A minimum version of log-rank test for testing the existence of cancer cure using relative survival data

Cancer survival is one of the most important measures to evaluate the effectiveness of treatment and early diagnosis. The ultimate goal of cancer research and patient care is the cure of cancer. As cancer treatments progress, cure becomes a reality for many cancers if patients are diagnosed early and get effective treatment. If a cure does exist for a certain type of cancer, it is useful to estimate the time of cure. For cancers that impose excess risk of mortality, it is informative to understand the difference in survival between cancer patients and the general cancer-free population. In population-based cancer survival studies, relative survival is the standard measure of excess mortality due to cancer. Cure is achieved when the survival of cancer patients is equivalent to that of the general population. This definition of cure is usually called the statistical cure, which is an important measure of burden due to cancer. In this paper, a minimum version of the log-rank test is proposed to test the equivalence of cancer patients' survival using the relative survival data. Performance of the proposed test is evaluated by simulation. Relative survival data from population-based cancer registries in SEER Program are used to examine patients' survival after diagnosis for various major cancer sites.     

A frailty mixture cure model with application to hospital admission data [corrected

Mixture cure models have been utilized to analyze survival data with possible cure. This paper considers the inclusion of frailty into the mixture cure model to model recurrent event data with a cure fraction. An attractive feature of the proposed model is the allowance for heterogeneity in risk among those individuals experiencing the event of interest in addition to the incorporation of a cured component. Maximum likelihood estimates can be obtained using the Expectation Maximization algorithm and standard errors are calculated from the Bootstrap method. The model is applied to hospital readmission data among colorectal cancer patients.     

Estimating the cure proportion of malignant melanoma,an alternative approach to assess long term survival: A population-based study

ObjectivesA large proportion of patients with cutaneous malignant melanoma (CMM) do not experience excess mortality due to their disease. This group of patients is referred to as the cure proportion. Few studies have examined the possibility of cure for CMM. The aim of this study was to estimate the cure proportion of patients with CMM in a Swedish population.MethodsWe undertook a population-based study of 5850 CMM patients in two Swedish health care regions during 1996–2005. We used flexible parametric cure models to estimate cure proportions and median survival times (MSTs) of uncured by stage, sex, age and anatomical site.ResultsDisease stage at diagnosis was the most important factor for the probability of cure, with a cure proportion of approximately 1.0 for stage IA. While the probability of cure decreased with older age, the influence of age was smaller on the MST of uncured. Differences in prognosis between males and females were mainly attributed to differences in cure as opposed to differences in MST of uncured.ConclusionsThis population-based study showed approximately 100% cure among stage IA disease. Almost 50% of patients had stage IA disease and the high cure proportion for this large patient group is reassuring.     

Mean residual life cure models for right-censored data with and without length-biased sampling

We propose a semiparametric mean residual life mixture cure model for right-censored survival data with a cured fraction. The model employs the proportional mean residual life model to describe the effects of covariates on the mean residual time of uncured subjects and the logistic regression model to describe the effects of covariates on the cure rate. We develop estimating equations to estimate the proposed cure model for the right-censored data with and without length-biased sampling, the latter is often found in prevalent cohort studies. In particular, we propose two estimating equations to estimate the effects of covariates in the cure rate and a method to combine them to improve the estimation efficiency. The consistency and asymptotic normality of the proposed estimates are established. The finite sample performance of the estimates is confirmed with simulations. The proposed estimation methods are applied to a clinical trial study on melanoma and a prevalent cohort study on early-onset type 2 diabetes mellitus.     

