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.     

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.     

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     

Frequentist and Bayesian approaches for a joint model for prostate cancer risk and longitudinal prostate-specific antigen data

The paper describes the use of frequentist and Bayesian shared-parameter joint models of longitudinal measurements of prostate-specific antigen (PSA) and the risk of prostate cancer (PCa). The motivating dataset corresponds to the screening arm of the Spanish branch of the European Randomized Screening for Prostate Cancer study. The results show that PSA is highly associated with the risk of being diagnosed with PCa and that there is an age-varying effect of PSA on PCa risk. Both the frequentist and Bayesian paradigms produced very close parameter estimates and subsequent 95% confidence and credibility intervals. Dynamic estimations of disease-free probabilities obtained using Bayesian inference highlight the potential of joint models to guide personalized risk-based screening strategies.     

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.     

Modeling survival in childhood cancer studies using two-stage non-mixture cure models

Non-mixture cure models (NMCMs) 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 cancer in studies when a proportion of those treated are completely cured. However, for many studies overall survival is also of interest. A two-stage NMCM that estimates the overall survival from a combination of two cure models, one from end of treatment to first recurrence and one from first recurrence to death, is proposed. The model is applied to two studies of Ewing's tumor in young patients. Caution needs to be exercised when extrapolating from cure models fitted to short follow-up times, but these data and associated simulations show how, when follow-up is limited, a two-stage model can give more stable estimates of the cure fraction than a one-stage model applied directly to overall survival.     

A general hazard model for lifetime data in the presence of cure rate

Historically, the cure rate model has been used for modeling time-to-event data within which a significant proportion of patients are assumed to be cured of illnesses, including breast cancer, non-Hodgkin lymphoma, leukemia, prostate cancer, melanoma, and head and neck cancer. Perhaps the most popular type of cure rate model is the mixture model introduced by Berkson and Gage [1]. In this model, it is assumed that a certain proportion of the patients are cured, in the sense that they do not present the event of interest during a long period of time and can found to be immune to the cause of failure under study. In this paper, we propose a general hazard model which accommodates comprehensive families of cure rate models as particular cases, including the model proposed by Berkson and Gage. The maximum-likelihood-estimation procedure is discussed. A simulation study analyzes the coverage probabilities of the asymptotic confidence intervals for the parameters. A real data set on children exposed to HIV by vertical transmission illustrates the methodology.     

A Bayesian finite mixture change-point model for assessing the risk of novice teenage drivers

The driving risk during the initial period after licensure for novice teenage drivers is typically the highest but decreases rapidly right after. The change-point of driving risk is a critical parameter for evaluating teenage driving risk, which also varies substantially among drivers. This paper presents latent class recurrent-event change-point models for detecting the change-points. The proposed model is applied to the Naturalist Teenage Driving Study, which continuously recorded the driving data of 42 novice teenage drivers for 18 months using advanced in-vehicle instrumentation. We propose a hierarchical BFMM to estimate the change-points by clusters of drivers with similar risk profiles. The model is based on a non-homogeneous Poisson process with piecewise-constant intensity functions. Latent variables which identify the membership of the subjects are used to detect potential clusters among subjects. Application to the Naturalistic Teenage Driving Study identifies three distinct clusters with change-points at 52.30, 108.99 and 150.20?hours of driving after first licensure, respectively. The overall intensity rate and the pattern of change also differ substantially among clusters. The results of this research provide more insight in teenagers' driving behaviour and will be critical to improve young drivers' safety education and parent management programs, as well as provide crucial reference for the GDL regulations to encourage safer driving.     

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.     

Analysis of cure rate survival data under proportional odds model

Due to significant progress in cancer treatments and management in survival studies involving time to relapse (or death), we often need survival models with cured fraction to account for the subjects enjoying prolonged survival. Our article presents a new proportional odds survival models with a cured fraction using a special hierarchical structure of the latent factors activating cure. This new model has same important differences with classical proportional odds survival models and existing cure-rate survival models. We demonstrate the implementation of Bayesian data analysis using our model with data from the SEER (Surveillance Epidemiology and End Results) database of the National Cancer Institute. Particularly aimed at survival data with cured fraction, we present a novel Bayes method for model comparisons and assessments, and demonstrate our new tool’s superior performance and advantages over competing tools.     

