Fitting semi-Markov models to interval-censored data with unknown initiation times

In a semi-Markov model, the hazard of making a transition between stages depends on the time spent in the current stage but is independent of time spent in other stages. If the initiation time (time of entry into the network) is not known for some persons and if transition time data are interval censored (i.e., if transition times are not known exactly but are known only to have occurred in some interval), then the length of time these persons spent in any stage is not known. We show how a semi-Markov model can still be fit to interval-censored data with missing initiation times. For the special case of models in which all persons enter the network at the same initial stage and proceed through the same succession of stages to a unique absorbing stage, we present discrete-time nonparametric maximum likelihood estimators of the waiting-time distributions for this type of data.     

Estimating the extent of tracking in interval-censored chain-of-events data

This paper describes a method for determining whether the times between a chain of successive events (which all individuals experience in the same order) are correlated, for data in which the exact event times are not observed. Such data arise when individuals are only observed occasionally to determine which events have occurred. In such data, the (unknown) event times are interval censored. In addition, some individuals may have experienced some of the events before their first observation and may be lost to follow-up before experiencing the last event. Using a frailty model proposed by Aalen (1988, Mathematical Scientist 13, 90-103) but which has never been used to analyze real data, we examine whether individuals who develop early markers of HIV infection can also be expected to develop antibody and other indicators of HIV infection more rapidly.     

Regression analysis of multivariate interval-censored failure time data with application to tumorigenicity experiments

This paper discusses multivariate interval-censored failure time data that occur when there exist several correlated survival times of interest and only interval-censored data are available for each survival time. Such data occur in many fields. One is tumorigenicity experiments, which usually concern different types of tumors, tumors occurring in different locations of animals, or together. For regression analysis of such data, we develop a marginal inference approach using the additive hazards model and apply it to a set of bivariate interval-censored data arising from a tumorigenicity experiment. Simulation studies are conducted for the evaluation of the presented approach and suggest that the approach performs well for practical situations.     

A goodness-of-fit test for the marginal Cox model for correlated interval-censored failure time data

The marginal Cox model approach is perhaps the most commonly used method in the analysis of correlated failure time data (Cai, 1999; Cai and Prentice, 1995; Lin, 1994; Wei, Lin and Weissfeld, 1989). It assumes that the marginal distributions for the correlated failure times can be described by the Cox model and leaves the dependence structure completely unspecified. This paper discusses the assessment of the marginal Cox model for correlated interval-censored data and a goodness-of-fit test is presented for the problem. The method is applied to a set of correlated interval-censored data arising from an AIDS clinical trial.     

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.     

Generalized log-rank tests for partly interval-censored failure time data

In this paper, we consider incomplete survival data: partly interval-censored failure time data where observed data include both exact and interval-censored observations on the survival time of interest. We present a class of generalized log-rank tests for this type of survival data and establish their asymptotic properties. The method is evaluated using simulation studies and illustrated by a set of real data from a diabetes study.     

A proportional hazards model for interval-censored failure time data

This paper develops a method for fitting the proportional hazards regression model when the data contain left-, right-, or interval-censored observations. Results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log-rank test for comparison of several survival curves. The method is used to analyze data from an animal tumorigenicity study and also a clinical trial.     

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.     

Gamma frailty model for linkage analysis with application to interval-censored migraine data

For many diseases, it seems that the age at onset is geneticallyinfluenced. Therefore, the age-at-onset data are often collectedin order to map the disease gene(s). The ages are often (right)censored or truncated, and therefore, many standard techniquesfor linkage analysis cannot be used. In this paper, we presenta correlated frailty model for censored survival data of siblings.The model is used for testing heritability for the age at onsetand linkage between the loci and the gene(s) that influence(s)the survival time. The model is applied to interval-censoredmigraine twin data. Heritability (obtained from the frailtiesrather than actual onset times) was estimated as 0.42; thisvalue was highly significant. The highest lod score, a scoreof 1.9, was found at the end of chromosome 19.     

A semiparametric model for regression analysis of interval-censored failure time data

Left-, right-, and interval-censored response time data arise in a variety of settings, including the analyses of data from laboratory animal carcinogenicity experiments, clinical trials, and longitudinal studies. For such incomplete data, the usual regression techniques such as the Cox (1972, Journal of the Royal Statistical Society, Series B 34, 187-220) proportional hazards model are inapplicable. In this paper, we present a method for regression analysis which accommodates interval-censored data. We present applications of this methodology to data sets from a study of breast cancer patients who were followed for cosmetic response to therapy, a small animal tumorigenicity study, and a clinical trial.     

Nonparametric estimation of a Markov 'illness-death' process from interval-censored observations, with application to diabetes survival data

The nonparametric estimation of the cumulative transition intensityfunctions in a threestate time-nonhomogeneous Markov processwith irreversible transitions, an ‘illness-death’model, is considered when times of the intermediate transition,e.g. onset of a disease, are interval-censored. The times of‘death’ are assumed to be known exactly or to beright-censored. In addition the observed process may be left-truncated.Data of this type arise when the process is sampled periodically.For example, when the patients are monitored through periodicexaminations the observations on times of change in their diseasestatus will be interval-censored. Under the sampling schemeconsidered here the Nelson–Aalen estimator (Aalen, 1978)for a cumulative transition intensity is not applicable. Inthe proposed method the maximum likelihood estimators of someof the transition intensities are derived from the estimatorsof the corresponding subdistribution functions. The maximumlikelihood estimators are shown to have a self-consistency property.The self-consistency algorithm is developed for the computationof the estimators. This approach generalises the results fromTurnbull (1976) and Frydman (1992). The methods are illustratedwith diabetes survival data.     

