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.     

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.     

Proportional Hazards Regression for the Analysis of Clustered Survival Data from Case–Cohort Studies

Summary Case–cohort sampling is a commonly used and efficient method for studying large cohorts. Most existing methods of analysis for case–cohort data have concerned the analysis of univariate failure time data. However, clustered failure time data are commonly encountered in public health studies. For example, patients treated at the same center are unlikely to be independent. In this article, we consider methods based on estimating equations for case–cohort designs for clustered failure time data. We assume a marginal hazards model, with a common baseline hazard and common regression coefficient across clusters. The proposed estimators of the regression parameter and cumulative baseline hazard are shown to be consistent and asymptotically normal, and consistent estimators of the asymptotic covariance matrices are derived. The regression parameter estimator is easily computed using any standard Cox regression software that allows for offset terms. The proposed estimators are investigated in simulation studies, and demonstrated empirically to have increased efficiency relative to some existing methods. The proposed methods are applied to a study of mortality among Canadian dialysis patients.     

Fast approximations of pseudo-observations in the context of right censoring and interval censoring

In the context of right-censored and interval-censored data, we develop asymptotic formulas to compute pseudo-observations for the survival function and the restricted mean survival time (RMST). These formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan–Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived involving the Hessian matrix and the score vector. Theoretical results that justify the use of pseudo-observations in regression are also derived. The formula is illustrated on the piecewise-constant-hazard model for the RMST. The proposed approximations are extremely accurate, even for small sample sizes, as illustrated by Monte Carlo simulations and real data. We also study the gain in terms of computation time, as compared to the original jackknife method, which can be substantial for a large dataset.     

Case-cohort designs and analysis for clustered failure time data

Case-cohort design is an efficient and economical design to study risk factors for infrequent disease in a large cohort. It involves the collection of covariate data from all failures ascertained throughout the entire cohort, and from the members of a random subcohort selected at the onset of follow-up. In the literature, the case-cohort design has been extensively studied, but was exclusively considered for univariate failure time data. In this article, we propose case-cohort designs adapted to multivariate failure time data. An estimation procedure with the independence working model approach is used to estimate the regression parameters in the marginal proportional hazards model, where the correlation structure between individuals within a cluster is left unspecified. Statistical properties of the proposed estimators are developed. The performance of the proposed estimators and comparisons of statistical efficiencies are investigated with simulation studies. A data example from the Translating Research into Action for Diabetes (TRIAD) study is used to illustrate the proposed methodology.     

Semiparametric additive time-varying coefficients model for longitudinal data with censored time origin

Statistical analysis of longitudinal data often involves modeling treatment effects on clinically relevant longitudinal biomarkers since an initial event (the time origin). In some studies including preventive HIV vaccine efficacy trials, some participants have biomarkers measured starting at the time origin, whereas others have biomarkers measured starting later with the time origin unknown. The semiparametric additive time-varying coefficient model is investigated where the effects of some covariates vary nonparametrically with time while the effects of others remain constant. Weighted profile least squares estimators coupled with kernel smoothing are developed. The method uses the expectation maximization approach to deal with the censored time origin. The Kaplan–Meier estimator and other failure time regression models such as the Cox model can be utilized to estimate the distribution and the conditional distribution of left censored event time related to the censored time origin. Asymptotic properties of the parametric and nonparametric estimators and consistent asymptotic variance estimators are derived. A two-stage estimation procedure for choosing weight is proposed to improve estimation efficiency. Numerical simulations are conducted to examine finite sample properties of the proposed estimators. The simulation results show that the theory and methods work well. The efficiency gain of the two-stage estimation procedure depends on the distribution of the longitudinal error processes. The method is applied to analyze data from the Merck 023/HVTN 502 Step HIV vaccine study.     

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.     

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.     

A Positive Stable Frailty Model for Clustered Failure Time Data with Covariate‐Dependent Frailty

Summary In this article, we propose a positive stable shared frailty Cox model for clustered failure time data where the frailty distribution varies with cluster‐level covariates. The proposed model accounts for covariate‐dependent intracluster correlation and permits both conditional and marginal inferences. We obtain marginal inference directly from a marginal model, then use a stratified Cox‐type pseudo‐partial likelihood approach to estimate the regression coefficient for the frailty parameter. The proposed estimators are consistent and asymptotically normal and a consistent estimator of the covariance matrix is provided. Simulation studies show that the proposed estimation procedure is appropriate for practical use with a realistic number of clusters. Finally, we present an application of the proposed method to kidney transplantation data from the Scientific Registry of Transplant Recipients.     

Frailty‐Based Competing Risks Model for Multivariate Survival Data

Summary In this work, we provide a new class of frailty‐based competing risks models for clustered failure times data. This class is based on expanding the competing risks model of Prentice et al. (1978, Biometrics 34 , 541–554) to incorporate frailty variates, with the use of cause‐specific proportional hazards frailty models for all the causes. Parametric and nonparametric maximum likelihood estimators are proposed. The main advantages of the proposed class of models, in contrast to the existing models, are: (1) the inclusion of covariates; (2) the flexible structure of the dependency among the various types of failure times within a cluster; and (3) the unspecified within‐subject dependency structure. The proposed estimation procedures produce the most efficient parametric and semiparametric estimators and are easy to implement. Simulation studies show that the proposed methods perform very well in practical situations.     

Nonparametric estimation of the joint distribution of a survival time subject to interval censoring and a continuous mark variable

This article considers three nonparametric estimators of the joint distribution function for a survival time and a continuous mark variable when the survival time is interval censored and the mark variable may be missing for interval-censored observations. Finite and large sample properties are described for the nonparametric maximum likelihood estimator (NPMLE) as well as estimators based on midpoint imputation (MIDMLE) and coarsening the mark variable (CMLE). The estimators are compared using data from a simulation study and a recent phase III HIV vaccine efficacy trial where the survival time is the time from enrollment to infection and the mark variable is the genetic distance from the infecting HIV sequence to the HIV sequence in the vaccine. Theoretical and empirical evidence are presented indicating the NPMLE and MIDMLE are inconsistent. Conversely, the CMLE is shown to be consistent in general and thus is preferred.     