Identifying the time of step change in the mean of autocorrelated processes

Control charts are used to detect changes in a process. Once a change is detected, knowledge of the change point would simplify the search for and identification of the special cause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process analysts. Change-point methods for the uncorrelated process have been studied extensively in the literature; however, less attention has been given to change-point methods for autocorrelated processes. Autocorrelation is common in practice and is often modeled via the class of autoregressive moving average (ARMA) models. In this article, a maximum likelihood estimator for the time of step change in the mean of covariance-stationary processes that fall within the general ARMA framework is developed. The estimator is intended to be used as an “add-on” following a signal from a phase II control chart. Considering first-order pure and mixed ARMA processes, Monte Carlo simulation is used to evaluate the performance of the proposed change-point estimator across a range of step change magnitudes following a genuine signal from a control chart. Results indicate that the estimator provides process analysts with an accurate and useful estimate of the last sample obtained from the unchanged process. Additionally, results indicate that if a change-point estimator designed for the uncorrelated process is applied to an autocorrelated process, the performance of the estimator can suffer dramatically.     

Bayesian Parametric Accelerated Failure Time Spatial Model and its Application to Prostate Cancer

Prostate cancer is the most common cancer diagnosed in American men and the second leading cause of death from malignancies. There are large geographical variation and racial disparities existing in the survival rate of prostate cancer. Much work on the spatial survival model is based on the proportional hazards model, but few focused on the accelerated failure time model. In this paper, we investigate the prostate cancer data of Louisiana from the SEER program and the violation of the proportional hazards assumption suggests the spatial survival model based on the accelerated failure time model is more appropriate for this data set. To account for the possible extra-variation, we consider spatially-referenced independent or dependent spatial structures. The deviance information criterion (DIC) is used to select a best fitting model within the Bayesian frame work. The results from our study indicate that age, race, stage and geographical distribution are significant in evaluating prostate cancer survival.     

Flexible Cure Rate Modeling Under Latent Activation Schemes

With rapid improvements in medical treatment and health care, many datasets dealing with time to relapse or death now reveal a substantial portion of patients who are cured (i.e., who never experience the event). Extended survival models called cure rate models account for the probability of a subject being cured and can be broadly classified into the classical mixture models of Berkson and Gage (BG type) or the stochastic tumor models pioneered by Yakovlev and extended to a hierarchical framework by Chen, Ibrahim, and Sinha (YCIS type). Recent developments in Bayesian hierarchical cure models have evoked significant interest regarding relationships and preferences between these two classes of models. Our present work proposes a unifying class of cure rate models that facilitates flexible hierarchical model-building while including both existing cure model classes as special cases. This unifying class enables robust modeling by accounting for uncertainty in underlying mechanisms leading to cure. Issues such as regressing on the cure fraction and propriety of the associated posterior distributions under different modeling assumptions are also discussed. Finally, we offer a simulation study and also illustrate with two datasets (on melanoma and breast cancer) that reveal our framework's ability to distinguish among underlying mechanisms that lead to relapse and cure.     

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.     

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.     

Multilevel modelling of the geographical distributions of diseases

Multilevel modelling is used on problems arising from the analysis of spatially distributed health data. We use three applications to demonstrate the use of multilevel modelling in this area. The first concerns small area all-cause mortality rates from Glasgow where spatial autocorrelation between residuals is examined. The second analysis is of prostate cancer cases in Scottish counties where we use a range of models to examine whether the incidence is higher in more rural areas. The third develops a multiple-cause model in which deaths from cancer and cardiovascular disease in Glasgow are examined simultaneously in a spatial model. We discuss some of the issues surrounding the use of complex spatial models and the potential for future developments.     

The odd log-logistic Lindley Poisson model for lifetime data

We define two new lifetime models called the odd log-logistic Lindley (OLL-L) and odd log-logistic Lindley Poisson (OLL-LP) distributions with various hazard rate shapes such as increasing, decreasing, upside-down bathtub, and bathtub. Various structural properties are derived. Certain characterizations of OLL-L distribution are presented. The maximum likelihood estimators of the unknown parameters are obtained. We propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest has a Poisson distribution and the time to event has an OLL-L distribution. The applicability of the new models is illustrated by means real datasets.     

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.     

Detecting multiple change points in piecewise constant hazard functions

The National Cancer Institute (NCI) suggests a sudden reduction in prostate cancer mortality rates, likely due to highly successful treatments and screening methods for early diagnosis. We are interested in understanding the impact of medical breakthroughs, treatments, or interventions, on the survival experience for a population. For this purpose, estimating the underlying hazard function, with possible time change points, would be of substantial interest, as it will provide a general picture of the survival trend and when this trend is disrupted. Increasing attention has been given to testing the assumption of a constant failure rate against a failure rate that changes at a single point in time. We expand the set of alternatives to allow for the consideration of multiple change-points, and propose a model selection algorithm using sequential testing for the piecewise constant hazard model. These methods are data driven and allow us to estimate not only the number of change points in the hazard function but where those changes occur. Such an analysis allows for better understanding of how changing medical practice affects the survival experience for a patient population. We test for change points in prostate cancer mortality rates using the NCI Surveillance, Epidemiology, and End Results dataset.