Application of the theory of finite mixtures for the estimation of 'cure' rates of treated cancer patients

Penalized likelihood estimation of a mixture cure Cox model with partly interval censoring-An application to thin melanoma

Time‐to‐event data in medical studies may involve some patients who are cured and will never experience the event of interest. In practice, those cured patients are right censored. However, when data contain a cured fraction, standard survival methods such as Cox proportional hazards models can produce biased results and therefore misleading interpretations. In addition, for some outcomes, the exact time of an event is not known; instead an interval of time in which the event occurred is recorded. This article proposes a new computational approach that can deal with both the cured fraction issues and the interval censoring challenge. To do so, we extend the traditional mixture cure Cox model to accommodate data with partly interval censoring for the observed event times. The traditional method for estimation of the model parameters is based on the expectation‐maximization (EM) algorithm, where the log‐likelihood is maximized through an indirect complete data log‐likelihood function. We propose in this article an alternative algorithm that directly optimizes the log‐likelihood function. Extensive Monte Carlo simulations are conducted to demonstrate the performance of the new method over the EM algorithm. The main advantage of the new algorithm is the generation of asymptotic variance matrices for all the estimated parameters. The new method is applied to a thin melanoma dataset to predict melanoma recurrence. Various inferences, including survival and hazard function plots with point‐wise confidence intervals, are presented. An R package is now available at Github and will be uploaded to R CRAN.

Exploring the existence of a stayer population with mover-stayer counting process models: application to joint damage in psoriatic arthritis

Many psoriatic arthritis patients do not progress to permanent joint damage in any of the 28 hand joints, even under prolonged follow‐up. This has led several researchers to fit models that estimate the proportion of stayers (those who do not have the propensity to experience the event of interest) and to characterize the rate of developing damaged joints in the movers (those who have the propensity to experience the event of interest). However, when fitted to the same data, the paper demonstrates that the choice of model for the movers can lead to widely varying conclusions on a stayer population, thus implying that, if interest lies in a stayer population, a single analysis should not generally be adopted. The aim of the paper is to provide greater understanding regarding estimation of a stayer population by comparing the inferences, performance and features of multiple fitted models to real and simulated data sets. The models for the movers are based on Poisson processes with patient level random effects and/or dynamic covariates, which are used to induce within‐patient correlation, and observation level random effects are used to account for time varying unobserved heterogeneity. The gamma, inverse Gaussian and compound Poisson distributions are considered for the random effects.

The application of cure models in the presence of competing risks: a tool for improved risk communication in population-based cancer patient survival

Quantifying cancer patient survival from the perspective of cure is clinically relevant. However, most cure models estimate cure assuming no competing causes of death. We use a relative survival framework to demonstrate how flexible parametric cure models can be used in combination with competing-risks theory to incorporate noncancer deaths. Under a model that incorporates statistical cure, we present the probabilities that cancer patients (1) have died from their cancer, (2) have died from other causes, (3) will eventually die from their cancer, or (4) will eventually die from other causes, all as a function of time since diagnosis. We further demonstrate how conditional probabilities can be used to update the prognosis among survivors (eg, at 1 or 5 years after diagnosis) by summarizing the proportion of patients who will not die from their cancer. The proposed method is applied to Swedish population-based data for persons diagnosed with melanoma, colon cancer, or acute myeloid leukemia between 1973 and 2007.

A cure rate survival model under a hybrid latent activation scheme

In lifetimes studies, the occurrence of an event (such as tumor detection or death) might be caused by one of many competing causes. Moreover, both the number of causes and the time-to-event associated with each cause are not usually observable. The number of causes can be zero, corresponding to a cure fraction. In this article, we propose a method of estimating the numerical characteristics of unobservable stages (such as initiation, promotion and progression) of carcinogenesis from data on tumor size at detection in the presence of latent competing causes. To this end, a general survival model for spontaneous carcinogenesis under a hybrid latent activation scheme has been developed to allow for a simple pattern of the dynamics of tumor growth. It is assumed that a tumor becomes detectable when its size attains some threshold level (proliferation of tumorais cells (or descendants) generated by the malignant cell), which is treated as a random variable. We assume the number of initiated cells and the number of malignant cells (competing causes) both to follow weighted Poisson distributions. The advantage of this model is that it incorporates into the analysis characteristics of the stage of tumor progression as well as the proportion of initiated cells that had been ‘promoted’ to the malignant ones and the proportion of malignant cells that die before tumor induction. The lifetimes corresponding to each competing cause are assumed to follow a Weibull distribution. Parameter estimation of the proposed model is discussed through the maximum likelihood estimation method. A simulation study has been carried out in order to examine the coverage probabilities of the confidence intervals. Finally, we illustrate the usefulness of the proposed model by applying it to a real data involving malignant melanoma.

