A Joint Modeling Approach for Longitudinal Data with Informative Observation Times and a Terminal Event

In this article, we propose a new joint modeling approach for the analysis of longitudinal data with informative observation times and a dependent terminal event. We specify a semiparametric mixed effects model for the longitudinal process, a proportional rate frailty model for the observation process, and a proportional hazards frailty model for the terminal event. The association among the three related processes is modeled via two latent variables. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the proposed estimators are established. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is illustrated.     

A joint modeling approach for analyzing marker data in the presence of a terminal event

In many medical studies, markers are contingent on recurrent events and the cumulative markers are usually of interest. However, the recurrent event process is often interrupted by a dependent terminal event, such as death. In this article, we propose a joint modeling approach for analyzing marker data with informative recurrent and terminal events. This approach introduces a shared frailty to specify the explicit dependence structure among the markers, the recurrent, and terminal events. Estimation procedures are developed for the model parameters and the degree of dependence, and a prediction of the covariate‐specific cumulative markers is provided. The finite sample performance of the proposed estimators is examined through simulation studies. An application to a medical cost study of chronic heart failure patients from the University of Virginia Health System is illustrated.     

Semiparametric frailty models for zero‐inflated event count data in the presence of informative dropout

Recurrent events data are commonly encountered in medical studies. In many applications, only the number of events during the follow‐up period rather than the recurrent event times is available. Two important challenges arise in such studies: (a) a substantial portion of subjects may not experience the event, and (b) we may not observe the event count for the entire study period due to informative dropout. To address the first challenge, we assume that underlying population consists of two subpopulations: a subpopulation nonsusceptible to the event of interest and a subpopulation susceptible to the event of interest. In the susceptible subpopulation, the event count is assumed to follow a Poisson distribution given the follow‐up time and the subject‐specific characteristics. We then introduce a frailty to account for informative dropout. The proposed semiparametric frailty models consist of three submodels: (a) a logistic regression model for the probability such that a subject belongs to the nonsusceptible subpopulation; (b) a nonhomogeneous Poisson process model with an unspecified baseline rate function; and (c) a Cox model for the informative dropout time. We develop likelihood‐based estimation and inference procedures. The maximum likelihood estimators are shown to be consistent. Additionally, the proposed estimators of the finite‐dimensional parameters are asymptotically normal and the covariance matrix attains the semiparametric efficiency bound. Simulation studies demonstrate that the proposed methodologies perform well in practical situations. We apply the proposed methods to a clinical trial on patients with myelodysplastic syndromes.     

Regression modeling of semicompeting risks data

Semicompeting risks data are often encountered in clinical trials with intermediate endpoints subject to dependent censoring from informative dropout. Unlike with competing risks data, dropout may not be dependently censored by the intermediate event. There has recently been increased attention to these data, in particular inferences about the marginal distribution of the intermediate event without covariates. In this article, we incorporate covariates and formulate their effects on the survival function of the intermediate event via a functional regression model. To accommodate informative censoring, a time-dependent copula model is proposed in the observable region of the data which is more flexible than standard parametric copula models for the dependence between the events. The model permits estimation of the marginal distribution under weaker assumptions than in previous work on competing risks data. New nonparametric estimators for the marginal and dependence models are derived from nonlinear estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. Graphical model checking techniques are presented for the assumed models. Nonparametric tests are developed accordingly, as are inferences for parametric submodels for the time-varying covariate effects and copula parameters. A novel time-varying sensitivity analysis is developed using the estimation procedures. Simulations and an AIDS data analysis demonstrate the practical utility of the methodology.     

Joint regression analysis for survival data in the presence of two sets of semi‐competing risks

In many clinical trials, multiple time‐to‐event endpoints including the primary endpoint (e.g., time to death) and secondary endpoints (e.g., progression‐related endpoints) are commonly used to determine treatment efficacy. These endpoints are often biologically related. This work is motivated by a study of bone marrow transplant (BMT) for leukemia patients, who may experience the acute graft‐versus‐host disease (GVHD), relapse of leukemia, and death after an allogeneic BMT. The acute GVHD is associated with the relapse free survival, and both the acute GVHD and relapse of leukemia are intermediate nonterminal events subject to dependent censoring by the informative terminal event death, but not vice versa, giving rise to survival data that are subject to two sets of semi‐competing risks. It is important to assess the impacts of prognostic factors on these three time‐to‐event endpoints. We propose a novel statistical approach that jointly models such data via a pair of copulas to account for multiple dependence structures, while the marginal distribution of each endpoint is formulated by a Cox proportional hazards model. We develop an estimation procedure based on pseudo‐likelihood and carry out simulation studies to examine the performance of the proposed method in finite samples. The practical utility of the proposed method is further illustrated with data from the motivating example.     

