Terminal Event Time Research Articles

Conditional modeling of recurrent event data with terminal event.

Recurrent event data with a terminal event arise in follow-up studies. The current literature has primarily focused on the effect of covariates on the recurrent event process using marginal estimating equation approaches or joint modeling approaches via frailties. In this article, we propose a conditional model for recurrent event data with a terminal event, which provides an intuitive interpretation of the effect of the terminal event: at an early time, the rate of recurrent events is nearly independent of the terminal event, but the dependence gets stronger as time goes close to the terminal event time. A two-stage likelihood-based approach is proposed to estimate parameters of interest. Asymptotic properties of the estimators are established. The finite-sample behavior of the proposed method is examined through simulation studies. A real data of colorectal cancer is analyzed by the proposed method for illustration.

A Symmetric Kernel Smoothing Estimation of the Time-Varying Coefficient for Medical Costs

In longitudinal studies, subjects are repeatedly observed at a set of distinct time points until the terminal event time. The time-varying coefficient model extends the parametric method and captures the dynamic trajectories of time-dependent covariate effects, thus enabling it to describe the potential relationship between the longitudinal variable and the observed time points. In this study, we propose a novel approach to the estimation of medical costs using a symmetric kernel smoothing method in the time-varying coefficient joint model. A smooth function of medical costs is derived by weighting the values of longitudinal data at all distinct observed time points via the combination of the kernel method and the inverse probability weighting method. For the simulation study, we first set up the true functions of time-varying coefficients; we then generated random samples for covariates and censored survival times. Subsequently, the longitudinal data of response variables could be produced. Further, numerical simulation experiments were conducted by using the proposed method and applying R code to the generated data. The estimated results for the parameters and non-parametric functions were compared with different settings. The numerical results illustrate that as the sample size increases, the bias and model-based standard errors decrease, and the performance improves with larger sample sizes. The estimates of functions in the model almost coincide with the true functions, as shown in the figures of the simulation study. Furthermore, the consistency of the obtained estimator is demonstrated via theoretical analysis, and a numerical simulation is performed to illustrate the performance of the proposed estimators. The proposed model is applied to a real-world data set acquired from a multicenter automatic defibrillator implantation trial (MADIT).

Kernel Estimation of Bivariate Time-Varying Coefficient Model for Longitudinal Data with Terminal Event

We propose a nonparametric bivariate time-varying coefficient model for longitudinal measurements with the occurrence of a terminal event that is subject to right censoring. The time-varying coefficients capture the longitudinal trajectories of covariate effects along with both the followup time and the residual lifetime. The proposed model extends the parametric conditional approach given terminal event time in recent literature, and thus avoids potential model misspecification. We consider a kernel smoothing method for estimating regression coefficients in our model and use cross-validation for bandwidth selection, applying undersmoothing in the final analysis to eliminate the asymptotic bias of the kernel estimator. We show that the kernel estimates follow a finite-dimensional normal distribution asymptotically under mild regularity conditions, and provide an easily computed sandwich covariance matrix estimator. We conduct extensive simulations that show desirable performance of the proposed approach, and apply the method to analyzing the medical cost data for patients with end-stage renal disease. Supplementary materials for this article are available online.

Causal inference for semi-competing risks data.

The causal effects of Apolipoprotein E $\epsilon4$ allele (APOE) on late-onset Alzheimer's disease (AD) and death are complicated to define because AD may occur under one intervention but not under the other, and because AD occurrence may affect age of death. In this article, this dual outcome scenario is studied using the semi-competing risks framework for time-to-event data. Two event times are of interest: a nonterminal event time (age at AD diagnosis), and a terminal event time (age at death). AD diagnosis time is observed only if it precedes death, which may occur before or after AD. We propose new estimands for capturing the causal effect of APOE on AD and death. Our proposal is based on a stratification of the population with respect to the order of the two events. We present a novel assumption utilizing the time-to-event nature of the data, which is more flexible than the often-invoked monotonicity assumption. We derive results on partial identifiability, suggest a sensitivity analysis approach, and give conditions under which full identification is possible. Finally, we present and implement nonparametric and semiparametric estimation methods under right-censored semi-competing risks data for studying the complex effect of APOE on AD and death.

