A new estimation method for the semiparametric accelerated failure time mixture cure model

The proportional hazard (PH) mixture cure model and the accelerated failure time (AFT) mixture cure model are usually used in analysing failure time data with long-term survivors. However, the semiparametric AFT mixture cure model has attracted less attention than the semiparametric PH mixture cure model because of the complexity of its estimation method. In this paper, we propose a new estimation method for the semiparametric AFT mixture cure model. This method employs the EM algorithm and the rank estimator of the AFT model to estimate the parameters of interest. The M-step in the EM algorithm, which incorporates the rank-like estimating equation, can be carried out easily using the linear programming method. To evaluate the performance of the proposed method, we conduct a simulation study. The results of the simulation study demonstrate that the proposed method performs better than the existing estimation method and the semiparametric AFT mixture cure model improves the identifiability of the parameters in comparison to the parametric AFT mixture cure model. To illustrate, we apply the model and the proposed method to a data set of failure times from bone marrow transplant patients.     

Mixture cure model with random effects for clustered interval-censored survival data

The mixture cure model is an effective tool for analysis of survival data with a cure fraction. This approach integrates the logistic regression model for the proportion of cured subjects and the survival model (either the Cox proportional hazards or accelerated failure time model) for uncured subjects. Methods based on the mixture cure model have been extensively investigated in the literature for data with exact failure/censoring times. In this paper, we propose a mixture cure modeling procedure for analyzing clustered and interval-censored survival time data by incorporating random effects in both the logistic regression and PH regression components. Under the generalized linear mixed model framework, we develop the REML estimation for the parameters, as well as an iterative algorithm for estimation of the survival function for interval-censored data. The estimation procedure is implemented via an EM algorithm. A simulation study is conducted to evaluate the performance of the proposed method in various practical situations. To demonstrate its usefulness, we apply the proposed method to analyze the interval-censored relapse time data from a smoking cessation study whose subjects were recruited from 51 zip code regions in the southeastern corner of Minnesota.     

Semiparametric accelerated failure time cure rate mixture models with competing risks

Modern medical treatments have substantially improved survival rates for many chronic diseases and have generated considerable interest in developing cure fraction models for survival data with a non‐ignorable cured proportion. Statistical analysis of such data may be further complicated by competing risks that involve multiple types of endpoints. Regression analysis of competing risks is typically undertaken via a proportional hazards model adapted on cause‐specific hazard or subdistribution hazard. In this article, we propose an alternative approach that treats competing events as distinct outcomes in a mixture. We consider semiparametric accelerated failure time models for the cause‐conditional survival function that are combined through a multinomial logistic model within the cure‐mixture modeling framework. The cure‐mixture approach to competing risks provides a means to determine the overall effect of a treatment and insights into how this treatment modifies the components of the mixture in the presence of a cure fraction. The regression and nonparametric parameters are estimated by a nonparametric kernel‐based maximum likelihood estimation method. Variance estimation is achieved through resampling methods for the kernel‐smoothed likelihood function. Simulation studies show that the procedures work well in practical settings. Application to a sarcoma study demonstrates the use of the proposed method for competing risk data with a cure fraction.     

Cure rate model with interval censored data

In cancer trials, a significant fraction of patients can be cured, that is, the disease is completely eliminated, so that it never recurs. In general, treatments are developed to both increase the patients' chances of being cured and prolong the survival time among non-cured patients. A cure rate model represents a combination of cure fraction and survival model, and can be applied to many clinical studies over several types of cancer. In this article, the cure rate model is considered in the interval censored data composed of two time points, which include the event time of interest. Interval censored data commonly occur in the studies of diseases that often progress without symptoms, requiring clinical evaluation for detection (Encyclopedia of Biostatistics. Wiley: New York, 1998; 2090-2095). In our study, an approximate likelihood approach suggested by Goetghebeur and Ryan (Biometrics 2000; 56:1139-1144) is used to derive the likelihood in interval censored data. In addition, a frailty model is introduced to characterize the association between the cure fraction and survival model. In particular, the positive association between the cure fraction and the survival time is incorporated by imposing a common normal frailty effect. The EM algorithm is used to estimate parameters and a multiple imputation based on the profile likelihood is adopted for variance estimation. The approach is applied to the smoking cessation study in which the event of interest is a smoking relapse and several covariates including an intensive care treatment are evaluated to be effective for both the occurrence of relapse and the non-smoking duration.     

