a conditional-marginal specification, in which the conditional distribution is univariate, and it absorbs the nonlinear parameter constraints. The marginal part of the prior specification is free of constraints. The prior distributions allow us to easily compute the full conditionals needed for Gibbs sampling, while incorporating the constraints. This class of shared gamma frailty models is illustrated with a real dataset.     

Joint Modelling of Repeated Measures and Survival Time Data

In many clinical trials both repeated measures data and event history data are simultaneously observed from the same subject. These two types of responses are usually correlated, because they are from the same subject. In this article, we propose a joint model for the combined analysis of repeated measures data and event history data in the framework of hierarchical generalized linear models. The correlation between repeated measures and event time is modelled by introducing a shared random effect. The model parameters are estimated using the hierarchical‐likelihood approach. The proposed model is illustrated using a real data set for the renal transplant patients.     

A joint model for longitudinal continuous and time‐to‐event outcomes with direct marginal interpretation

Joint modeling of various longitudinal sequences has received quite a bit of attention in recent times. This paper proposes a so‐called marginalized joint model for longitudinal continuous and repeated time‐to‐event outcomes on the one hand and a marginalized joint model for bivariate repeated time‐to‐event outcomes on the other. The model has several appealing features. It flexibly allows for association among measurements of the same outcome at different occasions as well as among measurements on different outcomes recorded at the same time. The model also accommodates overdispersion. The time‐to‐event outcomes are allowed to be censored. While the model builds upon the generalized linear mixed model framework, it is such that model parameters enjoy a direct marginal interpretation. All of these features have been considered before, but here we bring them together in a unified, flexible framework. The model framework's properties are scrutinized using a simulation study. The models are applied to data from a chronic heart failure study and to a so‐called comet assay, encountered in preclinical research. Almost surprisingly, the models can be fitted relatively easily using standard statistical software.     

A varying‐coefficient generalized odds rate model with time‐varying exposure: An application to fitness and cardiovascular disease mortality

Varying‐coefficient models have become a common tool to determine whether and how the association between an exposure and an outcome changes over a continuous measure. These models are complicated when the exposure itself is time‐varying and subjected to measurement error. For example, it is well known that longitudinal physical fitness has an impact on cardiovascular disease (CVD) mortality. It is not known, however, how the effect of longitudinal physical fitness on CVD mortality varies with age. In this paper, we propose a varying‐coefficient generalized odds rate model that allows flexible estimation of age‐modified effects of longitudinal physical fitness on CVD mortality. In our model, the longitudinal physical fitness is measured with error and modeled using a mixed‐effects model, and its associated age‐varying coefficient function is represented by cubic B‐splines. An expectation‐maximization algorithm is developed to estimate the parameters in the joint models of longitudinal physical fitness and CVD mortality. A modified pseudoadaptive Gaussian‐Hermite quadrature method is adopted to compute the integrals with respect to random effects involved in the E‐step. The performance of the proposed method is evaluated through extensive simulation studies and is further illustrated with an application to cohort data from the Aerobic Center Longitudinal Study.     

Mixture Hazard Models for Lifetime Data

We propose a general family of mixture hazard models to analyze lifetime data associated with bathtub and multimodal hazard functions. With this model we have a great flexibility for fitting lifetime data. Its version with covariates has the proportional hazard and the accelerated failure time models as special cases. A Bayesian analysis is presented for the model using informative priors, using sampling‐based approaches to perform the Bayesian computations. A real example with a medical data illustrates the methodology.     

An estimator for the proportional hazards model with multiple longitudinal covariates measured with error

In many longitudinal studies, it is of interest to characterize the relationship between a time-to-event (e.g. survival) and several time-dependent and time-independent covariates. Time-dependent covariates are generally observed intermittently and with error. For a single time-dependent covariate, a popular approach is to assume a joint longitudinal data-survival model, where the time-dependent covariate follows a linear mixed effects model and the hazard of failure depends on random effects and time-independent covariates via a proportional hazards relationship. Regression calibration and likelihood or Bayesian methods have been advocated for implementation; however, generalization to more than one time-dependent covariate may become prohibitive. For a single time-dependent covariate, Tsiatis and Davidian (2001) have proposed an approach that is easily implemented and does not require an assumption on the distribution of the random effects. This technique may be generalized to multiple, possibly correlated, time-dependent covariates, as we demonstrate. We illustrate the approach via simulation and by application to data from an HIV clinical trial.     

Bayesian informative dropout model for longitudinal binary data with random effects using conditional and joint modeling approaches

Dropouts are common in longitudinal study. If the dropout probability depends on the missing observations at or after dropout, this type of dropout is called informative (or nonignorable) dropout (ID). Failure to accommodate such dropout mechanism into the model will bias the parameter estimates. We propose a conditional autoregressive model for longitudinal binary data with an ID model such that the probabilities of positive outcomes as well as the drop‐out indicator in each occasion are logit linear in some covariates and outcomes. This model adopting a marginal model for outcomes and a conditional model for dropouts is called a selection model. To allow for the heterogeneity and clustering effects, the outcome model is extended to incorporate mixture and random effects. Lastly, the model is further extended to a novel model that models the outcome and dropout jointly such that their dependency is formulated through an odds ratio function. Parameters are estimated by a Bayesian approach implemented using the user‐friendly Bayesian software WinBUGS. A methadone clinic dataset is analyzed to illustrate the proposed models. Result shows that the treatment time effect is still significant but weaker after allowing for an ID process in the data. Finally the effect of drop‐out on parameter estimates is evaluated through simulation studies.     

Latent-Model Robustness in Joint Models for a Primary Endpoint and a Longitudinal Process

Summary .  Joint modeling of a primary response and a longitudinal process via shared random effects is widely used in many areas of application. Likelihood-based inference on joint models requires model specification of the random effects. Inappropriate model specification of random effects can compromise inference. We present methods to diagnose random effect model misspecification of the type that leads to biased inference on joint models. The methods are illustrated via application to simulated data, and by application to data from a study of bone mineral density in perimenopausal women and data from an HIV clinical trial.