The fused Kolmogorov–Smirnov screening for ultra-high dimensional semi-competing risks data

Semi-competing risks data, including probably correlated non-terminal event and terminal event times, are frequently encountered in medical research. Work on semi-competing risks data has mainly focused on the situations without or only with low-dimensional covariates. However, high and ultra-high dimensional data have been very common in modern scientific research. In this article, we propose a model-free feature screening procedure for ultra-high dimensional semi-competing risks data to discover features contributing separately or jointly to non-terminal and terminal event times via Kolmogorov–Smirnov statistics. This new approach could be used for contexts with discrete, categorical and continuous covariates, which is achieved through the technique of slice-and-fuse. It enjoys several desirable advantages inherited in the Kolmogorov–Smirnov statistics. Under rather mild conditions, we show that the newly suggested method possesses sure screening property. Monte-Carlo simulation studies are conducted to investigate the finite sample properties of our proposed procedure and make comparisons with existing methods, while a real data example is also offered for illustration.

Semiparametric copula-based regression modeling of semi-competing risks data

Semi-competing risks data often arise in medical studies where the terminal event (e.g., death) censors the non terminal event (e.g., cancer recurrence), but the non terminal event does not prevent the subsequent occurrence of the terminal event. This article considers regression modeling of semi-competing risks data to assess the covariate effects on the respective non terminal and terminal event times. We propose a copula-based framework for semi-competing risks regression with time-varying coefficients, where the dependence between the non terminal and terminal event times is characterized by a copula and the time-varying covariate effects are imposed on two marginal regression models. We develop a two-stage inferential procedure for estimating the association parameter in the copula model and time-varying regression parameters. We evaluate the finite sample performance of the proposed method through simulation studies and illustrate the method through an application to Surveillance, Epidemiology, and End Results–Medicare data for elderly women diagnosed with early-stage breast cancer and initially treated with breast-conserving surgery.

Prognostic Factors Associated with Curing in Patients with Breast Cancer: A Joint Frailty Model.

Background:Recurrent event data arise frequently in longitudinal medical studies. In many situations, there are a large portion of patients without any recurrences, manifesting the “zero-inflated” nature of the data. Moreover, there often exists a terminal event which may be correlated with the recurrent events. The goal of this study is to extend the application of joint frailty model to identify the prognostic factors associated with curing in patients with breast cancer.Methods:As a prospective study, medical records of women who had been attended to Cancer Research Center, Shahid Beheshti University of Medical Sciences from January 1998 to February 2016 were reviewed. Finally, after an initial review of medical records, 711 patients were included in the study and analyzed. A checklist that included items drawn from the demographic background of patients was provided in the study. Two joint frailty models for zero-inflated recurrent events, combining a logistic model for “structural zero” status (Yes/No) and a joint frailty proportional hazards model for recurrent and terminal event times were performed to identify factors associated with BCS.Results:The mean age of patients was 38.2 years. The numbers of subjects with 1, 2, 3, and 4 recurrent events were 392, 207, 97, and 15, respectively. The median follow-up time was 6.87 years. There were 137 (19.2%) deaths from cancer during the follow-up. Among the 574 patients who were censored, 418 had no tumor recurrence. Thus, there may exist a large portion of “cured” subjects. We can see that the radiation (OR = 6.02, CI = (3.87, 8.61)) and tumor size interaction with radiation (OR = 1.065, CI = (1.002–1.26)) were significant in the cure model (P < 0.05) which means that patients with smaller tumor sizes were more likely to be cured by radiation therapy.Conclusions:Our proposed models can help investigators to evaluate which treatment will result in a higher fraction of cured subjects. This is usually an important research question in biomedical studies.

Two-sample test based on empirical likelihood ratio under semi-competing risks data

This article considers the two-sample testing problem of the survival function of the non terminal event time under semi-competing risks data. The empirical likelihood function is constructed for the survival function estimation of the non terminal event time, then maximize it by the PSO (Particle swarm optimization) algorithm to obtain the MLE. For the testing problem, the article develops the empirical likelihood ratio test to compare the two survival curves. From simulation studies, it shows the performance of the proposed approaches is good. Finally, a real data analysis is presented for illustration.

Meta-analysis of individual patient data with semi-competing risks under the Weibull joint frailty–copula model

In meta-analysis of individual patient data with semi-competing risks, the joint frailty–copula model has been proposed, where frailty terms account for the between-study heterogeneity and copulas account for dependence between terminal and nonterminal event times. In the previous works, the baseline hazard functions in the joint frailty–copula model are estimated by the nonparametric model or the penalized spline model, which requires complex maximization schemes and resampling-based interval estimation. In this article, we propose the Weibull distribution for the baseline hazard functions under the joint frailty–copula model. We show that the Weibull model constitutes a conjugate model for the gamma frailty, leading to explicit expressions for the moments, survival functions, hazard functions, quantiles, and mean residual lifetimes. These results facilitate the parameter interpretation of prognostic inference. We propose a maximum likelihood estimation method and make our computer programs available in the R package, joint.Cox. We also show that the delta method is feasible to calculate interval estimates, which is a useful alternative to the resampling-based method. We conduct simulation studies to examine the accuracy of the proposed methods. Finally, we use the data on ovarian cancer patients to illustrate the proposed method.