Estimation of intervention effects using recurrent event time data in the presence of event dependence and a cured fraction

Recurrent event data with a fraction of subjects having zero event are often seen in randomized clinical trials. Those with zero event may belong to a cured (or non‐susceptible) fraction. Event dependence refers to the situation that a person's past event history affects his future event occurrences. In the presence of event dependence, an intervention may have an impact on the event rate in the non‐cured through two pathways—a primary effect directly on the outcome event and a secondary effect mediated through event dependence. The primary effect combined with the secondary effect is the total effect. We propose a frailty mixture model and a two‐step estimation procedure for the estimation of the effect of an intervention on the probability of cure and the total effect on event rate in the non‐cured. A summary measure of intervention effects is derived. The performance of the proposed model is evaluated by simulation. Data on respiratory exacerbations from a randomized, placebo‐controlled trial are re‐analyzed for illustration. Copyright © 2014 John Wiley & Sons, Ltd.     

Semiparametric mixture cure model analysis with competing risks data: Application to vascular access thrombosis data

The article is motivated by a nephrology study in Taiwan, which enrolled hemodialysis patients who suffered from vascular access thrombosis. After treatment, some patients were cured of thrombosis, while some may experience recurrence of either type (acute or nonacute) of vascular access thrombosis. Our major interest is to estimate the cumulative incidence probability of time to the first recurrence of acute thrombosis after therapy. Since the occurrence of one type of vascular access thrombosis precludes occurrence of the other type, patients are subject to competing risks. To account for the presence of competing risks and cured patients, we develop a mixture model approach to the regression analysis of competing-risks data with a cure fraction. We make inference about the effects of factors on both the cure rate and cumulative incidence function (CIF) for a failure of interest, which are separately specified in the logistic regression model and semiparametric regression model with time-varying and time-invariant effects. Based on two-stage method, we develop novel estimation equations using the inverse probability censoring weight techniques. The asymptotic properties of the estimators are rigorously studied and the plug-in variance estimators can be obtained for constructing interval estimators. We also propose a lack-of-fit test for assessing the adequacy of the proposed model and several tests for time-varying effects. The simulation studies and vascular access thrombosis data analysis are conducted to illustrate the proposed method.     

The use of Gaussian quadrature for estimation in frailty proportional hazards models

In this paper, we propose a novel Gaussian quadrature estimation method in various frailty proportional hazards models. We approximate the unspecified baseline hazard by a piecewise constant one, resulting in a parametric model that can be fitted conveniently by Gaussian quadrature tools in standard software such as SAS Proc NLMIXED. We first apply our method to simple frailty models for correlated survival data (e.g. recurrent or clustered failure times), then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death. Simulation studies show that our method compares favorably with the well-received penalized partial likelihood method and the Monte Carlo EM (MCEM) method, for both normal and Gamma frailty models. We apply our method to three real data examples: (1) the time to blindness of both eyes in a diabetic retinopathy study, (2) the joint analysis of recurrent opportunistic diseases in the presence of death for HIV-infected patients, and (3) the joint modeling of local, distant tumor recurrences and patients survival in a soft tissue sarcoma study. The proposed method greatly simplifies the implementation of the (joint) frailty models and makes them much more accessible to general statistical practitioners.     

Semiparametric estimation of the cure fraction in population-based cancer survival analysis

With rapid development in medical research, the treatment of diseases including cancer has progressed dramatically and those survivors may die from causes other than the one under study, especially among elderly patients. Motivated by the Surveillance, Epidemiology, and End Results (SEER) female breast cancer study, background mortality is incorporated into the mixture cure proportional hazards (MCPH) model to improve the cure fraction estimation in population-based cancer studies. Here, that patients are “cured” is defined as when the mortality rate of the individuals in diseased group returns to the same level as that expected in the general population, where the population level mortality is presented by the mortality table of the United States. The semiparametric estimation method based on the EM algorithm for the MCPH model with background mortality (MCPH+BM) is further developed and validated via comprehensive simulation studies. Real data analysis shows that the proposed semiparametric MCPH+BM model may provide more accurate estimation in population-level cancer study.     

A semi-parametric accelerated failure time cure model

A cure model is a useful approach for analysing failure time data in which some subjects could eventually experience, and others never experience, the event of interest. A cure model has two components: incidence which indicates whether the event could eventually occur and latency which denotes when the event will occur given the subject is susceptible to the event. In this paper, we propose a semi-parametric cure model in which covariates can affect both the incidence and the latency. A logistic regression model is proposed for the incidence, and the latency is determined by an accelerated failure time regression model with unspecified error distribution. An EM algorithm is developed to fit the model. The procedure is applied to a data set of tonsil cancer patients treated with radiation therapy.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.     