Estimating the proportion cured of cancer: Some practical advice for users

BackgroundCure models can provide improved possibilities for inference if used appropriately, but there is potential for misleading results if care is not taken. In this study, we compared five commonly used approaches for modelling cure in a relative survival framework and provide some practical advice on the use of these approaches.Patients and methodsData for colon, female breast, and ovarian cancers were used to illustrate these approaches. The proportion cured was estimated for each of these three cancers within each of three age groups. We then graphically assessed the assumption of cure and the model fit, by comparing the predicted relative survival from the cure models to empirical life table estimates.ResultsWhere both cure and distributional assumptions are appropriate (e.g., for colon or ovarian cancer patients aged <75 years), all five approaches led to similar estimates of the proportion cured. The estimates varied slightly when cure was a reasonable assumption but the distributional assumption was not (e.g., for colon cancer patients ≥75 years). Greater variability in the estimates was observed when the cure assumption was not supported by the data (breast cancer).ConclusionsIf the data suggest cure is not a reasonable assumption then we advise against fitting cure models. In the scenarios where cure was reasonable, we found that flexible parametric cure models performed at least as well, or better, than the other modelling approaches. We recommend that, regardless of the model used, the underlying assumptions for cure and model fit should always be graphically assessed.     

An Empirical Comparison of Parametric and Semiparametric Cure Models

Parametric and semiparametric cure models have been proposed for cure proportion estimation in cancer clinical research. In this paper, several parametric and semiparametric models are compared, and their estimation methods are discussed within the framework of the EM algorithm. We show that the semiparametric PH cure model can achieve efficiency levels similar to those of parametric cure models, provided that the failure time distribution is well specified and uncured patients have an increasing hazard rate. Therefore the semiparametric model is a viable alternative to parametric cure models. When the hazard rate of uncured patients is rapidly decreasing, the estimates from the semiparametric cure model tend to have large variations and biases. However, all other models also tend to have large variations and biases in this case.     

Cure by age and stage at diagnosis for colorectal cancer patients in North West England, 1997–2004: A population-based study

Background: Stage and age at diagnosis are important prognostic factors for patients with colorectal cancer. However, the proportion cured by stage and age is unknown in England. Materials and methods: This population-based study includes 29,563 adult patients who were diagnosed and registered with colorectal cancer during 1997–2004 and followed till 2007 in North West England. Multiple imputation was used to provide more reliable estimates of stage at diagnosis, when these data were missing. Cure mixture models were used to estimate the proportion ‘cured’ and the median survival of the uncured by age and stage. Results: For both colon and rectal cancer the proportion of patients cured and median survival time of the uncured decreased with advancing stage and increasing age. Patients aged under 65 years had the highest proportion cured and longest median survival of the uncured. Conclusion: Cure of colorectal cancer patients is dependent on stage and age at diagnosis with younger patients or those with less advanced disease having a better prognosis. Further efforts are required, in order to reduce the proportion of patients presenting with stage III and IV disease and ultimately increase the chance of cure.     

Generalized parametric cure models for relative survival

Cure models are used in time-to-event analysis when not all individuals are expected to experience the event of interest, or when the survival of the considered individuals reaches the same level as the general population. These scenarios correspond to a plateau in the survival and relative survival function, respectively. The main parameters of interest in cure models are the proportion of individuals who are cured, termed the cure proportion, and the survival function of the uncured individuals. Although numerous cure models have been proposed in the statistical literature, there is no consensus on how to formulate these. We introduce a general parametric formulation of mixture cure models and a new class of cure models, termed latent cure models, together with a general estimation framework and software, which enable fitting of a wide range of different models. Through simulations, we assess the statistical properties of the models with respect to the cure proportion and the survival of the uncured individuals. Finally, we illustrate the models using survival data on colon cancer, which typically display a plateau in the relative survival. As demonstrated in the simulations, mixture cure models which are not guaranteed to be constant after a finite time point, tend to produce accurate estimates of the cure proportion and the survival of the uncured. However, these models are very unstable in certain cases due to identifiability issues, whereas LC models generally provide stable results at the price of more biased estimates.     

