Joint modelling of repeated transitions in follow-up data--a case study on breast cancer data

In longitudinal studies where time to a final event is the ultimate outcome often information is available about intermediate events the individuals may experience during the observation period. Even though many extensions of the Cox proportional hazards model have been proposed to model such multivariate time-to-event data these approaches are still very rarely applied to real datasets. The aim of this paper is to illustrate the application of extended Cox models for multiple time-to-event data and to show their implementation in popular statistical software packages. We demonstrate a systematic way of jointly modelling similar or repeated transitions in follow-up data by analysing an event-history dataset consisting of 270 breast cancer patients, that were followed-up for different clinical events during treatment in metastatic disease. First, we show how this methodology can also be applied to non Markovian stochastic processes by representing these processes as "conditional" Markov processes. Secondly, we compare the application of different Cox-related approaches to the breast cancer data by varying their key model components (i.e. analysis time scale, risk set and baseline hazard function). Our study showed that extended Cox models are a powerful tool for analysing complex event history datasets since the approach can address many dynamic data features such as multiple time scales, dynamic risk sets, time-varying covariates, transition by covariate interactions, autoregressive dependence or intra-subject correlation.     

Analysis of longitudinal data in the presence of informative observational times and a dependent terminal event, with application to medical cost data

Summary .   In longitudinal observational studies, repeated measures are often taken at informative observation times. Also, there may exist a dependent terminal event such as death that stops the follow-up. For example, patients in poorer health are more likely to seek medical treatment and their medical cost for each visit tends to be higher. They are also subject to a higher mortality rate. In this article, we propose a random effects model of repeated measures in the presence of both informative observation times and a dependent terminal event. Three submodels are used, respectively, for (1) the intensity of recurrent observation times, (2) the amount of repeated measure at each observation time, and (3) the hazard of death. Correlated random effects are incorporated to join the three submodels. The estimation can be conveniently accomplished by Gaussian quadrature techniques, e.g., SAS Proc NLMIXED . An analysis of the cost-accrual process of chronic heart failure patients from the clinical data repository at the University of Virginia Health System is presented to illustrate the proposed method.     

Locally efficient estimation of the quality-adjusted lifetime distribution with right-censored data and covariates

Zhao and Tsiatis (1997) consider the problem of estimation of the distribution of the quality-adjusted lifetime when the chronological survival time is subject to right censoring. The quality-adjusted lifetime is typically defined as a weighted sum of the times spent in certain states up until death or some other failure time. They propose an estimator and establish the relevant asymptotics under the assumption of independent censoring. In this paper we extend the data structure with a covariate process observed until the end of follow-up and identify the optimal estimation problem. Because of the curse of dimensionality, no globally efficient nonparametric estimators, which have a good practical performance at moderate sample sizes, exist. Given a correctly specified model for the hazard of censoring conditional on the observed quality-of-life and covariate processes, we propose a closed-form one-step estimator of the distribution of the quality-adjusted lifetime whose asymptotic variance attains the efficiency bound if we can correctly specify a lower-dimensional working model for the conditional distribution of quality-adjusted lifetime given the observed quality-of-life and covariate processes. The estimator remains consistent and asymptotically normal even if this latter submodel is misspecified. The practical performance of the estimators is illustrated with a simulation study. We also extend our proposed one-step estimator to the case where treatment assignment is confounded by observed risk factors so that this estimator can be used to test a treatment effect in an observational study.     

Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events

The observation of repeated events for subjects in cohort studies could be terminated by loss to follow-up, end of study, or a major failure event such as death. In this context, the major failure event could be correlated with recurrent events, and the usual assumption of noninformative censoring of the recurrent event process by death, required by most statistical analyses, can be violated. Recently, joint modeling for 2 survival processes has received considerable attention because it makes it possible to study the joint evolution over time of 2 processes and gives unbiased and efficient parameters. The most commonly used estimation procedure in the joint models for survival events is the expectation maximization algorithm. We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of the continuous hazard functions in a general joint frailty model with right censoring and delayed entry. The simulation study demonstrates that this semiparametric approach yields satisfactory results in this complex setting. As an illustration, such an approach is applied to a prospective cohort with recurrent events of follicular lymphomas, jointly modeled with death.     

