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.     

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo‐likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite‐sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non‐proportionality. The relative merits of these methods are discussed in concluding remarks. Copyright © 2014 John Wiley & Sons, Ltd.     

A semiparametric marginal mixture cure model for clustered survival data

We consider a marginal model for the regression analysis of clustered failure time data with a cure fraction. We propose to use novel generalized estimating equations in an expectation–maximization algorithm to estimate regression parameters in a semiparametric proportional hazards mixture cure model. The dependence among the cure statuses and among the survival times of uncured patients within clusters are modeled by working correlation matrices in the estimating equations. We use a bootstrap method to obtain the variances of the estimates. We report a simulation study to demonstrate a substantial efficiency gain of the proposed method over an existing marginal method. Finally, we apply the model and the proposed method to a set of data from a multi‐institutional study of tonsil cancer patients treated with a radiation therapy. Copyright © 2012 John Wiley & Sons, Ltd.     

半参数治愈模型在长期生存者资料分析中的应用

目的 介绍长期生存者资料生存分析模型与方法 .方法 以SARS病人为例阐述半参数治愈模型原理与方法 ,并将长期生存者资料半参数治愈模型与Cox回归模型得到的结果 进行对比分析.结果 Cox比例风险回归模型得到四个协变量有统计学意义;半参数治愈模型比例风险回归部分得到一个有意义的协变量,logistic回归部分得到三个协变量有统计学意义.结论 在对长期生存者存在的资料分析时,半参数治愈模型比传统的Cox比例风险回归模型更具优势,不仅模型形式简明,参数估计解释合理,而且可从多角度提供更多有价值的信息,是一种适用范围更广,实用性更强的统计分析方法 .     

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.     

Induced smoothing for rank‐based regression with recurrent gap time data

Various semiparametric regression models have recently been proposed for the analysis of gap times between consecutive recurrent events. Among them, the semiparametric accelerated failure time (AFT) model is especially appealing owing to its direct interpretation of covariate effects on the gap times. In general, estimation of the semiparametric AFT model is challenging because the rank‐based estimating function is a nonsmooth step function. As a result, solutions to the estimating equations do not necessarily exist. Moreover, the popular resampling‐based variance estimation for the AFT model requires solving rank‐based estimating equations repeatedly and hence can be computationally cumbersome and unstable. In this paper, we extend the induced smoothing approach to the AFT model for recurrent gap time data. Our proposed smooth estimating function permits the application of standard numerical methods for both the regression coefficients estimation and the standard error estimation. Large‐sample properties and an asymptotic variance estimator are provided for the proposed method. Simulation studies show that the proposed method outperforms the existing nonsmooth rank‐based estimating function methods in both point estimation and variance estimation. The proposed method is applied to the data analysis of repeated hospitalizations for patients in the Danish Psychiatric Center Register.     

A flexible family of transformation cure rate models

In this paper, we introduce a flexible family of cure rate models, mainly motivated by the biological derivation of the classical promotion time cure rate model and assuming that a metastasis‐competent tumor cell produces a detectable‐tumor mass only when a specific number of distinct biological factors affect the cell. Special cases of the new model are, among others, the promotion time (proportional hazards), the geometric (proportional odds), and the negative binomial cure rate model. In addition, our model generalizes specific families of transformation cure rate models and some well‐studied destructive cure rate models. Exact likelihood inference is carried out by the aid of the expectation?maximization algorithm; a profile likelihood approach is exploited for estimating the parameters of the model while model discrimination problem is analyzed by the aid of the likelihood ratio test. A simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data‐set. Copyright © 2017 John Wiley & Sons, Ltd.     

