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.     

Testing for dependence between failure time and visit compliance with interval-censored data

Interval-censored failure-time data arise when subjects miss prescheduled visits at which the failure is to be assessed. The resulting intervals in which the failure is known to have occurred are overlapping. Most approaches to the analysis of these data assume that the visit-compliance process is ignorable with respect to likelihood analysis of the failure-time distribution. While this assumption offers considerable simplification, it is not always plausible. Here we test for dependence between the failure- and visit-compliance processes, applicable to studies in which data collection continues after the occurrence of the failure. We do not make any of the assumptions made by previous authors about the joint distribution of the visit-compliance process, a covariate process, and the failure time. Instead, we consider conditional models of the true failure history given the current visit compliance at each visit time, allowing for correlation across visit times. Because failure status is not known at some visit times due to missed visits, only models of the observed failure history given current visit compliance are estimable. We describe how the parameters from these models can be used to test for a negative association and how bounds on unestimable parameters provided by the observed data are needed additionally to infer a positive association. We illustrate the method with data from an AIDS study and we investigate the power of the test through a simulation study.     

A proportional hazards model for multivariate interval-censored failure time data

This paper focuses on the methodology developed for analyzing a multivariate interval-censored data set from an AIDS observational study. A purpose of the study was to determine the natural history of the opportunistic infection cytomeglovirus (CMV) in an HIV-infected individual. For this observational study, laboratory tests were performed at scheduled clinic visits to test for the presence of the CMV virus in the blood and in the urine (called CMV shedding in the blood and urine). The study investigators were interested in determining whether the stage of HIV disease at study entry was predictive of an increased risk for CMV shedding in either the blood or the urine. If all patients had made each clinic visit, the data would be multivariate grouped failure time data and published methods could be used. However, many patients missed several visits, and when they returned, their lab tests indicated a change in their blood and/or urine CMV shedding status, resulting in interval-censored failure time data. This paper outlines a method for applying the proportional hazards model to the analysis of multivariate interval-censored failure time data from a study of CMV in HIV-infected patients.     

Discrete-time nonparametric estimation for semi-Markov models of chain-of-events data subject to interval censoring and truncation

Chain-of-events data are longitudinal observations on a succession of events that can only occur in a prescribed order. One goal in an analysis of this type of data is to determine the distribution of times between the successive events. This is difficult when individuals are observed periodically rather than continuously because the event times are then interval censored. Chain-of-events data may also be subject to truncation when individuals can only be observed if a certain event in the chain (e.g., the final event) has occurred. We provide a nonparametric approach to estimate the distributions of times between successive events in discrete time for data such as these under the semi-Markov assumption that the times between events are independent. This method uses a self-consistency algorithm that extends Turnbull's algorithm (1976, Journal of the Royal Statistical Society, Series B 38, 290-295). The quantities required to carry out the algorithm can be calculated recursively for improved computational efficiency. Two examples using data from studies involving HIV disease are used to illustrate our methods.     

Fitting the log-F accelerated failure time model with incomplete covariate data

Data obtained from studies in the health sciences often have incompletely observed covariates as well as censored outcomes. In this paper, we present methods for fitting the log-F accelerated failure time model with incomplete continuous and/or categorical time-independent covariates using the Gibbs sampler. A general location model that allows different covariance structures across cells is specified for the covariates, and ignorable missingness of the covariates is assumed. Techniques that accommodate standard assumptions of ignorable censoring as well as certain types of nonignorable censoring are developed. We compare our approach to traditional complete-case analysis in an application to data obtained from a study of melanoma. The comparison indicates that substantial gains in efficiency are possible with our approach.     

Regression analysis of doubly censored failure time data with frailty

In doubly censored failure time data, the survival time of interest is defined as the elapsed time between an initial event and a subsequent event, and the occurrences of both events cannot be observed exactly. Instead, only right- or interval-censored observations on the occurrence times are available. For the analysis of such data, a number of methods have been proposed under the assumption that the survival time of interest is independent of the occurrence time of the initial event. This article investigates a different situation where the independence may not be true with the focus on regression analysis of doubly censored data. Cox frailty models are applied to describe the effects of covariates and an EM algorithm is developed for estimation. Simulation studies are performed to investigate finite sample properties of the proposed method and an illustrative example from an acquired immune deficiency syndrome (AIDS) cohort study is provided.     

Modeling longitudinal data with nonparametric multiplicative random effects jointly with survival data

Summary .   In clinical studies, longitudinal biomarkers are often used to monitor disease progression and failure time. Joint modeling of longitudinal and survival data has certain advantages and has emerged as an effective way to mutually enhance information. Typically, a parametric longitudinal model is assumed to facilitate the likelihood approach. However, the choice of a proper parametric model turns out to be more elusive than models for standard longitudinal studies in which no survival endpoint occurs. In this article, we propose a nonparametric multiplicative random effects model for the longitudinal process, which has many applications and leads to a flexible yet parsimonious nonparametric random effects model. A proportional hazards model is then used to link the biomarkers and event time. We use B-splines to represent the nonparametric longitudinal process, and select the number of knots and degrees based on a version of the Akaike information criterion (AIC). Unknown model parameters are estimated through maximizing the observed joint likelihood, which is iteratively maximized by the Monte Carlo Expectation Maximization (MCEM) algorithm. Due to the simplicity of the model structure, the proposed approach has good numerical stability and compares well with the competing parametric longitudinal approaches. The new approach is illustrated with primary biliary cirrhosis (PBC) data, aiming to capture nonlinear patterns of serum bilirubin time courses and their relationship with survival time of PBC patients.     

