A class of semiparametric cure models with current status data

Current status data occur in many biomedical studies where we only know whether the event of interest occurs before or after a particular time point. In practice, some subjects may never experience the event of interest, i.e., a certain fraction of the population is cured or is not susceptible to the event of interest. We consider a class of semiparametric transformation cure models for current status data with a survival fraction. This class includes both the proportional hazards and the proportional odds cure models as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the maximum likelihood estimators for the regression coefficients are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in finite samples. For illustration, we provide an application of the models to a study on the calcification of the hydrogel intraocular lenses.

     

Standard Exponential Cure Rate Model with Noninformative or Informative Uniform-Exponential Censoring

In survival data analysis it is frequent the occurrence of a significant amount of censoring to the right indicating that there may be a proportion of individuals in the study for which the event of interest will never happen. This fact is not considered by the ordinary survival theory. Consequently, the survival models with a cure fraction have been receiving a lot of attention in the recent years. In this article, we consider the standard mixture cure rate model where a fraction p ₀ of the population is of individuals cured or immune and the remaining 1 ? p ₀ are not cured. We assume an exponential distribution for the survival time and an uniform-exponential for the censoring time. In a simulation study, the impact caused by the informative uniform-exponential censoring on the coverage probabilities and lengths of asymptotic confidence intervals is analyzed by using the Fisher information and observed information matrices.     

A new threshold regression model for survival data with a cure fraction

Due to the fact that certain fraction of the population suffering a particular type of disease get cured because of advanced medical treatment and health care system, we develop a general class of models to incorporate a cure fraction by introducing the latent number N of metastatic-competent tumor cells or infected cells caused by bacteria or viral infection and the latent antibody level R of immune system. Various properties of the proposed models are carefully examined and a Markov chain Monte Carlo sampling algorithm is developed for carrying out Bayesian computation for model fitting and comparison. A real data set from a prostate cancer clinical trial is analyzed in detail to demonstrate the proposed methodology.     

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

Causal Proportional Hazards Models and Time-constant Exposure in Randomized Clinical Trials

The last decade saw enormous progress in the development of causal inference tools to account for noncompliance in randomized clinical trials. With survival outcomes, structural accelerated failure time (SAFT) models enable causal estimation of effects of observed treatments without making direct assumptions on the compliance selection mechanism. The traditional proportional hazards model has however rarely been used for causal inference. The estimator proposed by Loeys and Goetghebeur (2003, Biometrics vol. 59 pp. 100–105) is limited to the setting of all or nothing exposure. In this paper, we propose an estimation procedure for more general causal proportional hazards models linking the distribution of potential treatment-free survival times to the distribution of observed survival times via observed (time-constant) exposures. Specifically, we first build models for observed exposure-specific survival times. Next, using the proposed causal proportional hazards model, the exposure-specific survival distributions are backtransformed to their treatment-free counterparts, to obtain – after proper mixing – the unconditional treatment-free survival distribution. Estimation of the parameter(s) in the causal model is then based on minimizing a test statistic for equality in backtransformed survival distributions between randomized arms.     

Estimating the causal effects in randomized trials for survival data with a cure fraction and non compliance

We consider causal inference in randomized studies for survival data with a cure fraction and all-or-none treatment non compliance. To describe the causal effects, we consider the complier average causal effect (CACE) and the complier effect on survival probability beyond time t (CESP), where CACE and CESP are defined as the difference of cure rate and non cured subjects’ survival probability between treatment and control groups within the complier class. These estimands depend on the distributions of survival times in treatment and control groups. Given covariates and latent compliance type, we model these distributions with transformation promotion time cure model whose parameters are estimated by maximum likelihood. Both the infinite dimensional parameter in the model and the mixture structure of the problem create some computational difficulties which are overcome by an expectation-maximization (EM) algorithm. We show the estimators are consistent and asymptotically normal. Some simulation studies are conducted to assess the finite-sample performance of the proposed approach. We also illustrate our method by analyzing a real data from the Healthy Insurance Plan of Greater New York.     

