On Estimating the Relationship between Longitudinal Measurements and Time‐to‐Event Data Using a Simple Two‐Stage Procedure

Summary Ye, Lin, and Taylor (2008, Biometrics 64 , 1238–1246) proposed a joint model for longitudinal measurements and time‐to‐event data in which the longitudinal measurements are modeled with a semiparametric mixed model to allow for the complex patterns in longitudinal biomarker data. They proposed a two‐stage regression calibration approach that is simpler to implement than a joint modeling approach. In the first stage of their approach, the mixed model is fit without regard to the time‐to‐event data. In the second stage, the posterior expectation of an individual's random effects from the mixed‐model are included as covariates in a Cox model. Although Ye et al. (2008) acknowledged that their regression calibration approach may cause a bias due to the problem of informative dropout and measurement error, they argued that the bias is small relative to alternative methods. In this article, we show that this bias may be substantial. We show how to alleviate much of this bias with an alternative regression calibration approach that can be applied for both discrete and continuous time‐to‐event data. Through simulations, the proposed approach is shown to have substantially less bias than the regression calibration approach proposed by Ye et al. (2008) . In agreement with the methodology proposed by Ye et al. (2008) , an advantage of our proposed approach over joint modeling is that it can be implemented with standard statistical software and does not require complex estimation techniques.     

Robust Joint Modeling of Longitudinal Measurements and Competing Risks Failure Time Data

Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow‐up. Our model consists of a linear mixed effects sub‐model for the longitudinal outcome and a proportional cause‐specific hazards frailty sub‐model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub‐model, we adopt a t ‐distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture

Shared random effects joint models are becoming increasingly popular for investigating the relationship between longitudinal and time‐to‐event data. Although appealing, such complex models are computationally intensive, and quick, approximate methods may provide a reasonable alternative. In this paper, we first compare the shared random effects model with two approximate approaches: a naïve proportional hazards model with time‐dependent covariate and a two‐stage joint model, which uses plug‐in estimates of the fitted values from a longitudinal analysis as covariates in a survival model. We show that the approximate approaches should be avoided since they can severely underestimate any association between the current underlying longitudinal value and the event hazard. We present classical and Bayesian implementations of the shared random effects model and highlight the advantages of the latter for making predictions. We then apply the models described to a study of abdominal aortic aneurysms (AAA) to investigate the association between AAA diameter and the hazard of AAA rupture. Out‐of‐sample predictions of future AAA growth and hazard of rupture are derived from Bayesian posterior predictive distributions, which are easily calculated within an MCMC framework. Finally, using a multivariate survival sub‐model we show that underlying diameter rather than the rate of growth is the most important predictor of AAA rupture.     

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.     

Joint Spatial Modeling of Recurrent Infection and Growth with Processes under Intermittent Observation

Summary In this article, we present a new statistical methodology for longitudinal studies in forestry, where trees are subject to recurrent infection, and the hazard of infection depends on tree growth over time. Understanding the nature of this dependence has important implications for reforestation and breeding programs. Challenges arise for statistical analysis in this setting with sampling schemes leading to panel data, exhibiting dynamic spatial variability, and incomplete covariate histories for hazard regression. In addition, data are collected at a large number of locations, which poses computational difficulties for spatiotemporal modeling. A joint model for infection and growth is developed wherein a mixed nonhomogeneous Poisson process, governing recurring infection, is linked with a spatially dynamic nonlinear model representing the underlying height growth trajectories. These trajectories are based on the von Bertalanffy growth model and a spatially varying parameterization is employed. Spatial variability in growth parameters is modeled through a multivariate spatial process derived through kernel convolution. Inference is conducted in a Bayesian framework with implementation based on hybrid Monte Carlo. Our methodology is applied for analysis in an 11‐year study of recurrent weevil infestation of white spruce in British Columbia.     

