Normal frailty probit model for clustered interval‐censored failure time data

Clustered interval‐censored data commonly arise in many studies of biomedical research where the failure time of interest is subject to interval‐censoring and subjects are correlated for being in the same cluster. A new semiparametric frailty probit regression model is proposed to study covariate effects on the failure time by accounting for the intracluster dependence. Under the proposed normal frailty probit model, the marginal distribution of the failure time is a semiparametric probit model, the regression parameters can be interpreted as both the conditional covariate effects given frailty and the marginal covariate effects up to a multiplicative constant, and the intracluster association can be summarized by two nonparametric measures in simple and explicit form. A fully Bayesian estimation approach is developed based on the use of monotone splines for the unknown nondecreasing function and a data augmentation using normal latent variables. The proposed Gibbs sampler is straightforward to implement since all unknowns have standard form in their full conditional distributions. The proposed method performs very well in estimating the regression parameters as well as the intracluster association, and the method is robust to frailty distribution misspecifications as shown in our simulation studies. Two real‐life data sets are analyzed for illustration.     

A bivariate joint frailty model with mixture framework for survival analysis of recurrent events with dependent censoring and cure fraction

In the study of multiple failure time data with recurrent clinical endpoints, the classical independent censoring assumption in survival analysis can be violated when the evolution of the recurrent events is correlated with a censoring mechanism such as death. Moreover, in some situations, a cure fraction appears in the data because a tangible proportion of the study population benefits from treatment and becomes recurrence free and insusceptible to death related to the disease. A bivariate joint frailty mixture cure model is proposed to allow for dependent censoring and cure fraction in recurrent event data. The latency part of the model consists of two intensity functions for the hazard rates of recurrent events and death, wherein a bivariate frailty is introduced by means of the generalized linear mixed model methodology to adjust for dependent censoring. The model allows covariates and frailties in both the incidence and the latency parts, and it further accounts for the possibility of cure after each recurrence. It includes the joint frailty model and other related models as special cases. An expectation-maximization (EM)-type algorithm is developed to provide residual maximum likelihood estimation of model parameters. Through simulation studies, the performance of the model is investigated under different magnitudes of dependent censoring and cure rate. The model is applied to data sets from two colorectal cancer studies to illustrate its practical value.     

Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

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.     

Analysis of clustered recurrent event data with application to hospitalization rates among renal failure patients

End-stage renal disease (commonly referred to as renal failure) is of increasing concern in the United States and many countries worldwide. Incidence rates have increased, while the supply of donor organs has not kept pace with the demand. Although renal transplantation has generally been shown to be superior to dialysis with respect to mortality, very little research has been directed towards comparing transplant and wait-list patients with respect to morbidity. Using national data from the Scientific Registry of Transplant Recipients, we compare transplant and wait-list hospitalization rates. Hospitalizations are subject to two levels of dependence. In addition to the dependence among within-patient events, patients are also clustered by listing center. We propose two marginal methods to analyze such clustered recurrent event data; the first model postulates a common baseline event rate, while the second features cluster-specific baseline rates. Our results indicate that kidney transplantation offers a significant decrease in hospitalization, but that the effect is negated by a waiting time (until transplant) of more than 2 years. Moreover, graft failure (GF) results in a significant increase in the hospitalization rate which is greatest in the first month post-GF, but remains significantly elevated up to 4 years later. We also compare results from the proposed models to those based on a frailty model, with the various methods compared and contrasted.     

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.     

Bayesian cure rate frailty models with application to a root canal therapy study

Due to natural or artificial clustering, multivariate survival data often arise in biomedical studies, for example, a dental study involving multiple teeth from each subject. A certain proportion of subjects in the population who are not expected to experience the event of interest are considered to be "cured" or insusceptible. To model correlated or clustered failure time data incorporating a surviving fraction, we propose two forms of cure rate frailty models. One model naturally introduces frailty based on biological considerations while the other is motivated from the Cox proportional hazards frailty model. We formulate the likelihood functions based on piecewise constant hazards and derive the full conditional distributions for Gibbs sampling in the Bayesian paradigm. As opposed to the Cox frailty model, the proposed methods demonstrate great potential in modeling multivariate survival data with a cure fraction. We illustrate the cure rate frailty models with a root canal therapy data set.     