Net survival in recurrence-free colon cancer patients

PurposeConditional net survival in recurrence-free patients (CNS-RF) provides relevant clinical information and has never been assessed yet in a non-selected colon cancer population. We aimed to estimate conditional 5-year net survival in recurrence-free patients with colon cancer in the population-based Digestive Cancer Registry of Burgundy (France).MethodsCNS-RF was estimated in the 3736 patients resected for cure for primary colon cancer between 1976 and 2006, using a flexible parametric model of net survival for every additional year survived at diagnosis and from 1 to 5 years thereafter.ResultsThe net probability of surviving 5 more years increased from 72% at diagnosis to 92% for recurrence-free patients who survived 5 years after diagnosis. CNS-RF was over 90% 3 years after diagnosis in patients aged 75 and below. CNS-RF was over 95% in patients diagnosed after 2000 who were recurrence-free 3, 4 or 5 years after diagnosis. CNS-RF was similar between patients with stage I and II disease from 2 years after diagnosis and patients with stage III disease from 5 years after diagnosis. The initial differences in net survival related to gross features, clinical presentation, number of harvested nodes in stage II, and number of involved nodes in stage III disappeared after 2 years.ConclusionsCNS-RF is a relevant measure of prognosis in patients who have already achieved a period of remission. Providing an updated estimation of prognosis in the years following diagnosis may improve the survivors’ quality of life and access to credit or insurance.     

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.     

The joint modeling of a longitudinal disease progression marker and the failure time process in the presence of cure

In this paper we present an extension of cure models: to incorporate a longitudinal disease progression marker. The model is motivated by studies of patients with prostate cancer undergoing radiation therapy. The patients are followed until recurrence of the prostate cancer or censoring, with the PSA marker measured intermittently. Some patients are cured by the treatment and are immune from recurrence. A joint-cure model is developed for this type of data, in which the longitudinal marker and the failure time process are modeled jointly, with a fraction of patients assumed to be immune from the endpoint. A hierarchical nonlinear mixed-effects model is assumed for the marker and a time-dependent Cox proportional hazards model is used to model the time to endpoint. The probability of cure is modeled by a logistic link. The parameters are estimated using a Monte Carlo EM algorithm. Importance sampling with an adaptively chosen t-distribution and variable Monte Carlo sample size is used. We apply the method to data from prostate cancer and perform a simulation study. We show that by incorporating the longitudinal disease progression marker into the cure model, we obtain parameter estimates with better statistical properties. The classification of the censored patients into the cure group and the susceptible group based on the estimated conditional recurrence probability from the joint-cure model has a higher sensitivity and specificity, and a lower misclassification probability compared with the standard cure model. The addition of the longitudinal data has the effect of reducing the impact of the identifiability problems in a standard cure model and can help overcome biases due to informative censoring.     

Two-component mixture cure rate model with spline estimated nonparametric components

Summary In some survival analysis of medical studies, there are often long-term survivors who can be considered as permanently cured. The goals in these studies are to estimate the noncured probability of the whole population and the hazard rate of the susceptible subpopulation. When covariates are present as often happens in practice, to understand covariate effects on the noncured probability and hazard rate is of equal importance. The existing methods are limited to parametric and semiparametric models. We propose a two-component mixture cure rate model with nonparametric forms for both the cure probability and the hazard rate function. Identifiability of the model is guaranteed by an additive assumption that allows no time-covariate interactions in the logarithm of hazard rate. Estimation is carried out by an expectation-maximization algorithm on maximizing a penalized likelihood. For inferential purpose, we apply the Louis formula to obtain point-wise confidence intervals for noncured probability and hazard rate. Asymptotic convergence rates of our function estimates are established. We then evaluate the proposed method by extensive simulations. We analyze the survival data from a melanoma study and find interesting patterns for this study.     

Breast Cancer Survival in Germany: A Population-Based High Resolution Study from Saarland

Population-based survival studies of breast cancer patients are commonly restricted to age- and stage-specific analyses. This study from Germany aimed at extending available population-based survival data on further prognostic cancer characteristics such as tumor grade, hormone receptor status and human epidermal growth factor receptor type 2 (HER2/neu) expression. Data from the population-based Saarland Cancer Registry including female patients diagnosed with invasive breast cancer between 2000 and 2009 were included. Period analysis methodology and regression modelling were used to obtain estimates of 5-year relative survival and tumor related excess risks in 2005-2009. Overall age standardized 5-year relative survival was 83%. In addition to age and stage, tumor grade and hormone receptor status were independent predictors of 5-year relative survival. Detailed analyses by age, stage, morphology, tumor grade, hormone receptor status and HER2/neu expression consistently revealed lower survival of patients with high grade, hormone receptor negative or HER2/neu positive cancers and patients aged 70 years or older. This high resolution study extends available population-based survival data of breast cancer patients. Particular efforts should be made to overcome the persisting large survival deficits, which were observed for elderly patients in all clinical subgroups.     