Cure fraction model with random effects for regional variation in cancer survival

Assessing regional differences in the survival of cancer patients is important but difficult when separate regions are small or sparsely populated. In this paper, we apply a mixture cure fraction model with random effects to cause‐specific survival data of female breast cancer patients collected by the population‐based Finnish Cancer Registry. Two sets of random effects were used to capture the regional variation in the cure fraction and in the survival of the non‐cured patients, respectively. This hierarchical model was implemented in a Bayesian framework using a Metropolis‐within‐Gibbs algorithm. To avoid poor mixing of the Markov chain, when the variance of either set of random effects was close to zero, posterior simulations were based on a parameter‐expanded model with tailor‐made proposal distributions in Metropolis steps. The random effects allowed the fitting of the cure fraction model to the sparse regional data and the estimation of the regional variation in 10‐year cause‐specific breast cancer survival with a parsimonious number of parameters. Before 1986, the capital of Finland clearly stood out from the rest, but since then all the 21 hospital districts have achieved approximately the same level of survival. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling smoking cessation data with alternating states and a cure fraction using frailty models

We propose a flexible parametric model to describe alternating states recurrent‐event data where there is a possibility of cure with each type of event. We begin by introducing a novel cure model in which a common frailty influences both the cure probability and the hazard function given not cured. We then extend our model to data with recurring events of two alternating types. We assume that each type of event has a gamma frailty, and we link the frailties by a Clayton copula. We illustrate the model with an analysis of data from two smoking cessation trials comparing bupropion and placebo, in which each subject potentially experienced a series of lapse and recovery events. Our analysis suggests that bupropion increases the probability of permanent cure and decreases the hazard of lapse, but does not affect the distribution of time to recovery during a lapse. The data suggest a positive but non‐significant association between the lapse and recovery frailties. A simulation study suggests that the estimates have little bias and that their 95 per cent confidence intervals have nearly nominal coverage in samples of practical size. Copyright © 2010 John Wiley & Sons, Ltd.     

A bivariate survival model with compound Poisson frailty

A correlated frailty model is suggested for analysis of bivariate time‐to‐event data. The model is an extension of the correlated power variance function (PVF) frailty model (correlated three‐parameter frailty model) (J. Epidemiol. Biostat. 1999; 4:53–60). It is based on a bivariate extension of the compound Poisson frailty model in univariate survival analysis (Ann. Appl. Probab. 1992; 4:951–972). It allows for a non‐susceptible fraction (of zero frailty) in the population, overcoming the common assumption in survival analysis that all individuals are susceptible to the event under study. The model contains the correlated gamma frailty model and the correlated inverse Gaussian frailty model as special cases. A maximum likelihood estimation procedure for the parameters is presented and its properties are studied in a small simulation study. This model is applied to breast cancer incidence data of Swedish twins. The proportion of women susceptible to breast cancer is estimated to be 15 per cent. Copyright © 2009 John Wiley & Sons, Ltd.     

A penalized likelihood approach for mixture cure models

Cure models have been developed to analyze failure time data with a cured fraction. For such data, standard survival models are usually not appropriate because they do not account for the possibility of cure. Mixture cure models assume that the studied population is a mixture of susceptible individuals, who may experience the event of interest, and non‐susceptible individuals that will never experience it. Important issues in mixture cure models are estimation of the baseline survival function for susceptibles and estimation of the variance of the regression parameters. The aim of this paper is to propose a penalized likelihood approach, which allows for flexible modeling of the hazard function for susceptible individuals using M‐splines. This approach also permits direct computation of the variance of parameters using the inverse of the Hessian matrix. Properties and limitations of the proposed method are discussed and an illustration from a cancer study is presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A smoothing expectation and substitution algorithm for the semiparametric accelerated failure time frailty model

This paper proposes an estimation procedure for the semiparametric accelerated failure time frailty model that combines smoothing with an Expectation and Maximization‐like algorithm for estimating equations. The resulting algorithm permits simultaneous estimation of the regression parameter, the baseline cumulative hazard, and the parameter indexing a general frailty distribution. We develop novel moment‐based estimators for the frailty parameter, including a generalized method of moments estimator. Standard error estimates for all parameters are easily obtained using a randomly weighted bootstrap procedure. For the commonly used gamma frailty distribution, the proposed algorithm is very easy to implement using widely available numerical methods. Simulation results demonstrate that the algorithm performs very well in this setting. We re‐analyz several previously analyzed data sets for illustrative purposes. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of clustered failure time data with cure fraction using copula