Sample size calculation for a proportional hazards mixture cure model with nonbinary covariates

Sample size calculation is a critical issue in clinical trials because a small sample size leads to a biased inference and a large sample size increases the cost. With the development of advanced medical technology, some patients can be cured of certain chronic diseases, and the proportional hazards mixture cure model has been developed to handle survival data with potential cure information. Given the needs of survival trials with potential cure proportions, a corresponding sample size formula based on the log-rank test statistic for binary covariates has been proposed by Wang et al. [25]. However, a sample size formula based on continuous variables has not been developed. Herein, we presented sample size and power calculations for the mixture cure model with continuous variables based on the log-rank method and further modified it by Ewell's method. The proposed approaches were evaluated using simulation studies for synthetic data from exponential and Weibull distributions. A program for calculating necessary sample size for continuous covariates in a mixture cure model was implemented in R.     

Generating data from improper distributions: application to Cox proportional hazards models with cure

Cure rate models are survival models characterized by improper survivor distributions which occur when the cumulative distribution function, say F, of the survival times does not sum up to 1 (i.e. F(+∞)<1). The first objective of this paper is to provide a general approach to generate data from any improper distribution. An application to times to event data randomly drawn from improper distributions with proportional hazards is investigated using the semi-parametric proportional hazards model with cure obtained as a special case of the nonlinear transformation models in [Tsodikov, Semiparametric models: A generalized self-consistency approach, J. R. Stat. Soc. Ser. B 65 (2003), pp. 759–774]. The second objective of this paper is to show by simulations that the bias, the standard error and the mean square error of the maximum partial likelihood (PL) estimator of the hazard ratio as well as the statistical power based on the PL estimator strongly depend on the proportion of subjects in the whole population who will never experience the event of interest.     

A dependent Dirichlet process model for survival data with competing risks

In this paper, we first propose a dependent Dirichlet process (DDP) model using a mixture of Weibull models with each mixture component resembling a Cox model for survival data. We then build a Dirichlet process mixture model for competing risks data without regression covariates. Next we extend this model to a DDP model for competing risks regression data by using a multiplicative covariate effect on subdistribution hazards in the mixture components. Though built on proportional hazards (or subdistribution hazards) models, the proposed nonparametric Bayesian regression models do not require the assumption of constant hazard (or subdistribution hazard) ratio. An external time-dependent covariate is also considered in the survival model. After describing the model, we discuss how both cause-specific and subdistribution hazard ratios can be estimated from the same nonparametric Bayesian model for competing risks regression. For use with the regression models proposed, we introduce an omnibus prior that is suitable when little external information is available about covariate effects. Finally we compare the models’ performance with existing methods through simulations. We also illustrate the proposed competing risks regression model with data from a breast cancer study. An R package “DPWeibull” implementing all of the proposed methods is available at CRAN.

     

Expanded renal transplantation: a competing risk model approach

Multi-state models (MSMs) are useful to analyze survival data when, besides the event of main interest, one or more intermediate states of the individual are identified. These models take the several existing states and the possible transitions among them into account. At the same time, covariate effects on each transition intensity may be investigated separately and, therefore, MSMs are more flexible than the standard Cox proportional hazards model. In this work, we use MSMs to investigate the impact of the quality of a transplanted kidney for a group of patients at the Hospital Universitario Central de Asturias. Specifically, we use an illness-death model to study the evolution of patients with kidney disease who received a renal transplant after a dialysis period. The intermediate state is defined as the failure of the received organ, while the terminating state is the death of the patient. In order to increase the potential number of organs available for transplant, the standards of quality for the transplanted kidneys were relaxed (the new criteria are labeled expanded criteria), and these ‘expanded kidneys’ were transplanted in appropriate candidates (older patients, with higher prevalence of diabetes mellitus). Results suggest that the expanded kidneys have a minor effect on survival, while both the kidney mortality and the risk of death increase with the patient's age and the serum creatinine and serum hemoglobin levels.     

Mixture cure rate models with accelerated failures and nonparametric form of covariate effects

Two-component mixture cure rate model is popular in cure rate data analysis with the proportional hazards and accelerated failure time (AFT) models being the major competitors for modelling the latency component. [Wang, L., Du, P., and Liang, H. (2012), ‘Two-Component Mixture Cure Rate Model with Spline Estimated Nonparametric Components’, Biometrics, 68, 726–735] first proposed a nonparametric mixture cure rate model where the latency component assumes proportional hazards with nonparametric covariate effects in the relative risk. Here we consider a mixture cure rate model where the latency component assumes AFTs with nonparametric covariate effects in the acceleration factor. Besides the more direct physical interpretation than the proportional hazards, our model has an additional scalar parameter which adds more complication to the computational algorithm as well as the asymptotic theory. We develop a penalised EM algorithm for estimation together with confidence intervals derived from the Louis formula. Asymptotic convergence rates of the parameter estimates are established. Simulations and the application to a melanoma study shows the advantages of our new method.     