Association Tests for a Censored Quantitative Trait and Candidate Genes in Structured Populations with Multilevel Genetic Relatedness

Summary Several statistical methods for detecting associations between quantitative traits and candidate genes in structured populations have been developed for fully observed phenotypes. However, many experiments are concerned with failure‐time phenotypes, which are usually subject to censoring. In this article, we propose statistical methods for detecting associations between a censored quantitative trait and candidate genes in structured populations with complex multiple levels of genetic relatedness among sampled individuals. The proposed methods correct for continuous population stratification using both population structure variables as covariates and the frailty terms attributable to kinship. The relationship between the time‐at‐onset data and genotypic scores at a candidate marker is modeled via a parametric Weibull frailty accelerated failure time (AFT) model as well as a semiparametric frailty AFT model, where the baseline survival function is flexibly modeled as a mixture of Polya trees centered around a family of Weibull distributions. For both parametric and semiparametric models, the frailties are modeled via an intrinsic Gaussian conditional autoregressive prior distribution with the kinship matrix being the adjacency matrix connecting subjects. Simulation studies and applications to the Arabidopsis thaliana line flowering time data sets demonstrated the advantage of the new proposals over existing approaches.     

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.     

Sieve Estimation for the Cox Model with Clustered Interval-Censored Failure Time Data

Clustered interval-censored failure time data occur when the failure times of interest are clustered into small groups and known only to lie in certain intervals. A number of methods have been proposed for regression analysis of clustered failure time data, but most of them apply only to clustered right-censored data. In this paper, a sieve estimation procedure is proposed for fitting a Cox frailty model to clustered interval-censored failure time data. In particular, a two-step algorithm for parameter estimation is developed and the asymptotic properties of the resulting sieve maximum likelihood estimators are established. The finite sample properties of the proposed estimators are investigated through a simulation study and the method is illustrated by the data arising from a lymphatic filariasis study.     

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.     

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.     

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.     

Joint analysis of panel count and interval-censored data using distribution-free frailty analysis

We propose a joint analysis of recurrent and nonrecurrent event data subject to general types of interval censoring. The proposed analysis allows for general semiparametric models, including the Box–Cox transformation and inverse Box–Cox transformation models for the recurrent and nonrecurrent events, respectively. A frailty variable is used to account for the potential dependence between the recurrent and nonrecurrent event processes, while leaving the distribution of the frailty unspecified. We apply the pseudolikelihood for interval-censored recurrent event data, usually termed as panel count data, and the sufficient likelihood for interval-censored nonrecurrent event data by conditioning on the sufficient statistic for the frailty and using the working assumption of independence over examination times. Large sample theory and a computation procedure for the proposed analysis are established. We illustrate the proposed methodology by a joint analysis of the numbers of occurrences of basal cell carcinoma over time and time to the first recurrence of squamous cell carcinoma based on a skin cancer dataset, as well as a joint analysis of the numbers of adverse events and time to premature withdrawal from study medication based on a scleroderma lung disease dataset.     

Estimation of competing risks with general missing pattern in failure types

In competing risks data, missing failure types (causes) is a very common phenomenon. In this work, we consider a general missing pattern in which, if a failure type is not observed, one observes a set of possible types containing the true type, along with the failure time. We first consider maximum likelihood estimation with missing-at-random assumption via the expectation maximization (EM) algorithm. We then propose a Nelson-Aalen type estimator for situations when certain information on the conditional probability of the true type given a set of possible failure types is available from the experimentalists. This is based on a least-squares type method using the relationships between hazards for different types and hazards for different combinations of missing types. We conduct a simulation study to investigate the performance of this method, which indicates that bias may be small, even for high proportion of missing data, for sufficiently large number of observations. The estimates are somewhat sensitive to misspecification of the conditional probabilities of the true types when the missing proportion is high. We also consider an example from an animal experiment to illustrate our methodology.     

