The definition of start time in cancer treatment studies analysed by non-mixture cure models

Non-mixture cure models are derived from a simplified representation of the biological process that takes place after treatment for cancer. These models are intended to represent the time from the end of treatment to the time of first recurrence of the cancer in studies when a proportion of those treated are completely cured. However, for many studies, other start times are more relevant. In a clinical trial, it may be more natural to model the time from randomisation rather than the time from the end of treatment and in an epidemiological study, the time from diagnosis might be more meaningful. Some simulations and two real studies of childhood cancer are presented to show that starting from time of diagnosis or randomisation can affect the estimates of the cure fraction. The susceptibility of different parametric kernels to errors caused by using start times other than the end of treatment is also assessed. Analysing failures on treatment and relapse after completing the treatment as two processes offers a simple way of overcoming many of these problems.     

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     

The negative binomial–beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.     

Accelerated hazards mixture cure model

We propose a new cure model for survival data with a surviving or cure fraction. The new model is a mixture cure model where the covariate effects on the proportion of cure and the distribution of the failure time of uncured patients are separately modeled. Unlike the existing mixture cure models, the new model allows covariate effects on the failure time distribution of uncured patients to be negligible at time zero and to increase as time goes by. Such a model is particularly useful in some cancer treatments when the treat effect increases gradually from zero, and the existing models usually cannot handle this situation properly. We develop a rank based semiparametric estimation method to obtain the maximum likelihood estimates of the parameters in the model. We compare it with existing models and methods via a simulation study, and apply the model to a breast cancer data set. The numerical studies show that the new model provides a useful addition to the cure model literature.     

The Dynamic Model for Cancer Relapse Based on Two-Stage Model of Carcinogenesis

In this article, the time from the start of chemotherapy randomization until cancer relapse is of primary interest. Here, cancer relapse refers to the appearance of the first observable malignant clone after therapy. A dynamic model for cancer relapse after chemotherapy is developed. The model differs from the traditional cure rate models in that it takes into consideration the growth kinetics of malignant tumors using a two-stage carcinogenesis model. The survival and hazard functions for cancer relapse time are derived, and a simulation study is performed to validate the underlying model.     

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.     

Cure rate model with bivariate interval censored data

A mixture model is proposed to analyze a bivariate interval censored data with cure rates. There exist two types of association related with bivariate failure times and bivariate cure rates, respectively. A correlation coefficient is adopted for the association of bivariate cure rates and a copula function is applied for bivariate survival times. The conditional expectation of unknown quantities attributable to interval censored data and cure rates are calculated in the E-step in ES (Expectation-Solving algorithm) and the marginal estimates and the association measures are estimated in the S-step through a two-stage procedure. A simulation study is performed to evaluate the suggested method and a real data from HIV patients is analyzed as a real data example.     

Modelling population-based cancer survival trends using join point models for grouped survival data

Summary.  In the USA cancer as a whole is the second leading cause of death and a major burden to health care; thus medical progress against cancer is a major public health goal. There are many individual studies to suggest that cancer treatment breakthroughs and early diagnosis have significantly improved the prognosis of cancer patients. To understand better the relationship between medical improvements and the survival experience for the patient population at large, it is useful to evaluate cancer survival trends on the population level, e.g. to find out when and how much the cancer survival rates changed. We analyse population-based grouped cancer survival data by incorporating join points into the survival models. A join point survival model facilitates the identification of trends with significant change-points in cancer survival, when related to cancer treatments or interventions. The Bayesian information criterion is used to select the number of join points. The performance of the join point survival models is evaluated with respect to cancer prognosis, join point locations, annual percentage changes in death rates by year of diagnosis and sample sizes through intensive simulation studies. The model is then applied to grouped relative survival data for several major cancer sites from the 'Surveillance, epidemiology and end results' programme of the National Cancer Institute. The change-points in the survival trends for several major cancer sites are identified and the potential driving forces behind such change-points are discussed.     

Review and implementation of cure models based on first hitting times for Wiener processes

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501–513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.     

Direct parametric inference for the cumulative incidence function

Summary.  In survival data that are collected from phase III clinical trials on breast cancer, a patient may experience more than one event, including recurrence of the original cancer, new primary cancer and death. Radiation oncologists are often interested in comparing patterns of local or regional recurrences alone as first events to identify a subgroup of patients who need to be treated by radiation therapy after surgery. The cumulative incidence function provides estimates of the cumulative probability of locoregional recurrences in the presence of other competing events. A simple version of the Gompertz distribution is proposed to parameterize the cumulative incidence function directly. The model interpretation for the cumulative incidence function is more natural than it is with the usual cause-specific hazard parameterization. Maximum likelihood analysis is used to estimate simultaneously parametric models for cumulative incidence functions of all causes. The parametric cumulative incidence approach is applied to a data set from the National Surgical Adjuvant Breast and Bowel Project and compared with analyses that are based on parametric cause-specific hazard models and nonparametric cumulative incidence estimation.     