A marginal regression model for multivariate failure time data with a surviving fraction

A marginal regression approach for correlated censored survival data has become a widely used statistical method. Examples of this approach in survival analysis include from the early work by Wei et al. (J Am Stat Assoc 84:1065–1073, 1989) to more recent work by Spiekerman and Lin (J Am Stat Assoc 93:1164–1175, 1998). This approach is particularly useful if a covariate’s population average effect is of primary interest and the correlation structure is not of interest or cannot be appropriately specified due to lack of sufficient information. In this paper, we consider a semiparametric marginal proportional hazard mixture cure model for clustered survival data with a surviving or “cure” fraction. Unlike the clustered data in previous work, the latent binary cure statuses of patients in one cluster tend to be correlated in addition to the possible correlated failure times among the patients in the cluster who are not cured. The complexity of specifying appropriate correlation structures for the data becomes even worse if the potential correlation between cure statuses and the failure times in the cluster has to be considered, and thus a marginal regression approach is particularly attractive. We formulate a semiparametric marginal proportional hazards mixture cure model. Estimates are obtained using an EM algorithm and expressions for the variance–covariance are derived using sandwich estimators. Simulation studies are conducted to assess finite sample properties of the proposed model. The marginal model is applied to a multi-institutional study of local recurrences of tonsil cancer patients who received radiation therapy. It reveals new findings that are not available from previous analyses of this study that ignored the potential correlation between patients within the same institution.     

A Nonparametric Comparison of Conditional Distributions with Nonnegligible Cure Fractions

Survival data with nonnegligible cure fractions are commonly encountered in clinical cancer clinical research. Recently, several authors (e.g. Kuk and Chen, Biometrika 79 (1992) 531; Maller and Zhou, Journal of Applied Probability, 30 (1993) 602; Peng and Dear, Biometrics, 56 (2000) 237; Sy and Taylor, Biometrics 56 (2000) 227) have proposed to use semiparametric cure models to analyze such data. Much of the existing work has been emphasized on cure detections and regression techniques. In contrast, this project focuses on the hypothesis testing in the presence of a cure fraction. Specifically, our interest lies in detecting whether there exists survival differences among noncured patients between treatment arms. For this purpose, we investigate the use of a modified Cramér-von Mises statistic for two-sample survival comparisons within the framework of cure models. Such a test has been studied by Tamura et al., (Statistics in Medicine 19, 2000, 2169) using bootstrap procedure. We will focus on developing asymptotic theory and convergent algorithms in this paper. We show that the limiting distributions of the Cramér-von Mises statistic under the null hypothesis can be represented by stochastic integrals and a weighted noncentral chi-squares. Both representations lead to concrete numerical schemes for computing the limiting distributions. The algorithms can be easily implemented for data analysis and significantly reduce computing time compared to the bootstrap approach. For illustrative purposes, we apply the proposed test to a published clinical trial.     

The log-exponentiated generalized gamma regression model for censored data

For the first time, we introduce a generalized form of the exponentiated generalized gamma distribution [Cordeiro et al. The exponentiated generalized gamma distribution with application to lifetime data, J. Statist. Comput. Simul. 81 (2011), pp. 827–842.] that is the baseline for the log-exponentiated generalized gamma regression model. The new distribution can accommodate increasing, decreasing, bathtub- and unimodal-shaped hazard functions. A second advantage is that it includes classical distributions reported in the lifetime literature as special cases. We obtain explicit expressions for the moments of the baseline distribution of the new regression model. The proposed model can be applied to censored data since it includes as sub-models several widely known regression models. It therefore can be used more effectively in the analysis of survival data. We obtain maximum likelihood estimates for the model parameters by considering censored data. We show that our extended regression model is very useful by means of two applications to real data.     