A flexible cure rate model with dependent censoring and a known cure threshold

We propose a flexible cure rate model that accommodates different censoring distributions for the cured and uncured groups and also allows for some individuals to be observed as cured when their survival time exceeds a known threshold. We model the survival times for the uncured group using an accelerated failure time model with errors distributed according to the seminonparametric distribution, potentially truncated at a known threshold. We suggest a straightforward extension of the usual expectation–maximization algorithm approach for obtaining estimates in cure rate models to accommodate the cure threshold and dependent censoring. We additionally suggest a likelihood ratio test for testing for the presence of dependent censoring in the proposed cure rate model. We show through numerical studies that our model has desirable properties and leads to approximately unbiased parameter estimates in a variety of scenarios. To demonstrate how our method performs in practice, we analyze data from a bone marrow transplantation study and a liver transplant study. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of linear transformation models with covariate measurement error and interval censoring

Among several semiparametric models, the Cox proportional hazard model is widely used to assess the association between covariates and the time-to-event when the observed time-to-event is interval-censored. Often, covariates are measured with error. To handle this covariate uncertainty in the Cox proportional hazard model with the interval-censored data, flexible approaches have been proposed. To fill a gap and broaden the scope of statistical applications to analyze time-to-event data with different models, in this paper, a general approach is proposed for fitting the semiparametric linear transformation model to interval-censored data when a covariate is measured with error. The semiparametric linear transformation model is a broad class of models that includes the proportional hazard model and the proportional odds model as special cases. The proposed method relies on a set of estimating equations to estimate the regression parameters and the infinite-dimensional parameter. For handling interval censoring and covariate measurement error, a flexible imputation technique is used. Finite sample performance of the proposed method is judged via simulation studies. Finally, the suggested method is applied to analyze a real data set from an AIDS clinical trial.     

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.     

A general semiparametric hazards regression model: efficient estimation and structure selection

We consider a general semiparametric hazards regression model that encompasses the Cox proportional hazards model and the accelerated failure time model for survival analysis. To overcome the nonexistence of the maximum likelihood, we derive a kernel‐smoothed profile likelihood function and prove that the resulting estimates of the regression parameters are consistent and achieve semiparametric efficiency. In addition, we develop penalized structure selection techniques to determine which covariates constitute the accelerated failure time model and which covariates constitute the proportional hazards model. The proposed method is able to estimate the model structure consistently and model parameters efficiently. Furthermore, variance estimation is straightforward. The proposed estimation performs well in simulation studies and is applied to the analysis of a real data set. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation method of the semiparametric mixture cure gamma frailty model

Mixture cure frailty model has been proposed to analyze censored survival data with a cured fraction and unobservable information among the uncured patients. Different from a usual mixture cure model, the frailty model is employed to model the latency component in the mixture cure frailty model. In this paper, we extend the mixture cure frailty model by incorporating covariates into both the cure rate and the latency distribution parts of the model and propose a semiparametric estimation method for the model. The Expectation Maximization (EM) algorithm and the multiple imputation method are employed to estimate parameters of interest. In the simulation study, we show that both estimation methods work well. To illustrate, we apply the model and the proposed methods to a data set of failure times from bone marrow transplant patients.     

An information criterion for marginal structural models

Marginal structural models were developed as a semiparametric alternative to the G‐computation formula to estimate causal effects of exposures. In practice, these models are often specified using parametric regression models. As such, the usual conventions regarding regression model specification apply. This paper outlines strategies for marginal structural model specification and considerations for the functional form of the exposure metric in the final structural model. We propose a quasi‐likelihood information criterion adapted from use in generalized estimating equations. We evaluate the properties of our proposed information criterion using a limited simulation study. We illustrate our approach using two empirical examples. In the first example, we use data from a randomized breastfeeding promotion trial to estimate the effect of breastfeeding duration on infant weight at 1 year. In the second example, we use data from two prospective cohorts studies to estimate the effect of highly active antiretroviral therapy on CD4 count in an observational cohort of HIV‐infected men and women. The marginal structural model specified should reflect the scientific question being addressed but can also assist in exploration of other plausible and closely related questions. In marginal structural models, as in any regression setting, correct inference depends on correct model specification. Our proposed information criterion provides a formal method for comparing model fit for different specifications. Copyright © 2012 John Wiley & Sons, Ltd.     

Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study

Cure models for clustered survival data have the potential for broad applicability. In this paper, we consider the mixture cure model with random effects and propose several estimation methods based on Gaussian quadrature, rejection sampling, and importance sampling to obtain the maximum likelihood estimates of the model for clustered survival data with a cure fraction. The methods are flexible to accommodate various correlation structures. A simulation study demonstrates that the maximum likelihood estimates of parameters in the model tend to have smaller biases and variances than the estimates obtained from the existing methods. We apply the model to a study of tonsil cancer patients clustered by treatment centers to investigate the effect of covariates on the cure rate and on the failure time distribution of the uncured patients. The maximum likelihood estimates of the parameters demonstrate strong correlation among the failure times of the uncured patients and weak correlation among cure statuses in the same center.     

A semiparametric probit model for case 2 interval‐censored failure time data

Interval‐censored data occur naturally in many fields and the main feature is that the failure time of interest is not observed exactly, but is known to fall within some interval. In this paper, we propose a semiparametric probit model for analyzing case 2 interval‐censored data as an alternative to the existing semiparametric models in the literature. Specifically, we propose to approximate the unknown nonparametric nondecreasing function in the probit model with a linear combination of monotone splines, leading to only a finite number of parameters to estimate. Both the maximum likelihood and the Bayesian estimation methods are proposed. For each method, regression parameters and the baseline survival function are estimated jointly. The proposed methods make no assumptions about the observation process and can be applicable to any interval‐censored data with easy implementation. The methods are evaluated by simulation studies and are illustrated by two real‐life interval‐censored data applications. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

An Expectation Maximization algorithm for fitting the generalized odds‐rate model to interval censored data

The generalized odds‐rate model is a class of semiparametric regression models, which includes the proportional hazards and proportional odds models as special cases. There are few works on estimation of the generalized odds‐rate model with interval censored data because of the challenges in maximizing the complex likelihood function. In this paper, we propose a gamma‐Poisson data augmentation approach to develop an Expectation Maximization algorithm, which can be used to fit the generalized odds‐rate model to interval censored data. The proposed Expectation Maximization algorithm is easy to implement and is computationally efficient. The performance of the proposed method is evaluated by comprehensive simulation studies and illustrated through applications to datasets from breast cancer and hemophilia studies. In order to make the proposed method easy to use in practice, an R package ‘ICGOR’ was developed. Copyright © 2016 John Wiley & Sons, Ltd.     

Penalized likelihood estimation for semiparametric mixed models,with application to alcohol treatment research

In this article, we implement a practical computational method for various semiparametric mixed effects models, estimating nonlinear functions by penalized splines. We approximate the integration of the penalized likelihood with respect to random effects with the use of adaptive Gaussian quadrature, which we can conveniently implement in SAS procedure NLMIXED. We carry out the selection of smoothing parameters through approximated generalized cross‐validation scores. Our method has two advantages: (1) the estimation is more accurate than the current available quasi‐likelihood method for sparse data, for example, binary data; and (2) it can be used in fitting more sophisticated models. We show the performance of our approach in simulation studies with longitudinal outcomes from three settings: binary, normal data after Box–Cox transformation, and count data with log‐Gamma random effects. We also develop an estimation method for a longitudinal two‐part nonparametric random effects model and apply it to analyze repeated measures of semicontinuous daily drinking records in a randomized controlled trial of topiramate. Copyright © 2012 John Wiley & Sons, Ltd.     

Semiparametric transformation models for panel count data with correlated observation and follow‐up times

The statistical analysis of panel count data has recently attracted a great deal of attention, and a number of approaches have been developed. However, most of these approaches are for situations where the observation and follow‐up processes are independent of the underlying recurrent event process unconditional or conditional on covariates. In this paper, we discuss a more general situation where both the observation and the follow‐up processes may be related with the recurrent event process of interest. For regression analysis, we present a class of semiparametric transformation models and develop some estimating equations for estimation of regression parameters. Numerical studies under different settings conducted for assessing the proposed methodology suggest that it works well for practical situations, and the approach is applied to a skin cancer study that motivated the study. Copyright © 2013 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.