Semiparametric Estimation of Longitudinal Medical Cost Trajectory

ABSTRACTEstimating the average monthly medical costs from disease diagnosis to a terminal event such as death for an incident cohort of patients is a topic of immense interest to researchers in health policy and health economics because patterns of average monthly costs over time reveal how medical costs vary across phases of care. The statistical challenges to estimating monthly medical costs longitudinally are multifold; the longitudinal cost trajectory (formed by plotting the average monthly costs from diagnosis to the terminal event) is likely to be nonlinear, with its shape depending on the time of the terminal event, which can be subject to right censoring. The goal of this article is to tackle this statistically challenging topic by estimating the conditional mean cost at any month t given the time of the terminal event s. The longitudinal cost trajectories with different terminal event times form a bivariate surface of t and s, under the constraint t ⩽ s. We propose to estimate this surface using bivariate penalized splines in an expectation-maximization algorithm that treats the censored terminal event times as missing data. We evaluate the proposed model and estimation method in simulations and apply the method to the medical cost data of an incident cohort of stage IV breast cancer patients from the Surveillance, Epidemiology, and End Results–Medicare Linked Database. Supplementary materials for this article are available online.

Joint analysis of recurrent event data with additive–multiplicative hazards model for the terminal event time

Recurrent event data are often collected in longitudinal follow-up studies. In this article, we propose a semiparametric method to model the recurrent and terminal events jointly. We present an additive–multiplicative hazards model for the terminal event and a proportional intensity model for the recurrent events, and a shared frailty is used to model the dependence between the recurrent and terminal events. We adopt estimating equation approaches for inference, and the asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies. An application to a medical cost study of chronic heart failure patients from the University of Virginia Health System is illustrated.

Joint analysis of recurrent event data with a dependent terminal event

Recurrent event data frequently occur in many longitudinal studies, and the observation on recurrent events could be stopped by a terminal event such as death. This paper considers joint modeling and analysis of recurrent event and terminal event data through a common subject-specific frailty, in which the proportional intensity model is used for modeling the recurrent event process and the additive hazards model is used for modeling the terminal event time. Estimating equation approaches are developed for parameter estimation and asymptotic properties of the resulting estimators are established. In addition, some procedures are presented for model checking. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a heart failure study is provided.

Conditional Modeling of Longitudinal Data With Terminal Event

ABSTRACTWe consider a random effects model for longitudinal data with the occurrence of an informative terminal event that is subject to right censoring. Existing methods for analyzing such data include the joint modeling approach using latent frailty and the marginal estimating equation approach using inverse probability weighting; in both cases the effect of the terminal event on the response variable is not explicit and thus not easily interpreted. In contrast, we treat the terminal event time as a covariate in a conditional model for the longitudinal data, which provides a straightforward interpretation while keeping the usual relationship of interest between the longitudinally measured response variable and covariates for times that are far from the terminal event. A two-stage semiparametric likelihood-based approach is proposed for estimating the regression parameters; first, the conditional distribution of the right-censored terminal event time given other covariates is estimated and then the likelihood function for the longitudinal event given the terminal event and other regression parameters is maximized. The method is illustrated by numerical simulations and by analyzing medical cost data for patients with end-stage renal disease. Desirable asymptotic properties are provided. Supplementary materials for this article are available online.

Quantile residual life regression based on semi-competing risks data

ABSTRACTThis paper investigates the quantile residual life regression based on semi-competing risk data. Because the terminal event time dependently censors the non-terminal event time, the inference on the non-terminal event time is not available without extra assumption. Therefore, we assume that the non-terminal event time and the terminal event time follow an Archimedean copula. Then, we apply the inverse probability weight technique to construct an estimating equation of quantile residual life regression coefficients. But, the estimating equation may not be continuous in coefficients. Thus, we apply the generalized solution approach to overcome this problem. Since the variance estimation of the proposed estimator is difficult to obtain, we use the bootstrap resampling method to estimate it. From simulations, it shows the performance of the proposed method is well. Finally, we analyze the Bone Marrow Transplant data for illustrations.