Socioeconomic status and cancer survival in Brazil: Analysis of population data from the municipalities of Aracaju and Curitiba, 1996–2012

IntroductionThe association between socioeconomic status and cancer prognosis has been demonstrated in several countries. Despite the existence of indirect evidence of this phenomenon in Brazil, few studies in this regard are available.ObjectivesThe objective of the present study is to analyse socioeconomic related survival gaps for patients diagnosed with breast, cervical, lung, prostate, and colorectal cancer in the cities of Aracaju (SE) and Curitiba (PR).MethodsUsing population-based data, we estimated net survival by tumour site, year of diagnosis, socioeconomic status and local of residence. Net survival estimation was done with multilevel parametric model allowing flexible spline functions do estimate excess mortality hazards.Results28,005 cases were included in survival analysis. Five-year net survival showed positive association with SES. Intermunicipal survival gaps favouring Aracaju where prominent for breast (reaching 16,1% in 5 years)ObjectivesStudy the impact of socioeconomic factors on cancer survival in two Brazilian capitals. Methods: Survival analysis using population-based cancer data including patients diagnosed with breast, lung, prostate, cervical and colorectal cancer between 1996 and 2012 in Aracaju and Curitiba. Outcomes were excessive mortality hazard (EMH) and 5- and 8-years net survival (NS). The association of race/skin color and socioeconomic level (SES) with EMH and net survival were analyzed using a multilevel regression model with flexible splines.Results28,005 cases were included, 6636 from Aracaju and 21,369 from Curitiba. NS for all diseases studied increased more prominently for Curitiba population. We observed NS gap between the populations of Aracaju and Curitiba that increased or remained stable during the study period, with emphasis on the growth of the difference in NS of lung and colon cancer (among men). Only for cervical cancer and prostate cancer there was a reduction in the intermunicipal gaps. 5-year NS for breast cancer in Aracaju ranged from 55.2% to 73.4% according to SES. In Curitiba this variation was from 66.5% to 83.8%.ConclusionThe results of the present study suggests widening of socioeconomic and regional inequalities in the survival of patients with colorectal, breast, cervical, lung and prostate cancers in Brazil during the 1990 s and 2000 s     

Partner status and survival after cancer: A competing risks analysis

ObjectiveThe survival benefits of having a partner for all cancers combined is well recognized, however its prognostic importance for individual cancer types, including competing mortality causes, is less clear. This study was undertaken to quantify the impact of partner status on survival due to cancer-specific and competing mortality causes.MethodsData were obtained from the population-based Queensland Cancer Registry on 176,050 incident cases of ten leading cancers diagnosed in Queensland (Australia) from 1996 to 2012. Flexible parametric competing-risks models were used to estimate cause-specific hazards and cumulative probabilities of death, adjusting for age, stage (breast, colorectal and melanoma only) and stratifying by sex.ResultsBoth unpartnered males and females had higher total cumulative probability of death than their partnered counterparts for each site. For example, the survival disadvantage for unpartnered males ranged from 3% to 30% with higher mortality burden from both the primary cancer and competing mortality causes. The cause-specific age-adjusted hazard ratios were also consistent with patients without a partner having increased mortality risk although the specific effect varied by site, sex and cause of death. For all combined sites, unpartnered males had a 46%, 18% and 44% higher risk of cancer-specific, other cancer and non-cancer mortality respectively with similar patterns for females. The higher mortality risk persisted after adjustment for stage.ConclusionsIt is important to better understand the mechanisms by which having a partner is beneficial following a cancer diagnosis, so that this can inform improvements in cancer management for all people with cancer.     

