A Bayesian Hierarchical Mixture Cure Modelling Framework to Utilize Multiple Survival Datasets for Long-Term Survivorship Estimates: A Case Study From Previously Untreated Metastatic Melanoma

Time to an event of interest over a lifetime is a central measure of the clinical benefit of an intervention used in a health technology assessment (HTA). Within the same trial, multiple end-points may also be considered. For example, overall and progression-free survival time for different drugs in oncology studies. A common challenge is when an intervention is only effective for some proportion of the population who are not clinically identifiable. Therefore, latent group membership as well as separate survival models for identified groups need to be estimated. However, follow-up in trials may be relatively short leading to substantial censoring. We present a general Bayesian hierarchical framework that can handle this complexity by exploiting the similarity of cure fractions between end-points; accounting for the correlation between them and improving the extrapolation beyond the observed data. Assuming exchangeability between cure fractions facilitates the borrowing of information between end-points. We undertake a comprehensive simulation study to evaluate the model performance under different scenarios. We also show the benefits of using our approach with a motivating example, the CheckMate 067 phase 3 trial consisting of patients with metastatic melanoma treated with first line therapy.

Marginal semiparametric accelerated failure time cure model for clustered survival data

The semiparametric accelerated failure time mixture cure model is an appealing alternative to the proportional hazards mixture cure model in analyzing failure time data with long-term survivors. However, this model was only proposed for independent survival data and it has not been extended to clustered or correlated survival data, partly due to the complexity of the estimation method for the model. In this paper, we consider a marginal semiparametric accelerated failure time mixture cure model for clustered right-censored failure time data with a potential cure fraction. We overcome the complexity of the existing semiparametric method by proposing a generalized estimating equations approach based on the expectation-maximization algorithm to estimate the regression parameters in the model. The correlation structures within clusters are modeled by working correlation matrices in the proposed generalized estimating equations. The large sample properties of the regression estimators are established. Numerical studies demonstrate that the proposed estimation method is easy to use and robust to the misspecification of working matrices and that higher efficiency is achieved when the working correlation structure is closer to the true correlation structure. We apply the proposed model and estimation method to a contralateral breast cancer study and reveal new insights when the potential correlation between patients is taken into account.

Multilevel joint frailty model for hierarchically clustered binary and survival data

Hierarchical data arise when observations are clustered into groups. Multilevel models are practically useful in these settings, but these models are elusive in the context of hierarchical data with mixed multivariate outcomes. In this article, we consider binary and survival outcomes and assume the hierarchical structure is induced by clustering of both outcomes within patients and clustering of patients within hospitals which frequently occur in multicenter studies. We introduce a multilevel joint frailty model that analyzes the outcomes simultaneously to jointly estimate their regression parameters and explicitly model within-patient correlation between the outcomes and within-hospital correlation separately for each outcome. Estimation is facilitated by a computationally efficient residual maximum likelihood method that further predicts cluster-specific frailties for both outcomes and circumvents the formidable challenges induced by multidimensional integration that complicates the underlying likelihood. The performance of the model and estimation procedure is investigated via extensive simulation studies. The practical utility of the model is illustrated through simultaneous modeling of disease-free survival and binary endpoint of platelet recovery in a multicenter allogeneic bone marrow transplantation dataset that motivates this study.

Mixture cure models with time-varying and multilevel frailties for recurrent event data

Many medical studies yield data on recurrent clinical events from populations which consist of a proportion of cured patients in the presence of those who experience the event at several times (uncured). A frailty mixture cure model has recently been postulated for such data, with an assumption that the random subject effect (frailty) of each uncured patient is constant across successive gap times between recurrent events. We propose two new models in a more general setting, assuming a multivariate time-varying frailty with an AR(1) correlation structure for each uncured patient and addressing multilevel recurrent event data originated from multi-institutional (multi-centre) clinical trials, using extra random effect terms to adjust for institution effect and treatment-by-institution interaction. To solve the difficulties in parameter estimation due to these highly complex correlation structures, we develop an efficient estimation procedure via an EM-type algorithm based on residual maximum likelihood (REML) through the generalised linear mixed model (GLMM) methodology. Simulation studies are presented to assess the performances of the models. Data sets from a colorectal cancer study and rhDNase multi-institutional clinical trial were analyzed to exemplify the proposed models. The results demonstrate a large positive AR(1) correlation among frailties across successive gap times, indicating a constant frailty may not be realistic in some situations. Comparisons of findings with existing frailty models are discussed.

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.

A Bayesian piecewise survival cure rate model for spatially clustered data

This paper proposes a Bayesian hierarchical cure rate survival model for spatially clustered time to event data. We consider a mixture cure rate model with covariates and a flexible (semi)parametric baseline survival distribution for uncured individuals. The spatial correlation structure is introduced in the form of frailties which follow a Multivariate Conditionally Autoregressive distribution on a pre-specified map. We obtain the usual posterior estimates, smoothed by regional level maps of spatial frailties and cure rates. A simulation study demonstrates that the parameters of the models with spatially correlated frailties have smaller relative biases and MSE than the ones obtained using simple frailty models. We apply our methodology to Hodgkin lymphoma cancer survival times for patients diagnosed in the state of Connecticut.

geecure: An R-package for marginal proportional hazards mixture cure models

BACKGROUND AND OBJECTIVE

Most of available software packages for mixture cure models to analyze survival data with a cured fraction assume independent survival times, and they are not suitable for correlated survival times, such as clustered survival data. The objective of this paper is to present a software package to fit a marginal mixture cure model to clustered survival data with a cured fraction.