Estimation of the survival function with redistribution algorithm under semi-competing risks data

This paper focuses on the estimation of the survival function of the non-terminal event time for semi-competing risks data. Without extra assumptions, we cannot make inference on the non-terminal event time because the non-terminal event time is dependently censored by the terminal event time. Thus, we utilize the Archimedean copula model to specify the dependency between the non-terminal event time and the terminal event time. Under the Archimedean copula assumption, we apply the redistribution method to estimate the survival function of the non-terminal event time and compare it with the copula-graphic estimator introduced by Lakhal et al. (2008). We also apply our suggested approach to analyze the Bone Marrow Transplant data.

Time-varying latent model for longitudinal data with informative observation and terminal event times

Longitudinal data often occur in follow-up studies, and in many situations, there may exist informative observation times and a dependent terminal event such as death that stops the follow-up. We propose a semiparametric mixed effect model with time-varying latent effects in the analysis of longitudinal data with informative observation times and a dependent terminal event. Estimating equation approaches are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is provided.

Analyzing semi-competing risks data with missing cause of informative terminal event.

Cancer studies frequently yield multiple event times that correspond to landmarks in disease progression, including non-terminal events (i.e., cancer recurrence) and an informative terminal event (i.e., cancer-related death). Hence, we often observe semi-competing risks data. Work on such data has focused on scenarios in which the cause of the terminal event is known. However, in some circumstances, the information on cause for patients who experience the terminal event is missing; consequently, we are not able to differentiate an informative terminal event from a non-informative terminal event. In this article, we propose a method to handle missing data regarding the cause of an informative terminal event when analyzing the semi-competing risks data. We first consider the nonparametric estimation of the survival function for the terminal event time given missing cause-of-failure data via the expectation-maximization algorithm. We then develop an estimation method for semi-competing risks data with missing cause of the terminal event, under a pre-specified semiparametric copula model. We conduct simulation studies to investigate the performance of the proposed method. We illustrate our methodology using data from a study of early-stage breast cancer. Copyright © 2016 John Wiley & Sons, Ltd.

Analyzing longitudinal data with informative observation and terminal event times

Longitudinal data often arise when subjects are followed over a period of time, and in many situations, there may exist informative observation times and a dependent terminal event such as death that stops the follow-up. In this article, we propose joint modeling and analysis of longitudinal data with possibly informative observation times and a dependent terminal event in which a common subject-specific latent variable is used to characterize the correlations. A borrow-strength estimation procedure is developed for parameter estimation, and both large-sample and finite-sample properties of the proposed estimators are established. In addition, some goodness-of-fit methods for assessing the adequacy of the model are provided. An application to a bladder cancer study is illustrated.

Flexible association modelling and prediction with semi‐competing risks data

AbstractSemi‐competing risks data involve a non‐terminal event time, such as time to disease progression, and a terminal event time such as time to death. Existing methods for handling semi‐competing risks data often assume that the underlying association between the two event times follows a pre‐specified copula with unknown association parameters, which often correspond to the strength of association. In this article we propose a flexible association model that does not require pre‐specifying a copula. Therefore our methods facilitate a convenient and robust evaluation of the underlying association pattern, as well as the association strength. Furthermore the proposed association model leads to a robust estimator for the conditional survival probability of the terminal event given the non‐terminal event. The methods were also extended to handle left‐truncation. Both the association and survival estimators were shown to feature desirable asymptotic properties and satisfactory numerical performance. Our methods were successfully applied to a diabetes data set to study the association between time to diabetic nephropathy and time to death, and to predict the mortality rate given the onset time of nephropathy. The Canadian Journal of Statistics 44: 361–374; 2016 © 2016 Statistical Society of Canada

A new joint model of recurrent event data with the additive hazards model for the terminal event time

Recurrent event data are frequently encountered in clinical and observational studies related to biomedical science, econometrics, reliability and demography. In some situations, recurrent events serve as important indicators for evaluating disease progression, health deterioration, or insurance risk. In statistical literature, non informative censoring is typically assumed when statistical methods and theories are developed for analyzing recurrent event data. In many applications, however, there may exist a terminal event, such as death, that stops the follow-up, and it is the correlation of this terminal event with the recurrent event process that is of interest. This work considers joint modeling and analysis of recurrent event and terminal event data, with the focus primarily on determining how the terminal event process and the recurrent event process are correlated (i.e. does the frequency of the recurrent event influence the risk of the terminal event). We propose a joint model of the recurrent event process and the terminal event, linked through a common subject-specific latent variable, in which the proportional intensity model is used for modeling the recurrent event process and the additive hazards model is used for modeling the terminal event time.