Multilevel Mixture Cure Models with Random Effects

This paper extends the multilevel survival model by allowing the existence of cured fraction in the model. Random effects induced by the multilevel clustering structure are specified in the linear predictors in both hazard function and cured probability parts. Adopting the generalized linear mixed model (GLMM) approach to formulate the problem, parameter estimation is achieved by maximizing a best linear unbiased prediction (BLUP) type log‐likelihood at the initial step of estimation, and is then extended to obtain residual maximum likelihood (REML) estimators of the variance component. The proposed multilevel mixture cure model is applied to analyze the (i) child survival study data with multilevel clustering and (ii) chronic granulomatous disease (CGD) data on recurrent infections as illustrations. A simulation study is carried out to evaluate the performance of the REML estimators and assess the accuracy of the standard error estimates.     

A nonparametric mixture model for cure rate estimation

Nonparametric methods have attracted less attention than their parametric counterparts for cure rate analysis. In this paper, we study a general nonparametric mixture model. The proportional hazards assumption is employed in modeling the effect of covariates on the failure time of patients who are not cured. The EM algorithm, the marginal likelihood approach, and multiple imputations are employed to estimate parameters of interest in the model. This model extends models and improves estimation methods proposed by other researchers. It also extends Cox's proportional hazards regression model by allowing a proportion of event-free patients and investigating covariate effects on that proportion. The model and its estimation method are investigated by simulations. An application to breast cancer data, including comparisons with previous analyses using a parametric model and an existing nonparametric model by other researchers, confirms the conclusions from the parametric model but not those from the existing nonparametric model.     

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.     

Role of age and tumour stage in the temporal pattern of 'cure' from stomach cancer: a population-based study in Osaka, Japan

Objectives: To evaluate progress in stomach cancer care in Japan since 1975. Design: Population-based study of data extracted from the Osaka Cancer Registry. Setting: Population-based cancer registry in the area of Osaka Prefecture. Participants: All 66,032 cases diagnosed with a stomach cancer in Osaka Prefecture, Japan between 1975 and 2000 and registered in the Osaka Cancer Registry. Main outcome measures: ‘Cure’ fraction and median survival time for ‘uncured’ patients were estimated with multivariable mixture ‘cure’ model. The role played by age and stage at diagnosis on the changes in ‘cure’ parameters between 1975 and 2000 was evaluated. Missing stage was handled by multiple imputation approach. Results: More than 50% of the patients diagnosed with a stomach cancer in 1996–2000 were estimated ‘cured’ from their cancer, corresponding to a 20% increase since 1975–1980. Median survival time for ‘uncured’ patients however remained unchanged at about 8 months. ‘Cure’ fraction was over 85% for localised tumours and 30% for regional tumours, but stayed as low as 2.5% for distant metastatic cancers. Improvement was underestimated by about 10% because of ageing of cancer patients. Changes in stage distribution explained up to 40% of the increase in ‘cure’ fraction among men and up to 13% in women. Overdiagnosis was unlikely to play any role in these patterns. Conclusions: ‘Cure’ fraction from stomach cancer dramatically increased in Osaka, Japan since 1975, partly because of earlier stage at diagnosis, but mostly due to improvement in treatment of stomach cancer patients. This study, based on a leading country in term of stomach cancer management, provides insightful results for other countries in which ‘cure’ fraction is usually much lower.     

Semiparametric estimation in copula models for bivariate sequential survival times

Sequentially observed survival times are of interest in many studies but there are difficulties in analyzing such data using nonparametric or semiparametric methods. First, when the duration of followup is limited and the times for a given individual are not independent, induced dependent censoring arises for the second and subsequent survival times. Non-identifiability of the marginal survival distributions for second and later times is another issue, since they are observable only if preceding survival times for an individual are uncensored. In addition, in some studies a significant proportion of individuals may never have the first event. Fully parametric models can deal with these features, but robustness is a concern. We introduce a new approach to address these issues. We model the joint distribution of the successive survival times by using copula functions, and provide semiparametric estimation procedures in which copula parameters are estimated without parametric assumptions on the marginal distributions. This provides more robust estimates and checks on the fit of parametric models. The methodology is applied to a motivating example involving relapse and survival following colon cancer treatment.