Clustered survival data in the presence of cure has received increasing attention. In this paper, we consider a semiparametric mixture cure model which incorporates a logistic regression model for the cure fraction and a semiparametric regression model for the failure time. We utilize Archimedean copula (AC) models to assess the strength of association for both susceptibility and failure times between susceptible individuals in the same cluster. Instead of using the full likelihood approach, we consider a composite likelihood function and a two-stage estimation procedure for both marginal and association parameters. A Jackknife procedure that takes out one cluster at a time is proposed for the variance estimation of the estimators. Akaike information criterion is applied to select the best model among ACs. Simulation studies are performed to validate our estimating procedures, and two real data sets are analyzed to demonstrate the practical use of our proposed method.     

Estimation of summary protective efficacy using a frailty mixture model for recurrent event time data

Recurrent event time data are common in experimental and observational studies. The analytic strategy needs to consider three issues: within‐subject event dependence, between‐subject heterogeneity in event rates, and the possibility of a nonsusceptible fraction. Motivated by the need to estimate the summary protective efficacy from recurrent event time data as seen in many infectious disease clinical trials, we propose a two‐part frailty mixture model that simultaneously accommodates all the three issues. In terms of vaccine action models, the proposed model is a combination of the ‘all‐or‐none’ and the ‘leaky’ models, and the summary protective efficacy is a unified measure of the vaccine's twofold effects in completely or partially protecting the vaccinated individuals against the study event. The model parameters of interest are estimated using the expectation‐maximization algorithm with their respective variances estimated using Louis's formula for the expectation‐maximization algorithm. The summary protective efficacy is estimated by a composite estimand with its variance estimated using the delta method. The performance of the proposed estimation approach is investigated by a simulation study. Data from a trial of malaria prophylaxis conducted in Ghana are reanalyzed. Copyright © 2012 John Wiley & Sons, Ltd.     

Regression analysis of failure time data with informative interval censoring

Interval censoring arises when a subject misses prescheduled visits at which the failure is to be assessed. Most existing approaches for analysing interval-censored failure time data assume that the censoring mechanism is independent of the true failure time. However, there are situations where this assumption may not hold. In this paper, we consider such a situation in which the dependence structure between the censoring variables and the failure time can be modelled through some latent variables and a method for regression analysis of failure time data is proposed. The method makes use of the proportional hazards frailty model and an EM algorithm is presented for estimation. Finite sample properties of the proposed estimators of regression parameters are examined through simulation studies and we illustrate the method with data from an AIDS study.     

Frailty models: Applications to biomedical and genetic studies

In survival analysis, frailty models are potential choices for modeling unexplained heterogeneity in a population. This tutorial presents an overview and general framework of frailty modeling and estimation for multiplicative hazards models in the context of biomedical and genetic studies. Other topics in frailty models, such as diagnostic methods for model adequacy and inference in frailty models, are also discussed. Examples of analyses using multivariate frailty models in a non-parametric hazards setting on biomedical datasets are provided, and the implications of choosing to use frailty and relevance to genetic applications are discussed.     

Variable selection in subdistribution hazard frailty models with competing risks data

The proportional subdistribution hazards model (i.e. Fine‐Gray model) has been widely used for analyzing univariate competing risks data. Recently, this model has been extended to clustered competing risks data via frailty. To the best of our knowledge, however, there has been no literature on variable selection method for such competing risks frailty models. In this paper, we propose a simple but unified procedure via a penalized h‐likelihood (HL) for variable selection of fixed effects in a general class of subdistribution hazard frailty models, in which random effects may be shared or correlated. We consider three penalty functions, least absolute shrinkage and selection operator (LASSO), smoothly clipped absolute deviation (SCAD) and HL, in our variable selection procedure. We show that the proposed method can be easily implemented using a slight modification to existing h‐likelihood estimation approaches. Numerical studies demonstrate that the proposed procedure using the HL penalty performs well, providing a higher probability of choosing the true model than LASSO and SCAD methods without losing prediction accuracy. The usefulness of the new method is illustrated using two actual datasets from multi‐center clinical trials. Copyright © 2014 John Wiley & Sons, Ltd.