Time‐Varying Latent Effect Model for Longitudinal Data with Informative Observation Times

Summary In analysis of longitudinal data, it is not uncommon that observation times of repeated measurements are subject‐specific and correlated with underlying longitudinal outcomes. Taking account of the dependence between observation times and longitudinal outcomes is critical under these situations to assure the validity of statistical inference. In this article, we propose a flexible joint model for longitudinal data analysis in the presence of informative observation times. In particular, the new procedure considers the shared random‐effect model and assumes a time‐varying coefficient for the latent variable, allowing a flexible way of modeling longitudinal outcomes while adjusting their association with observation times. Estimating equations are developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug‐in method. One additional advantage of the procedure is that it provides a unified framework to test whether the effect of the latent variable is zero, constant, or time‐varying. Simulation studies show that the proposed approach is appropriate for practical use. An application to a bladder cancer data is also given to illustrate the methodology.     

Joint Modeling of Multivariate Longitudinal Data and Competing Risks Using Multiphase Sub-models

In many clinical studies that involve follow-up, it is common to observe one or more sequences of longitudinal measurements, as well as one or more time to event outcomes. A competing risks situation arises when the probability of occurrence of one event is altered/hindered by another time to event. Recently, there has been much attention paid to the joint analysis of a single longitudinal response and a single time to event outcome, when the missing data mechanism in the longitudinal process is non-ignorable. We, in this paper, propose an extension where multiple longitudinal responses are jointly modeled with competing risks (multiple time to events). Our shared parameter joint model consists of a system of multiphase non-linear mixed effects sub-models for the multiple longitudinal responses, and a system of cause-specific non-proportional hazards frailty sub-models for competing risks, with associations among multiple longitudinal responses and competing risks modeled using latent parameters. The joint model is applied to a data set of patients who are on mechanical circulatory support and are awaiting heart transplant, using readily available software. While on the mechanical circulatory support, patient liver and renal functions may worsen and these in turn may influence one of the two possible competing outcomes: (i) death before transplant; (ii) transplant. In one application, we propose a system of multiphase cause-specific non-proportional hazard sub-model where frailty can be time varying. Performance under different scenarios was assessed using simulation studies. By using the proposed joint modeling of the multiphase sub-models, one can identify: (i) non-linear trends in multiple longitudinal outcomes; (ii) time-varying hazards and cumulative incidence functions of the competing risks; (iii) identify risk factors for the both types of outcomes, where the effect may or may not change with time; and (iv) assess the association between multiple longitudinal and competing risks outcomes, where the association may or may not change with time.     

Spatial multistate transitional models for longitudinal event data

Summary .   Follow-up medical studies often collect longitudinal data on patients. Multistate transitional models are useful for analysis in such studies where at any point in time, individuals may be said to occupy one of a discrete set of states and interest centers on the transition process between states. For example, states may refer to the number of recurrences of an event, or the stage of a disease. We develop a hierarchical modeling framework for the analysis of such longitudinal data when the processes corresponding to different subjects may be correlated spatially over a region. Continuous-time Markov chains incorporating spatially correlated random effects are introduced. Here, joint modeling of both spatial dependence as well as dependence between different transition rates is required and a multivariate spatial approach is employed. A proportional intensities frailty model is developed where baseline intensity functions are modeled using parametric Weibull forms, piecewise-exponential formulations, and flexible representations based on cubic B-splines. The methodology is developed within the context of a study examining invasive cardiac procedures in Quebec. We consider patients admitted for acute coronary syndrome throughout the 139 local health units of the province and examine readmission and mortality rates over a 4-year period.     

Mixtures of varying coefficient models for longitudinal data with discrete or continuous nonignorable dropout

The analysis of longitudinal repeated measures data is frequently complicated by missing data due to informative dropout. We describe a mixture model for joint distribution for longitudinal repeated measures, where the dropout distribution may be continuous and the dependence between response and dropout is semiparametric. Specifically, we assume that responses follow a varying coefficient random effects model conditional on dropout time, where the regression coefficients depend on dropout time through unspecified nonparametric functions that are estimated using step functions when dropout time is discrete (e.g., for panel data) and using smoothing splines when dropout time is continuous. Inference under the proposed semiparametric model is hence more robust than the parametric conditional linear model. The unconditional distribution of the repeated measures is a mixture over the dropout distribution. We show that estimation in the semiparametric varying coefficient mixture model can proceed by fitting a parametric mixed effects model and can be carried out on standard software platforms such as SAS. The model is used to analyze data from a recent AIDS clinical trial and its performance is evaluated using simulations.     