Semiparametric Estimation in Transformation Models with Cure Fraction

Semiparametric transformation model has been extensively investigated in the literature. The model, however, has little dealt with survival data with cure fraction. In this article, we consider a class of semi-parametric transformation models, where an unknown transformation of the survival times with cure fraction is assumed to be linearly related to the covariates and the error distributions are parametrically specified by an extreme value distribution with unknown parameters. Estimators for the coefficients of covariates are obtained from pseudo Z-estimator procedures allowing censored observations. We show that the estimators are consistent and asymptotically normal. The bootstrap estimation of the variances of the estimators is also investigated.     

A general long-term aging model with different underlying activation mechanisms: Modeling,Bayesian estimation,and case influence diagnostics

In this paper we propose a general cure rate aging model. Our approach enables different underlying activation mechanisms which lead to the event of interest. The number of competing causes of the event of interest is assumed to follow a logarithmic distribution. The model is parameterized in terms of the cured fraction which is then linked to covariates. We explore the use of Markov chain Monte Carlo methods to develop a Bayesian analysis for the proposed model. Moreover, some discussions on the model selection to compare the fitted models are given, as well as case deletion influence diagnostics are developed for the joint posterior distribution based on the ψ-divergence, which has several divergence measures as particular cases, such as the Kullback–Leibler (K-L), J-distance, L₁ norm, and χ²-square divergence measures. Simulation studies are performed and experimental results are illustrated based on a real malignant melanoma data.     

Estimating the cure fraction in population-based cancer studies by using finite mixture models

Summary.  The cure fraction (the proportion of patients who are cured of disease) is of interest to both patients and clinicians and is a useful measure to monitor trends in survival of curable disease. The paper extends the non-mixture and mixture cure fraction models to estimate the proportion cured of disease in population-based cancer studies by incorporating a finite mixture of two Weibull distributions to provide more flexibility in the shape of the estimated relative survival or excess mortality functions. The methods are illustrated by using public use data from England and Wales on survival following diagnosis of cancer of the colon where interest lies in differences between age and deprivation groups. We show that the finite mixture approach leads to improved model fit and estimates of the cure fraction that are closer to the empirical estimates. This is particularly so in the oldest age group where the cure fraction is notably lower. The cure fraction is broadly similar in each deprivation group, but the median survival of the 'uncured' is lower in the more deprived groups. The finite mixture approach overcomes some of the limitations of the more simplistic cure models and has the potential to model the complex excess hazard functions that are seen in real data.     

Bayesian and likelihood inference for cure rates based on defective inverse Gaussian regression models

Failure time models are considered when there is a subpopulation of individuals that is immune, or not susceptible, to an event of interest. Such models are of considerable interest in biostatistics. The most common approach is to postulate a proportion p of immunes or long-term survivors and to use a mixture model [5]. This paper introduces the defective inverse Gaussian model as a cure model and examines the use of the Gibbs sampler together with a data augmentation algorithm to study Bayesian inferences both for the cured fraction and the regression parameters. The results of the Bayesian and likelihood approaches are illustrated on two real data sets.     

A Bayesian approach to mixture cure models with spatial frailties for population‐based cancer relative survival data

As the treatments of cancer progress, a certain number of cancers are curable if diagnosed early. In population‐based cancer survival studies, cure is said to occur when mortality rate of the cancer patients returns to the same level as that expected for the general cancer‐free population. The estimates of cure fraction are of interest to both cancer patients and health policy makers. Mixture cure models have been widely used because the model is easy to interpret by separating the patients into two distinct groups. Usually parametric models are assumed for the latent distribution for the uncured patients. The estimation of cure fraction from the mixture cure model may be sensitive to misspecification of latent distribution. We propose a Bayesian approach to mixture cure model for population‐based cancer survival data, which can be extended to county‐level cancer survival data. Instead of modeling the latent distribution by a fixed parametric distribution, we use a finite mixture of the union of the lognormal, loglogistic, and Weibull distributions. The parameters are estimated using the Markov chain Monte Carlo method. Simulation study shows that the Bayesian method using a finite mixture latent distribution provides robust inference of parameter estimates. The proposed Bayesian method is applied to relative survival data for colon cancer patients from the Surveillance, Epidemiology, and End Results (SEER) Program to estimate the cure fractions. The Canadian Journal of Statistics 40: 40–54; 2012 © 2012 Statistical Society of Canada     

Flexible competing risks regression modeling and goodness-of-fit

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496–509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine–Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data.