Bayesian estimators for conditional hazard functions

This article introduces a new Bayesian approach to the analysis of right-censored survival data. The hazard rate of interest is modeled as a product of conditionally independent stochastic processes corresponding to (1) a baseline hazard function and (2) a regression function representing the temporal influence of the covariates. These processes jump at times that form a time-homogeneous Poisson process and have a pairwise dependency structure for adjacent values. The two processes are assumed to be conditionally independent given their jump times. Features of the posterior distribution, such as the mean covariate effects and survival probabilities (conditional on the covariate), are evaluated using the Metropolis-Hastings-Green algorithm. We illustrate our methodology by an application to nasopharynx cancer survival data.     

Applying the Cox proportional hazards model when the change time of a binary time-varying covariate is interval censored

This paper develops methodology for estimation of the effect of a binary time-varying covariate on failure times when the change time of the covariate is interval censored. The motivating example is a study of cytomegalovirus (CMV) disease in patients with human immunodeficiency virus (HIV) disease. We are interested in determining whether CMV shedding predicts an increased hazard for developing active CMV disease. Since a clinical screening test is needed to detect CMV shedding, the time that shedding begins is only known to lie in an interval bounded by the patient's last negative and first positive tests. In a Cox proportional hazards model with a time-varying covariate for CMV shedding, the partial likelihood depends on the covariate status of every individual in the risk set at each failure time. Due to interval censoring, this is not always known. To solve this problem, we use a Monte Carlo EM algorithm with a Gibbs sampler embedded in the E-step. We generate multiple completed data sets by drawing imputed exact shedding times based on the joint likelihood of the shedding times and event times under the Cox model. The method is evaluated using a simulation study and is applied to the data set described above.     

A Bayesian Semiparametric Survival Model with Longitudinal Markers

Summary We consider inference for data from a clinical trial of treatments for metastatic prostate cancer. Patients joined the trial with diverse prior treatment histories. The resulting heterogeneous patient population gives rise to challenging statistical inference problems when trying to predict time to progression on different treatment arms. Inference is further complicated by the need to include a longitudinal marker as a covariate. To address these challenges, we develop a semiparametric model for joint inference of longitudinal data and an event time. The proposed approach includes the possibility of cure for some patients. The event time distribution is based on a nonparametric Pólya tree prior. For the longitudinal data we assume a mixed effects model. Incorporating a regression on covariates in a nonparametric event time model in general, and for a Pólya tree model in particular, is a challenging problem. We exploit the fact that the covariate itself is a random variable. We achieve an implementation of the desired regression by factoring the joint model for the event time and the longitudinal outcome into a marginal model for the event time and a regression of the longitudinal outcomes on the event time, i.e., we implicitly model the desired regression by modeling the reverse conditional distribution.     

Comparison of procedures to assess non-linear and time-varying effects in multivariable models for survival data

The focus of many medical applications is to model the impact of several factors on time to an event. A standard approach for such analyses is the Cox proportional hazards model. It assumes that the factors act linearly on the log hazard function (linearity assumption) and that their effects are constant over time (proportional hazards (PH) assumption). Variable selection is often required to specify a more parsimonious model aiming to include only variables with an influence on the outcome. As follow-up increases the effect of a variable often gets weaker, which means that it varies in time. However, spurious time-varying effects may also be introduced by mismodelling other parts of the multivariable model, such as omission of an important covariate or an incorrect functional form of a continuous covariate. These issues interact. To check whether the effect of a variable varies in time several tests for non-PH have been proposed. However, they are not sufficient to derive a model, as appropriate modelling of the shape of time-varying effects is required. In three examples we will compare five recently published strategies to assess whether and how the effects of covariates from a multivariable model vary in time. For practical use we will give some recommendations.     

Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths

For observational longitudinal studies of geriatric populations, outcomes such as disability or cognitive functioning are often censored by death. Statistical analysis of such data may explicitly condition on either vital status or survival time when summarizing the longitudinal response. For example a pattern-mixture model characterizes the mean response at time t conditional on death at time S = s (for s > t), and thus uses future status as a predictor for the time t response. As an alternative, we define regression conditioning on being alive as a regression model that conditions on survival status, rather than a specific survival time. Such models may be referred to as partly conditional since the mean at time t is specified conditional on being alive (S > t), rather than using finer stratification (S = s for s > t). We show that naive use of standard likelihood-based longitudinal methods and generalized estimating equations with non-independence weights may lead to biased estimation of the partly conditional mean model. We develop a taxonomy for accommodation of both dropout and death, and describe estimation for binary longitudinal data that applies selection weights to estimating equations with independence working correlation. Simulation studies and an analysis of monthly disability status illustrate potential bias in regression methods that do not explicitly condition on survival.     

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.     

Flexible Estimation of Differences in Treatment-Specific Recurrent Event Means in the Presence of a Terminating Event

Summary .  In this article, we consider the setting where the event of interest can occur repeatedly for the same subject (i.e., a recurrent event; e.g., hospitalization) and may be stopped permanently by a terminating event (e.g., death). Among the different ways to model recurrent/terminal event data, the marginal mean (i.e., averaging over the survival distribution) is of primary interest from a public health or health economics perspective. Often, the difference between treatment-specific recurrent event means will not be constant over time, particularly when treatment-specific differences in survival exist. In such cases, it makes more sense to quantify treatment effect based on the cumulative difference in the recurrent event means, as opposed to the instantaneous difference in the rates. We propose a method that compares treatments by separately estimating the survival probabilities and recurrent event rates given survival, then integrating to get the mean number of events. The proposed method combines an additive model for the conditional recurrent event rate and a proportional hazards model for the terminating event hazard. The treatment effects on survival and on recurrent event rate among survivors are estimated in constructing our measure and explain the mechanism generating the difference under study. The example that motivates this research is the repeated occurrence of hospitalization among kidney transplant recipients, where the effect of expanded criteria donor (ECD) compared to non-ECD kidney transplantation on the mean number of hospitalizations is of interest.     

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

For randomized clinical trials where the endpoint of interest is a time-to-event subject to censoring, estimating the treatment effect has mostly focused on the hazard ratio from the Cox proportional hazards model. Since the model’s proportional hazards assumption is not always satisfied, a useful alternative, the so-called additive hazards model, may instead be used to estimate a treatment effect on the difference of hazard functions. Still, the hazards difference may be difficult to grasp intuitively, particularly in a clinical setting of, e.g., patient counseling, or resource planning. In this paper, we study the quantiles of a covariate’s conditional survival function in the additive hazards model. Specifically, we estimate the residual time quantiles, i.e., the quantiles of survival times remaining at a given time t, conditional on the survival times greater than t, for a specific covariate in the additive hazards model. We use the estimates to translate the hazards difference into the difference in residual time quantiles, which allows a more direct clinical interpretation. We determine the asymptotic properties, assess the performance via Monte-Carlo simulations, and demonstrate the use of residual time quantiles in two real randomized clinical trials.     

Generalized additive models with interval-censored data and time-varying covariates: application to human immunodeficiency virus infection in hemophiliacs

We describe a method for extending smooth nonparametric modeling methods to time-to-event data where the event may be known only to lie within a window of time. Maximum penalized likelihood is used to fit a discrete proportional hazards model that also models the baseline hazard, and left-truncation and time-varying covariates are accommodated. The implementation follows generalized additive modeling conventions, allowing both parametric and smooth terms and specifying the amount of smoothness in terms of the effective degrees of freedom. We illustrate the method on a well-known interval-censored data set on time of human immunodeficiency virus infection in a multicenter study of hemophiliacs. The ability to examine time-varying covariates, not available with previous methods, allows detection and modeling of nonproportional hazards and use of a time-varying covariate that fits the data better and is more plausible than a fixed alternative.     