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.     

Joint models for multivariate longitudinal and multivariate survival data

Joint modeling of longitudinal and survival data is becoming increasingly essential in most cancer and AIDS clinical trials. We propose a likelihood approach to extend both longitudinal and survival components to be multidimensional. A multivariate mixed effects model is presented to explicitly capture two different sources of dependence among longitudinal measures over time as well as dependence between different variables. For the survival component of the joint model, we introduce a shared frailty, which is assumed to have a positive stable distribution, to induce correlation between failure times. The proposed marginal univariate survival model, which accommodates both zero and nonzero cure fractions for the time to event, is then applied to each marginal survival function. The proposed multivariate survival model has a proportional hazards structure for the population hazard, conditionally as well as marginally, when the baseline covariates are specified through a specific mechanism. In addition, the model is capable of dealing with survival functions with different cure rate structures. The methodology is specifically applied to the International Breast Cancer Study Group (IBCSG) trial to investigate the relationship between quality of life, disease-free survival, and overall survival.     

Integrating ecophysiological models into species distribution projections of European reptile range shifts in response to climate change

Uncertainty in projections of global change impacts on biodiversity over the 21st century is high. Improved predictive accuracy is needed, highlighting the importance of using different types of models when predicting species range shifts. However, this is still rarely done. Our approach integrates the outputs of a spatially‐explicit physiologically inspired model of extinction and correlative species distribution models to assess climate‐change induced range shifts of three European reptile species (Lacerta lepida, Iberolacerta monticola, and Hemidactylus turcicus) in the coming decades. We integrated the two types of models by mapping and quantifying agreement and disagreement between their projections. We analyzed the relationships between climate change and projected range shifts. Agreement between model projections varied greatly between species and depended on whether or not they consider dispersal ability. Under our approach, the reliability of predictions is greatest where the predictions of these different types of models converge, and in this way uncertainty is reduced; sites where this convergence occurs are characterized by both current high temperatures and significant future temperature increase, suggesting they may become hotspots of local extinctions. Moreover, this approach can be readily implemented with other types of models.     

A Likelihood-Based Approach with Shared Latent Random Parameters for the Longitudinal Binary and Informative Censoring Processes

Longitudinal studies with binary outcomes characterized by informative right censoring are commonly encountered in clinical, basic, behavioral, and health sciences. Approaches developed to analyze data with binary outcomes were mainly tailored to clustered or longitudinal data with missing completely at random or at random. Studies that focused on informative right censoring with binary outcomes are characterized by their imbedded computational complexity and difficulty of implementation. Here we present a new maximum likelihood-based approach with repeated binary measures modeled in a generalized linear mixed model as a function of time and other covariates. The longitudinal binary outcome and the censoring process determined by the number of times a subject is observed share latent random variables (random intercept and slope) where these subject-specific random effects are common to both models. A simulation study and sensitivity analysis were conducted to test the model under different assumptions and censoring settings. Our results showed accuracy of the estimates generated under this model when censoring was fully informative or partially informative with dependence on the slopes. A successful implementation was undertaken on a cohort of renal transplant patients with blood urea nitrogen as a binary outcome measured over time to indicate normal and abnormal kidney function until the emanation of graft rejection that eventuated in informative right censoring. In addition to its novelty and accuracy, an additional key feature and advantage of the proposed model is its viability of implementation on available analytical tools and widespread application on any other longitudinal dataset with informative censoring.

     

Bayesian Semiparametric Frailty Selection in Multivariate Event Time Data

Biomedical studies often collect multivariate event time data from multiple clusters (either subjects or groups) within each of which event times for individuals are correlated and the correlation may vary in different classes. In such survival analyses, heterogeneity among clusters for shared and specific classes can be accommodated by incorporating parametric frailty terms into the model. In this article, we propose a Bayesian approach to relax the parametric distribution assumption for shared and specific‐class frailties by using a Dirichlet process prior while also allowing for the uncertainty of heterogeneity for different classes. Multiple cluster‐specific frailty selections rely on variable selection‐type mixture priors by applying mixtures of point masses at zero and inverse gamma distributions to the variance of log frailties. This selection allows frailties with zero variance to effectively drop out of the model. A reparameterization of log‐frailty terms is performed to reduce the potential bias of fixed effects due to variation of the random distribution and dependence among the parameters resulting in easy interpretation and faster Markov chain Monte Carlo convergence. Simulated data examples and an application to a lung cancer clinical trial are used for illustration.     