Two‐stage estimation for multivariate recurrent event data with a dependent terminal event

Recurrent event data arise in longitudinal follow‐up studies, where each subject may experience the same type of events repeatedly. The work in this article is motivated by the data from a study of repeated peritonitis for patients on peritoneal dialysis. Due to the aspects of medicine and cost, the peritonitis cases were classified into two types: Gram‐positive and non‐Gram‐positive peritonitis. Further, since the death and hemodialysis therapy preclude the occurrence of recurrent events, we face multivariate recurrent event data with a dependent terminal event. We propose a flexible marginal model, which has three characteristics: first, we assume marginal proportional hazard and proportional rates models for terminal event time and recurrent event processes, respectively; second, the inter‐recurrences dependence and the correlation between the multivariate recurrent event processes and terminal event time are modeled through three multiplicative frailties corresponding to the specified marginal models; third, the rate model with frailties for recurrent events is specified only on the time before the terminal event. We propose a two‐stage estimation procedure for estimating unknown parameters. We also establish the consistency of the two‐stage estimator. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is applied to the peritonitis cohort data that motivated this study.     

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.     

Bayesian models for multivariate current status data with informative censoring

Multivariate current status data, consist of indicators of whether each of several events occur by the time of a single examination. Our interest focuses on inferences about the joint distribution of the event times. Conventional methods for analysis of multiple event-time data cannot be used because all of the event times are censored and censoring may be informative. Within a given subject, we account for correlated event times through a subject-specific latent variable, conditional upon which the various events are assumed to occur independently. We also assume that each event contributes independently to the hazard of censoring. Nonparametric step functions are used to characterize the baseline distributions of the different event times and of the examination times. Covariate and subject-specific effects are incorporated through generalized linear models. A Markov chain Monte Carlo algorithm is described for estimation of the posterior distributions of the unknowns. The methods are illustrated through application to multiple tumor site data from an animal carcinogenicity study.     

Use of summary measures to adjust for informative missingness in repeated measures data with random effects

We discuss how to apply the conditional informative missing model of Wu and Bailey (1989, Biometrics 45, 939-955) to the setting where the probability of missing a visit depends on the random effects of the primary response in a time-dependent fashion. This includes the case where the probability of missing a visit depends on the true value of the primary response. Summary measures for missingness that are weighted sums of the indicators of missed visits are derived for these situations. These summary measures are then incorporated as covariates in a random effects model for the primary response. This approach is illustrated by analyzing data collected from a trial of heroin addicts where missed visits are informative about drug test results. Simulations of realistic experiments indicate that these time-dependent summary measures also work well under a variety of informative censoring models. These summary measures can achieve large reductions in estimation bias and mean squared errors relative to those obtained by using other summary measures.     

Joint modeling and analysis of longitudinal data with informative observation times

Summary .  In analysis of longitudinal data, it is often assumed that observation times are predetermined and are the same across study subjects. Such an assumption, however, is often violated in practice. As a result, the observation times may be highly irregular. It is well known that if the sampling scheme is correlated with the outcome values, the usual statistical analysis may yield bias. In this article, we propose joint modeling and analysis of longitudinal data with possibly informative observation times via latent variables. A two-step estimation procedure is developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, and that the asymptotic variance can be consistently estimated using the bootstrap method. Simulation studies and a real data analysis demonstrate that our method performs well with realistic sample sizes and is appropriate for practical use.     

Modeling repeated count data subject to informative dropout

In certain diseases, outcome is the number of morbid events over the course of follow-up. In epilepsy, e.g., daily seizure counts are often used to reflect disease severity. Follow-up of patients in clinical trials of such diseases is often subject to censoring due to patients dying or dropping out. If the sicker patients tend to be censored in such trials, estimates of the treatment effect that do not incorporate the censoring process may be misleading. We extend the shared random effects approach of Wu and Carroll (1988, Biometrics 44, 175-188) to the setting of repeated counts of events. Three strategies are developed. The first is a likelihood-based approach for jointly modeling the count and censoring processes. A shared random effect is incorporated to introduce dependence between the two processes. The second is a likelihood-based approach that conditions on the dropout times in adjusting for informative dropout. The third is a generalized estimating equations (GEE) approach, which also conditions on the dropout times but makes fewer assumptions about the distribution of the count process. Estimation procedures for each of the approaches are discussed, and the approaches are applied to data from an epilepsy clinical trial. A simulation study is also conducted to compare the various approaches. Through analyses and simulations, we demonstrate the flexibility of the likelihood-based conditional model for analyzing data from the epilepsy trial.     