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.     

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.     

Buckley-James-type estimator with right-censored and length-biased data

We present a natural generalization of the Buckley-James-type estimator for traditional survival data to right-censored length-biased data under the accelerated failure time (AFT) model. Length-biased data are often encountered in prevalent cohort studies and cancer screening trials. Informative right censoring induced by length-biased sampling creates additional challenges in modeling the effects of risk factors on the unbiased failure times for the target population. In this article, we evaluate covariate effects on the failure times of the target population under the AFT model given the observed length-biased data. We construct a Buckley-James-type estimating equation, develop an iterative computing algorithm, and establish the asymptotic properties of the estimators. We assess the finite-sample properties of the proposed estimators against the estimators obtained from the existing methods. Data from a prevalent cohort study of patients with dementia are used to illustrate the proposed methodology.     

Order-restricted inference for clustered ROC data with application to fingerprint matching accuracy

Receiver operating characteristic (ROC) curve is commonly used to evaluate and compare the accuracy of classification methods or markers. Estimating ROC curves has been an important problem in various fields including biometric recognition and diagnostic medicine. In real applications, classification markers are often developed under two or more ordered conditions, such that a natural stochastic ordering exists among the observations. Incorporating such a stochastic ordering into estimation can improve statistical efficiency (Davidov and Herman, 2012). In addition, clustered and correlated data arise when multiple measurements are gleaned from the same subject, making estimation of ROC curves complicated due to within-cluster correlations. In this article, we propose to model the ROC curve using a weighted empirical process to jointly account for the order constraint and within-cluster correlation structure. The algebraic properties of resulting summary statistics of the ROC curve such as its area and partial area are also studied. The algebraic expressions reduce to the ones by Davidov and Herman (2012) for independent observations. We derive asymptotic properties of the proposed order-restricted estimators and show that they have smaller mean-squared errors than the existing estimators. Simulation studies also demonstrate better performance of the newly proposed estimators over existing methods for finite samples. The proposed method is further exemplified with the fingerprint matching data from the National Institute of Standards and Technology Special Database 4.     

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.     

Prediction errors for state occupation and transition probabilities in multi‐state models

In this paper, we consider the estimation of prediction errors for state occupation probabilities and transition probabilities for multistate time‐to‐event data. We study prediction errors based on the Brier score and on the Kullback–Leibler score and prove their properness. In the presence of right‐censored data, two classes of estimators, based on inverse probability weighting and pseudo‐values, respectively, are proposed, and consistency properties of the proposed estimators are investigated. The second part of the paper is devoted to the estimation of dynamic prediction errors for state occupation probabilities for multistate models, conditional on being alive, and for transition probabilities. Cross‐validated versions are proposed. Our methods are illustrated on the CSL1 randomized clinical trial comparing prednisone versus placebo for liver cirrhosis patients.     

A marginal mixed baseline hazards model for multivariate failure time data

In multivariate failure time data analysis, a marginal regression modeling approach is often preferred to avoid assumptions on the dependence structure among correlated failure times. In this paper, a marginal mixed baseline hazards model is introduced. Estimating equations are proposed for the estimation of the marginal hazard ratio parameters. The proposed estimators are shown to be consistent and asymptotically Gaussian with a robust covariance matrix that can be consistently estimated. Simulation studies indicate the adequacy of the proposed methodology for practical sample sizes. The methodology is illustrated with a data set from the Framingham Heart Study.     

Quantile Inference with Multivariate Failure Time Data

Quantiles, especially the medians, of survival times are often used as summary statistics to compare the survival experiences between different groups. Quantiles are robust against outliers and preferred over the mean. Multivariate failure time data often arise in biomedical research. For example, in clinical trials, each patient in the study may experience multiple events which may be of the same type or distinct types, while in family studies of genetic diseases or litter matched mice studies, failure times for subjects in the same cluster may be correlated. In this article, we propose nonparametric procedures for the estimation of quantiles with multivariate failure time data. We show that the proposed estimators asymptotically follow a multivariate normal distribution. The asymptotic variance‐covariance matrix of the estimated quantiles is estimated based on the kernel smoothing and bootstrap techniques. Simulation results show that the proposed estimators perform well in finite samples. The methods are illustrated with the burn‐wound infection data and the Diabetic Retinopathy Study (DRS) data.     

Accelerated failure time modeling via nonparametric mixtures

An accelerated failure time (AFT) model assuming a log-linear relationship between failure time and a set of covariates can be either parametric or semiparametric, depending on the distributional assumption for the error term. Both classes of AFT models have been popular in the analysis of censored failure time data. The semiparametric AFT model is more flexible and robust to departures from the distributional assumption than its parametric counterpart. However, the semiparametric AFT model is subject to producing biased results for estimating any quantities involving an intercept. Estimating an intercept requires a separate procedure. Moreover, a consistent estimation of the intercept requires stringent conditions. Thus, essential quantities such as mean failure times might not be reliably estimated using semiparametric AFT models, which can be naturally done in the framework of parametric AFT models. Meanwhile, parametric AFT models can be severely impaired by misspecifications. To overcome this, we propose a new type of the AFT model using a nonparametric Gaussian-scale mixture distribution. We also provide feasible algorithms to estimate the parameters and mixing distribution. The finite sample properties of the proposed estimators are investigated via an extensive stimulation study. The proposed estimators are illustrated using a real dataset.