Estimating the turning point of the log-logistic hazard function in the presence of long-term survivors with an application for uterine cervical cancer data

The hazard function plays an important role in cancer patient survival studies, as it quantifies the instantaneous risk of death of a patient at any given time. Often in cancer clinical trials, unimodal hazard functions are observed, and it is of interest to detect (estimate) the turning point (mode) of hazard function, as this may be an important measure in patient treatment strategies with cancer. Moreover, when patient cure is a possibility, estimating cure rates at different stages of cancer, in addition to their proportions, may provide a better summary of the effects of stages on survival rates. Therefore, the main objective of this paper is to consider the problem of estimating the mode of hazard function of patients at different stages of cervical cancer in the presence of long-term survivors. To this end, a mixture cure rate model is proposed using the log-logistic distribution. The model is conveniently parameterized through the mode of the hazard function, in which cancer stages can affect both the cured fraction and the mode. In addition, we discuss aspects of model inference through the maximum likelihood estimation method. A Monte Carlo simulation study assesses the coverage probability of asymptotic confidence intervals.     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

Bayesian gamma frailty models for survival data with semi-competing risks and treatment switching

Motivated from a colorectal cancer study, we propose a class of frailty semi-competing risks survival models to account for the dependence between disease progression time, survival time, and treatment switching. Properties of the proposed models are examined and an efficient Gibbs sampling algorithm using the collapsed Gibbs technique is developed. A Bayesian procedure for assessing the treatment effect is also proposed. The deviance information criterion (DIC) with an appropriate deviance function and Logarithm of the pseudomarginal likelihood (LPML) are constructed for model comparison. A simulation study is conducted to examine the empirical performance of DIC and LPML and as well as the posterior estimates. The proposed method is further applied to analyze data from a colorectal cancer study.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

A new approach for joint modelling of longitudinal measurements and survival times with a cure fraction

The joint analysis of longitudinal measurements and survival data is useful in clinical trials and other medical studies. In this paper, we consider a joint model which assumes a linear mixed $tt$ model for longitudinal measurements and a promotion time cure model for survival data and links these two models through a latent variable. A semiparametric inference procedure with an EM algorithm implementation is developed for the parameters in the joint model. The proposed procedure is evaluated in a simulation study and applied to analyze the quality of life and time to recurrence data from a clinical trial on women with early breast cancer. The Canadian Journal of Statistics 40: 207–224; 2012 © 2012 Statistical Society of Canada     

Jointly modeling longitudinal proportional data and survival times with an application to the quality of life data in a breast cancer trial

Motivated by the joint analysis of longitudinal quality of life data and recurrence free survival times from a cancer clinical trial, we present in this paper two approaches to jointly model the longitudinal proportional measurements, which are confined in a finite interval, and survival data. Both approaches assume a proportional hazards model for the survival times. For the longitudinal component, the first approach applies the classical linear mixed model to logit transformed responses, while the second approach directly models the responses using a simplex distribution. A semiparametric method based on a penalized joint likelihood generated by the Laplace approximation is derived to fit the joint model defined by the second approach. The proposed procedures are evaluated in a simulation study and applied to the analysis of breast cancer data motivated this research.     

Change point-cure models with application to estimating the change-point effect of age of diagnosis among prostate cancer patients

Previous research on prostate cancer survival trends in the United States National Cancer Institute's Surveillance Epidemiology and End Results database has indicated a potential change-point in the age of diagnosis of prostate cancer around age 50. Identifying a change-point value in prostate cancer survival and cure could have important policy and health care management implications. Statistical analysis of this data has to address two complicating features: (1) change-point models are not smooth functions and so present computational and theoretical difficulties; and (2) models for prostate cancer survival need to account for the fact that many men diagnosed with prostate cancer can be effectively cured of their disease with early treatment. We develop a cure survival model that allows for change-point effects in covariates to investigate a potential change-point in the age of diagnosis of prostate cancer. Our results do not indicate that age under 50 is associated with increased hazard of death from prostate cancer.     

Cure rate survival models with missing covariates: a simulation study

In this paper we study the cure rate survival model involving a competitive risk structure with missing categorical covariates. A parametric distribution that can be written as a sequence of one-dimensional conditional distributions is specified for the missing covariates. We consider the missing data at random situation so that the missing covariates may depend only on the observed ones. Parameter estimates are obtained by using the EM algorithm via the method of weights. Extensive simulation studies are conducted and reported to compare estimates efficiency with and without missing data. As expected, the estimation approach taking into consideration the missing covariates presents much better efficiency in terms of mean square errors than the complete case situation. Effects of increasing cured fraction and censored observations are also reported. We demonstrate the proposed methodology with two real data sets. One involved the length of time to obtain a BS degree in Statistics, and another about the time to breast cancer recurrence.     

Joint analysis of longitudinal measurements and survival times with a cure fraction based on partly linear mixed and semiparametric cure models

In a joint analysis of longitudinal quality of life (QoL) scores and relapse-free survival (RFS) times from a clinical trial on early breast cancer conducted by the Canadian Cancer Trials Group, we observed a complicated trajectory of QoL scores and existence of long-term survivors. Motivated by this observation, we proposed in this paper a flexible joint model for the longitudinal measurements and survival times. A partly linear mixed effect model is used to capture the complicated but smooth trajectory of longitudinal measurements and approximated by B-splines and a semiparametric mixture cure model with the B-spline baseline hazard to model survival times with a cure fraction. These two models are linked by shared random effects to explore the dependence between longitudinal measurements and survival times. A semiparametric inference procedure with an EM algorithm is proposed to estimate the parameters in the joint model. The performance of proposed procedures are evaluated by simulation studies and through the application to the analysis of data from the clinical trial which motivated this research.     

SEMIPARAMETRIC ADDITIVE RISKS REGRESSION FOR TWO-STAGE DESIGN SURVIVAL STUDIES

In this article we study a semiparametric additive risks model (McKeague and Sasieni (1994)) for two-stage design survival data where accurate information is available only on second stage subjects, a subset of the first stage study. We derive two-stage estimators by combining data from both stages. Large sample inferences are developed. As a by-product, we also obtain asymptotic properties of the single stage estimators of McKeague and Sasieni (1994) when the semiparametric additive risks model is misspecified. The proposed two-stage estimators are shown to be asymptotically more efficient than the second stage estimators. They also demonstrate smaller bias and variance for finite samples. The developed methods are illustrated using small intestine cancer data from the SEER (Surveillance, Epidemiology, and End Results) Program.