Variable selection for nonparametric additive Cox model with interval-censored data

The standard Cox model is perhaps the most commonly used model for regression analysis of failure time data but it has some limitations such as the assumption on linear covariate effects. To relax this, the nonparametric additive Cox model, which allows for nonlinear covariate effects, is often employed, and this paper will discuss variable selection and structure estimation for this general model. For the problem, we propose a penalized sieve maximum likelihood approach with the use of Bernstein polynomials approximation and group penalization. To implement the proposed method, an efficient group coordinate descent algorithm is developed and can be easily carried out for both low- and high-dimensional scenarios. Furthermore, a simulation study is performed to assess the performance of the presented approach and suggests that it works well in practice. The proposed method is applied to an Alzheimer's disease study for identifying important and relevant genetic factors.     

A two-sample comparison for multiple ordered event data

A longitudinal study is conducted to compare the process of particular disease between two groups. The process of the disease is monitored according to which of several ordered events occur. In the paper, the sojourn time between two successive events is considered as the outcome of interest. The group effects on the sojourn times of the multiple events are parameterized by scale changes in a semiparametric accelerated failure time model where the dependence structure among the multivariate sojourn times is unspecified. Suppose that the sojourn times are subject to dependent censoring and the censoring times are observed for all subjects. A log-rank-type estimating approach by rescaling the sojourn times and the dependent censoring times into the same distribution is constructed to estimate the group effects and the corresponding estimators are consistent and asymptotically normal. Without the dependent censoring, the independent censoring times in general are not available for the uncensored data. In order to complete the censoring information, pseudo-censoring times are generated from the corresponding nonparametrically estimated survival function in each group, and we can still obtained unbiased estimating functions for the group effects. A real application and a simulation study are conducted to illustrate the proposed methods.     

Semiparametric regression analysis of interval-censored data

We propose a semiparametric approach to the proportional hazards regression analysis of interval-censored data. An EM algorithm based on an approximate likelihood leads to an M-step that involves maximizing a standard Cox partial likelihood to estimate regression coefficients and then using the Breslow estimator for the unknown baseline hazards. The E-step takes a particularly simple form because all incomplete data appear as linear terms in the complete-data log likelihood. The algorithm of Turnbull (1976, Journal of the Royal Statistical Society, Series B 38, 290-295) is used to determine times at which the hazard can take positive mass. We found multiple imputation to yield an easily computed variance estimate that appears to be more reliable than asymptotic methods with small to moderately sized data sets. In the right-censored survival setting, the approach reduces to the standard Cox proportional hazards analysis, while the algorithm reduces to the one suggested by Clayton and Cuzick (1985, Applied Statistics 34, 148-156). The method is illustrated on data from the breast cancer cosmetics trial, previously analyzed by Finkelstein (1986, Biometrics 42, 845-854) and several subsequent authors.     

A semiparametric joint model for cluster size and subunit-specific interval-censored outcomes

Clustered data frequently arise in biomedical studies, where observations, or subunits, measured within a cluster are associated. The cluster size is said to be informative, if the outcome variable is associated with the number of subunits in a cluster. In most existing work, the informative cluster size issue is handled by marginal approaches based on within-cluster resampling, or cluster-weighted generalized estimating equations. Although these approaches yield consistent estimation of the marginal models, they do not allow estimation of within-cluster associations and are generally inefficient. In this paper, we propose a semiparametric joint model for clustered interval-censored event time data with informative cluster size. We use a random effect to account for the association among event times of the same cluster as well as the association between event times and the cluster size. For estimation, we propose a sieve maximum likelihood approach and devise a computationally-efficient expectation-maximization algorithm for implementation. The estimators are shown to be strongly consistent, with the Euclidean components being asymptotically normal and achieving semiparametric efficiency. Extensive simulation studies are conducted to evaluate the finite-sample performance, efficiency and robustness of the proposed method. We also illustrate our method via application to a motivating periodontal disease dataset.     

Marginal proportional hazards models for clustered interval-censored data with time-dependent covariates

The Botswana Combination Prevention Project was a cluster-randomized HIV prevention trial whose follow-up period coincided with Botswana's national adoption of a universal test and treat strategy for HIV management. Of interest is whether, and to what extent, this change in policy modified the preventative effects of the study intervention. To address such questions, we adopt a stratified proportional hazards model for clustered interval-censored data with time-dependent covariates and develop a composite expectation maximization algorithm that facilitates estimation of model parameters without placing parametric assumptions on either the baseline hazard functions or the within-cluster dependence structure. We show that the resulting estimators for the regression parameters are consistent and asymptotically normal. We also propose and provide theoretical justification for the use of the profile composite likelihood function to construct a robust sandwich estimator for the variance. We characterize the finite-sample performance and robustness of these estimators through extensive simulation studies. Finally, we conclude by applying this stratified proportional hazards model to a re-analysis of the Botswana Combination Prevention Project, with the national adoption of a universal test and treat strategy now modeled as a time-dependent covariate.