METHODS

We developed an R package geecure that fits the marginal proportional hazards mixture cure (PHMC) models to clustered right-censored survival data with a cured fraction. The dependence among the cure statuses and among the survival times of uncured patients within a cluster are modeled by working correlation matrices through the generalized estimating equations, and the Expectation-Solution algorithm is used to estimate the parameters. The variances of the estimated regression parameters are estimated by either a sandwich method or a bootstrap method.

RESULTS

The package geecure can fit the marginal PHMC model where the cumulative baseline hazard function is either a two-parameter Weibull distribution or specified nonparametrically. Fitting the parametric PHMC model with the Weibull baseline hazard function on average takes less time than fitting the semiparametric PHMC model does. Two variance estimation methods are comparable in the simulation study. The sandwich method takes much less time than the bootstrap method in variance estimation.

CONCLUSIONS

The package geecure provides an easy access to the marginal PHMC models for clustered survival data with a cured fraction in routine survival analysis. It is easy to use and will make the wide applications of the marginal PHMC models possible.

Modeling clustered long-term survivors using marginal mixture cure model

There is a great deal of recent interests in modeling right-censored clustered survival time data with a possible fraction of cured subjects who are nonsusceptible to the event of interest using marginal mixture cure models. In this paper, we consider a semiparametric marginal mixture cure model for such data and propose to extend an existing generalized estimating equation approach by a new unbiased estimating equation for the regression parameters in the latency part of the model. The large sample properties of the regression effect estimators in both incidence and the latency parts are established. The finite sample properties of the estimators are studied in simulation studies. The proposed method is illustrated with a bone marrow transplantation data and a tonsil cancer data.

Association measures for bivariate failure times in the presence of a cure fraction

This paper proposes a new joint model for pairs of failure times in the presence of a cure fraction. The proposed model relaxes some of the assumptions required by the existing approaches. This allows us to add some flexibility to the dependence structure and to widen the range of association measures that can be defined. A numerically stable iterative algorithm based on estimating equations is proposed to estimate the parameters. The estimators are shown to be consistent and asymptotically normal. Simulations show that they have good finite-sample properties. The added flexibility of the proposal is illustrated with an application to data from a diabetes retinopathy study.

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.

Using cure models and multiple imputation to utilize recurrence as an auxiliary variable for overall survival

BACKGROUND

Intermediate outcome variables can often be used as auxiliary variables for the true outcome of interest in randomized clinical trials. For many cancers, time to recurrence is an informative marker in predicting a patient's overall survival outcome and could provide auxiliary information for the analysis of survival times.

PURPOSE

To investigate whether models linking recurrence and death combined with a multiple imputation procedure for censored observations can result in efficiency gains in the estimation of treatment effects and be used to shorten trial lengths.

METHODS

Recurrence and death times are modeled using data from 12 trials in colorectal cancer. Multiple imputation is used as a strategy for handling missing values arising from censoring. The imputation procedure uses a cure model for time to recurrence and a time-dependent Weibull proportional hazards model for time to death. Recurrence times are imputed, and then death times are imputed conditionally on recurrence times. To illustrate these methods, trials are artificially censored 2 years after the last accrual, the imputation procedure implemented, and a log-rank test and Cox model used to analyze and compare these new data with the original data.

RESULTS

The results show modest, but consistent gains in efficiency in the analysis using the auxiliary information in recurrence times. Comparison of analyses show the treatment effect estimates and log-rank test results from the 2-year censored imputed data to be in between the estimates from the original data and the artificially censored data, indicating that the procedure was able to recover some of the lost information due to censoring.

LIMITATIONS

The models used are all fully parametric, requiring distributional assumptions of the data.

CONCLUSIONS

The proposed models may be useful in improving the efficiency in estimation of treatment effects in cancer trials and shortening trial length.

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 sequential threshold cure model for genetic analysis of time-to-event data

In analysis of time-to-event data, classical survival models ignore the presence of potential nonsusceptible (cured) individuals, which, if present, will invalidate the inference procedures. Existence of nonsusceptible individuals is particularly relevant under challenge testing with specific pathogens, which is a common procedure in aquaculture breeding schemes. A cure model is a survival model accounting for a fraction of nonsusceptible individuals in the population. This study proposes a mixed cure model for time-to-event data, measured as sequential binary records. In a simulation study survival data were generated through 2 underlying traits: susceptibility and endurance (risk of dying per time-unit), associated with 2 sets of underlying liabilities. Despite considerable phenotypic confounding, the proposed model was largely able to distinguish the 2 traits. Furthermore, if selection is for improved susceptibility rather than endurance, the error of applying a classical survival model was nonnegligible. The difference was most pronounced for scenarios with substantial underlying genetic variation in endurance and when the 2 underlying traits were lowly genetically correlated. In the presence of nonsusceptible individuals, the method provides a novel and more accurate tool for utilization of time-to-event data, and has also been proven successful when applied to zero-inflated longitudinal binary data.