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.     

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.     

The right response at the right time: Exploring helminth immune modulation in sticklebacks by experimental coinfection

Parasites are one of the strongest selective agents in nature. They select for hosts that evolve counter‐adaptive strategies to cope with infection. Helminth parasites are special because they can modulate their hosts’ immune responses. This phenomenon is important in epidemiological contexts, where coinfections may be affected. How different types of hosts and helminths interact with each other is insufficiently investigated. We used the three‐spined stickleback (Gasterosteus aculeatus) – Schistocephalus solidus model to study mechanisms and temporal components of helminth immune modulation. Sticklebacks from two contrasting populations with either high resistance (HR) or low resistance (LR) against S. solidus, were individually exposed to S. solidus strains with characteristically high growth (HG) or low growth (LG) in G. aculeatus. We determined the susceptibility to another parasite, the eye fluke Diplostomum pseudospathaceum, and the expression of 23 key immune genes at three time points after S. solidus infection. D. pseudospathaceum infection rates and the gene expression responses depended on host and S. solidus type and changed over time. Whereas the effect of S. solidus type was not significant after three weeks, T regulatory responses and complement components were upregulated at later time points if hosts were infected with HG S. solidus. HR hosts showed a well orchestrated immune response, which was absent in LR hosts. Our results emphasize the role of regulatory T cells and the timing of specific immune responses during helminth infections. This study elucidates the importance to consider different coevolutionary trajectories and ecologies when studying host‐parasite interactions.     

Epidemiology in evolutionary time: the case of Wolbachia horizontal transmission between arthropod host species

Wolbachia are bacterial endosymbionts that manipulate the reproduction of their arthropod hosts. Although theory suggests that infections are frequently lost within host species due to the evolution of resistance, Wolbachia infect a huge number of species worldwide. This apparent paradox suggests that horizontal transmission between host species has been a key factor in shaping the global Wolbachia pandemic. Because Wolbachia infections are thus acquired and lost like any other infection, we use a standard epidemiological model to analyse Wolbachia horizontal transmission dynamics over evolutionary time. Conceptually modifying the model, we apply it not to transmission between individuals but between species. Because, on evolutionary timescales, infections spread frequently between closely related species and occasionally over large phylogenetic distances, we represent the set of host species as a small‐world network that satisfies both requirements. Our model reproduces the effect of basic epidemiological parameters, which demonstrates the validity of our approach. We find that the ratio between transmission rate and recovery rate is crucial for determining the proportion of infected species (incidence) and that, in a given host network, the incidence may still be increasing over evolutionary time. Our results also point to the importance of occasional transmission over long phylogenetic distances for the observed high incidence levels of Wolbachia. In conclusion, we are able to explain why Wolbachia are so abundant among arthropods, although selection for resistance within hosts often leads to infection loss. Furthermore, our unorthodox approach of using epidemiology in evolutionary time can be applied to all symbionts that use horizontal transmission to infect new hosts.     

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.     

Bayesian latent multi‐state modeling for nonequidistant longitudinal electronic health records

Large amounts of longitudinal health records are now available for dynamic monitoring of the underlying processes governing the observations. However, the health status progression across time is not typically observed directly: records are observed only when a subject interacts with the system, yielding irregular and often sparse observations. This suggests that the observed trajectories should be modeled via a latent continuous‐time process potentially as a function of time‐varying covariates. We develop a continuous‐time hidden Markov model to analyze longitudinal data accounting for irregular visits and different types of observations. By employing a specific missing data likelihood formulation, we can construct an efficient computational algorithm. We focus on Bayesian inference for the model: this is facilitated by an expectation‐maximization algorithm and Markov chain Monte Carlo methods. Simulation studies demonstrate that these approaches can be implemented efficiently for large data sets in a fully Bayesian setting. We apply this model to a real cohort where patients suffer from chronic obstructive pulmonary disease with the outcome being the number of drugs taken, using health care utilization indicators and patient characteristics as covariates.