A frailty model for informative censoring

To account for the correlation between failure and censoring, we propose a new frailty model for clustered data. In this model, the risk to be censored is affected by the risk of failure. This model allows flexibility in the direction and degree of dependence between failure and censoring. It includes the traditional frailty model as a special case. It allows censoring by some causes to be analyzed as informative while treating censoring by other causes as noninformative. It can also analyze data for competing risks. To fit the model, the EM algorithm is used with Markov chain Monte Carlo simulations in the E-steps. Simulation studies and analysis of data for kidney disease patients are provided. Consequences of incorrectly assuming noninformative censoring are investigated.     

An approach to model clustered survival data with dependent censoring

In this study we introduce a likelihood-based method, via the Weibull and piecewise exponential distributions, capable of accommodating the dependence between failure and censoring times. The methodology is developed for the analysis of clustered survival data and it assumes that failure and censoring times are mutually independent conditional on a latent frailty. The dependent censoring mechanism is accounted through the frailty effect and this is accomplished by means of a key parameter accommodating the correlation between failure and censored observations. The full specification of the likelihood in our work simplifies the inference procedures with respect to Huang and Wolfe since it reduces the computation burden of working with the profile likelihood. In addition, the assumptions made for the baseline distributions lead to models with continuous survival functions. In order to carry out inferences, we devise a Monte Carlo EM algorithm. The performance of the proposed models is investigated through a simulation study. Finally, we explore a real application involving patients from the Dialysis Outcomes and Practice Patterns Study observed between 1996 and 2015.     

Semiparametric transformation models for current status data with informative censoring

Current status data arise due to only one feasible examination such that the failure time of interest occurs before or after the examination time. If the examination time is intrinsically related to the failure time of interest, the examination time is referred to as an informative censoring time. Such data may occur in many fields, for example, epidemiological surveys and animal carcinogenicity experiments. To avoid severely misleading inferences resulted from ignoring informative censoring, we propose a class of semiparametric transformation models with log‐normal frailty for current status data with informative censoring. A shared frailty is used to account for the correlation between the failure time and censoring time. The expectation‐maximization (EM) algorithm combining a sieve method for approximating an infinite‐dimensional parameter is employed to estimate all parameters. To investigate finite sample properties of the proposed method, simulation studies are conducted, and a data set from a rodent tumorigenicity experiment is analyzed for illustrative purposes.     

Identifiability assumptions for missing covariate data in failure time regression models

Methods in the literature for missing covariate data in survival models have relied on the missing at random (MAR) assumption to render regression parameters identifiable. MAR means that missingness can depend on the observed exit time, and whether or not that exit is a failure or a censoring event. By considering ways in which missingness of covariate X could depend on the true but possibly censored failure time T and the true censoring time C, we attempt to identify missingness mechanisms which would yield MAR data. We find that, under various reasonable assumptions about how missingness might depend on T and/or C, additional strong assumptions are needed to obtain MAR. We conclude that MAR is difficult to justify in practical applications. One exception arises when missingness is independent of T, and C is independent of the value of the missing X. As alternatives to MAR, we propose two new missingness assumptions. In one, the missingness depends on T but not on C; in the other, the situation is reversed. For each, we show that the failure time model is identifiable. When missingness is independent of T, we show that the naive complete record analysis will yield a consistent estimator of the failure time distribution. When missingness is independent of C, we develop a complete record likelihood function and a corresponding estimator for parametric failure time models. We propose analyses to evaluate the plausibility of either assumption in a particular data set, and illustrate the ideas using data from the literature on this problem.     

Regression analysis for recurrent events data under dependent censoring

Summary Recurrent events data are commonly seen in longitudinal follow‐up studies. Dependent censoring often occurs due to death or exclusion from the study related to the disease process. In this article, we assume flexible marginal regression models on the recurrence process and the dependent censoring time without specifying their dependence structure. The proposed model generalizes the approach by Ghosh and Lin (2003, Biometrics 59, 877–885). The technique of artificial censoring provides a way to maintain the homogeneity of the hypothetical error variables under dependent censoring. Here we propose to apply this technique to two Gehan‐type statistics. One considers only order information for pairs whereas the other utilizes additional information of observed censoring times available for recurrence data. A model‐checking procedure is also proposed to assess the adequacy of the fitted model. The proposed estimators have good asymptotic properties. Their finite‐sample performances are examined via simulations. Finally, the proposed methods are applied to analyze the AIDS linked to the intravenous experiences cohort data.     

Semiparametric regression estimation in the presence of dependent censoring

We propose a semiparametric estimation procedure for estimatingthe regression of an outcome Y, measured at the end of a fixedfollow-up period, on baseline explanatory variables X, measuredprior to start of follow-up, in the presence of dependent censoringgiven X. The proposed estimators are consistent when the dataare ‘missing at random’ but not ‘missing completelyat random’ (Rubin, 1976), and do not require full specificationof the complete data likelihood. Specifically, we assume thatthe probability of censoring at time t is independent of theoutcome Y conditional on the recorded history up to t of a vectorof time-dependent covariates that are correlated with Y. Ourestimators can be used to adjust for dependent censoring andnonrandom noncompliance in randomised trials studying the effectof a treatment on the mean of a response variable of interest.Even with independent censoring, our methods allow the investigatorto increase efficiency by exploiting the correlation of theoutcome with a vector of time-dependent covariates.