Analysis of multivariate recurrent event data with time‐dependent covariates and informative censoring

Multivariate recurrent event data are usually encountered in many clinical and longitudinal studies in which each study subject may experience multiple recurrent events. For the analysis of such data, most existing approaches have been proposed under the assumption that the censoring times are noninformative, which may not be true especially when the observation of recurrent events is terminated by a failure event. In this article, we consider regression analysis of multivariate recurrent event data with both time‐dependent and time‐independent covariates where the censoring times and the recurrent event process are allowed to be correlated via a frailty. The proposed joint model is flexible where both the distributions of censoring and frailty variables are left unspecified. We propose a pairwise pseudolikelihood approach and an estimating equation‐based approach for estimating coefficients of time‐dependent and time‐independent covariates, respectively. The large sample properties of the proposed estimates are established, while the finite‐sample properties are demonstrated by simulation studies. The proposed methods are applied to the analysis of a set of bivariate recurrent event data from a study of platelet transfusion reactions.     

Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

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.

     

Modeling multivariate survival data by a semiparametric random effects proportional odds model

In this article, the focus is on the analysis of multivariate survival time data with various types of dependence structures. Examples of multivariate survival data include clustered data and repeated measurements from the same subject, such as the interrecurrence times of cancer tumors. A random effect semiparametric proportional odds model is proposed as an alternative to the proportional hazards model. The distribution of the random effects is assumed to be multivariate normal and the random effect is assumed to act additively to the baseline log-odds function. This class of models, which includes the usual shared random effects model, the additive variance components model, and the dynamic random effects model as special cases, is highly flexible and is capable of modeling a wide range of multivariate survival data. A unified estimation procedure is proposed to estimate the regression and dependence parameters simultaneously by means of a marginal-likelihood approach. Unlike the fully parametric case, the regression parameter estimate is not sensitive to the choice of correlation structure of the random effects. The marginal likelihood is approximated by the Monte Carlo method. Simulation studies are carried out to investigate the performance of the proposed method. The proposed method is applied to two well-known data sets, including clustered data and recurrent event times data.     

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.     

Multivariate frailty models for two types of recurrent events with a dependent terminal event: Application to breast cancer data

Individuals may experience more than one type of recurrent event and a terminal event during the life course of a disease. Follow‐up may be interrupted for several reasons, including the end of a study, or patients lost to follow‐up, which are noninformative censoring events. Death could also stop the follow‐up, hence, it is considered as a dependent terminal event. We propose a multivariate frailty model that jointly analyzes two types of recurrent events with a dependent terminal event. Two estimation methods are proposed: a semiparametrical approach using penalized likelihood estimation where baseline hazard functions are approximated by M‐splines, and another one with piecewise constant baseline hazard functions. Finally, we derived martingale residuals to check the goodness‐of‐fit. We illustrate our proposals with a real dataset on breast cancer. The main objective was to model the dependency between the two types of recurrent events (locoregional and metastatic) and the terminal event (death) after a breast cancer.     

Analysis of times to repeated events in two-arm randomized trials with noncompliance and dependent censoring

This article develops randomization-based methods for times to repeated events in two-arm randomized trials with noncompliance and dependent censoring. Structural accelerated failure time models are assumed to capture causal effects on repeated event times and dependent censoring time, but the dependence structure among repeated event times and dependent censoring time is unspecified. Artificial censoring techniques to accommodate nonrandom noncompliance and dependent censoring are proposed. Estimation of the acceleration parameters are based on rank-based estimating functions. A simulation study is conducted to evaluate the performance of the developed methods. An illustration of the methods using data from an acute myeloid leukemia trial is provided.     

Joint inference for nonlinear mixed-effects models and time to event at the presence of missing data

In many longitudinal studies, the individual characteristics associated with the repeated measures may be possible covariates of the time to an event of interest, and thus, it is desirable to model the time-to-event process and the longitudinal process jointly. Statistical analyses may be further complicated in such studies with missing data such as informative dropouts. This article considers a nonlinear mixed-effects model for the longitudinal process and the Cox proportional hazards model for the time-to-event process. We provide a method for simultaneous likelihood inference on the 2 models and allow for nonignorable data missing. The approach is illustrated with a recent AIDS study by jointly modeling HIV viral dynamics and time to viral rebound.