A marginalized conditional linear model for longitudinal binary data when informative dropout occurs in continuous time

Within the pattern-mixture modeling framework for informative dropout, conditional linear models (CLMs) are a useful approach to deal with dropout that can occur at any point in continuous time (not just at observation times). However, in contrast with selection models, inferences about marginal covariate effects in CLMs are not readily available if nonidentity links are used in the mean structures. In this article, we propose a CLM for long series of longitudinal binary data with marginal covariate effects directly specified. The association between the binary responses and the dropout time is taken into account by modeling the conditional mean of the binary response as well as the dependence between the binary responses given the dropout time. Specifically, parameters in both the conditional mean and dependence models are assumed to be linear or quadratic functions of the dropout time; and the continuous dropout time distribution is left completely unspecified. Inference is fully Bayesian. We illustrate the proposed model using data from a longitudinal study of depression in HIV-infected women, where the strategy of sensitivity analysis based on the extrapolation method is also demonstrated.     

A hot-deck multiple imputation procedure for gaps in longitudinal recurrent event histories

We propose a regression-based hot-deck multiple imputation method for gaps of missing data in longitudinal studies, where subjects experience a recurrent event process and a terminal event. Examples are repeated asthma episodes and death, or menstrual periods and menopause, as in our motivating application. Research interest concerns the onset time of a marker event, defined by the recurrent event process, or the duration from this marker event to the final event. Gaps in the recorded event history make it difficult to determine the onset time of the marker event, and hence, the duration from onset to the final event. Simple approaches such as jumping gap times or dropping cases with gaps have obvious limitations. We propose a procedure for imputing information in the gaps by substituting information in the gap from a matched individual with a completely recorded history in the corresponding interval. Predictive mean matching is used to incorporate information on longitudinal characteristics of the repeated process and the final event time. Multiple imputation is used to propagate imputation uncertainty. The procedure is applied to an important data set for assessing the timing and duration of the menopausal transition. The performance of the proposed method is assessed by a simulation study.     

Time-varying functional regression for predicting remaining lifetime distributions from longitudinal trajectories

A recurring objective in longitudinal studies on aging and longevity has been the investigation of the relationship between age-at-death and current values of a longitudinal covariate trajectory that quantifies reproductive or other behavioral activity. We propose a novel technique for predicting age-at-death distributions for situations where an entire covariate history is included in the predictor. The predictor trajectories up to current time are represented by time-varying functional principal component scores, which are continuously updated as time progresses and are considered to be time-varying predictor variables that are entered into a class of time-varying functional regression models that we propose. We demonstrate for biodemographic data how these methods can be applied to obtain predictions for age-at-death and estimates of remaining lifetime distributions, including estimates of quantiles and of prediction intervals for remaining lifetime. Estimates and predictions are obtained for individual subjects, based on their observed behavioral trajectories, and include a dimension-reduction step that is implemented by projecting on a single index. The proposed techniques are illustrated with data on longitudinal daily egg-laying for female medflies, predicting remaining lifetime and age-at-death distributions from individual event histories observed up to current time.     

Bayesian approaches to joint cure-rate and longitudinal models with applications to cancer vaccine trials

Complex issues arise when investigating the association between longitudinal immunologic measures and time to an event, such as time to relapse, in cancer vaccine trials. Unlike many clinical trials, we may encounter patients who are cured and no longer susceptible to the time-to-event endpoint. If there are cured patients in the population, there is a plateau in the survival function, S(t), after sufficient follow-up. If we want to determine the association between the longitudinal measure and the time-to-event in the presence of cure, existing methods for jointly modeling longitudinal and survival data would be inappropriate, since they do not account for the plateau in the survival function. The nature of the longitudinal data in cancer vaccine trials is also unique, as many patients may not exhibit an immune response to vaccination at varying time points throughout the trial. We present a new joint model for longitudinal and survival data that accounts both for the possibility that a subject is cured and for the unique nature of the longitudinal data. An example is presented from a cancer vaccine clinical trial.     

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.