Estimations of the joint distribution of failure time and failure type with dependent truncation

In biomedical studies involving survival data, the observation of failure times is sometimes accompanied by a variable which describes the type of failure event (Kalbeisch and Prentice, 2002). This paper considers two specific challenges which are encountered in the joint analysis of failure time and failure type. First, because the observation of failure times is subject to left truncation, the sampling bias extends to the failure type which is associated with the failure time. An analytical challenge is to deal with such sampling bias. Second, in case that the joint distribution of failure time and failure type is allowed to have a temporal trend, it is of interest to estimate the joint distribution of failure time and failure type nonparametrically. This paper develops statistical approaches to address these two analytical challenges on the basis of prevalent survival data. The proposed approaches are examined through simulation studies and illustrated by using a real data set.     

Analysis of multivariate recurrent event data with time‐dependent covariates and informative censoring

Multivariate recurrent event data are usually encountered in many clinical and longitudinal studies in which each study subject may experience multiple recurrent events. For the analysis of such data, most existing approaches have been proposed under the assumption that the censoring times are noninformative, which may not be true especially when the observation of recurrent events is terminated by a failure event. In this article, we consider regression analysis of multivariate recurrent event data with both time‐dependent and time‐independent covariates where the censoring times and the recurrent event process are allowed to be correlated via a frailty. The proposed joint model is flexible where both the distributions of censoring and frailty variables are left unspecified. We propose a pairwise pseudolikelihood approach and an estimating equation‐based approach for estimating coefficients of time‐dependent and time‐independent covariates, respectively. The large sample properties of the proposed estimates are established, while the finite‐sample properties are demonstrated by simulation studies. The proposed methods are applied to the analysis of a set of bivariate recurrent event data from a study of platelet transfusion reactions.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

A joint frailty model for survival and gap times between recurrent events

Therapy for patients with a recurrent disease focuses on delaying disease recurrence and prolonging survival. A common analysis approach for such data is to estimate the distribution of disease-free survival, that is, the time to the first disease recurrence or death, whichever happens first. However, treating death similarly as disease recurrence may give misleading results. Also considering only the first recurrence and ignoring subsequent ones can result in loss of statistical power. We use a joint frailty model to simultaneously analyze disease recurrences and survival. Separate parameters for disease recurrence and survival are used in the joint model to distinguish treatment effects on these two types of events. The correlation between disease recurrences and survival is taken into account by a shared frailty. The effect of disease recurrence on survival can also be estimated by this model. The EM algorithm is used to fit the model, with Markov chain Monte Carlo simulations in the E-steps. The method is evaluated by simulation studies and illustrated through a study of patients with heart failure. Sensitivity analysis for the parametric assumption of the frailty distribution is assessed by simulations.     

A General Class of Semiparametric Transformation Frailty Models for Nonproportional Hazards Survival Data

Summary We propose a semiparametrically efficient estimation of a broad class of transformation regression models for nonproportional hazards data. Classical transformation models are to be viewed from a frailty model paradigm, and the proposed method provides a unified approach that is valid for both continuous and discrete frailty models. The proposed models are shown to be flexible enough to model long‐term follow‐up survival data when the treatment effect diminishes over time, a case for which the PH or proportional odds assumption is violated, or a situation in which a substantial proportion of patients remains cured after treatment. Estimation of the link parameter in frailty distribution, considered to be unknown and possibly dependent on a time‐independent covariates, is automatically included in the proposed methods. The observed information matrix is computed to evaluate the variances of all the parameter estimates. Our likelihood‐based approach provides a natural way to construct simple statistics for testing the PH and proportional odds assumptions for usual survival data or testing the short‐ and long‐term effects for survival data with a cure fraction. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two medical studies are provided.