Estimating the personal cure rate of cancer patients using population-based grouped cancer survival data

Cancer patients are subject to multiple competing risks of death and may die from causes other than the cancer diagnosed. The probability of not dying from the cancer diagnosed, which is one of the patients' main concerns, is sometimes called the 'personal cure' rate. Two approaches of modelling competing-risk survival data, namely the cause-specific hazards approach and the mixture model approach, have been used to model competing-risk survival data. In this article, we first show the connection and differences between crude cause-specific survival in the presence of other causes and net survival in the absence of other causes. The mixture survival model is extended to population-based grouped survival data to estimate the personal cure rate. Using the colorectal cancer survival data from the Surveillance, Epidemiology and End Results Programme, we estimate the probabilities of dying from colorectal cancer, heart disease, and other causes by age at diagnosis, race and American Joint Committee on Cancer stage.

A finite mixture survival model to characterize risk groups of neuroblastoma

Neuroblastoma is a childhood cancer with patients experiencing heterogeneous survival outcomes despite aggressive treatment. Disease outcomes range from early death to spontaneous regression of the tumor followed by cure. Owing to this heterogeneity, it is of interest to identify patients with similar types of neuroblastoma so that specific types of treatment can be developed. Oncologists are especially interested in identifying patients who will be cured so that the minimum amount of a potentially toxic treatment can be given to this group of patients. We analyze a large cohort of neuroblastoma patients and develop a finite mixture model that uses covariates to predict the probability of being in a cure group or other (one or more) risk groups. A prediction method is developed that uses the estimated probabilities to assign a patient to different risk groups. The robustness of the model and the prediction method is examined via simulation by looking at misclassification rates under misspecified models.

Bayesian Hierarchical Multiresolution Hazard Model for the Study of Time-Dependent Failure Patterns in Early Stage Breast Cancer

The multiresolution estimator, developed originally in engineering applications as a wavelet-based method for density estimation, has been recently extended and adapted for estimation of hazard functions (Bouman et al. 2005, 2007). Using the multiresolution hazard (MRH) estimator in the Bayesian framework, we are able to incorporate any a priori desired shape and amount of smoothness in the hazard function. The MRH method's main appeal is in its relatively simple estimation and inference procedures, making it possible to obtain simultaneous confidence bands on the hazard function over the entire time span of interest. Moreover, these confidence bands properly reflect the multiple sources of uncertainty, such as multiple centers or heterogeneity in the patient population. Also, rather than the commonly employed approach of estimating covariate effects and the hazard function separately, the Bayesian MRH method estimates all of these parameters jointly, thus resulting in properly adjusted inference about any of the quantities.In this paper, we extend the previously proposed MRH methods (Bouman et al. 2005, 2007) into the hierarchical multiresolution hazard setting (HMRH), to accommodate the case of separate hazard rate functions within each of several strata as well as some common covariate effects across all strata while accounting for within-stratum correlation. We apply this method to examine patterns of tumor recurrence after treatment for early stage breast cancer, using data from two large-scale randomized clinical trials that have substantially influenced breast cancer treatment standards. We implement the proposed model to estimate the recurrence hazard and explore how the shape differs between patients grouped by a key tumor characteristic (estrogen receptor status) and treatment types, after adjusting for other important patient characteristics such as age, tumor size and progesterone level. We also comment on whether the hazards exhibit nonmonotonic patterns consistent with recent hypotheses suggesting multiple hazard change-points at specific time landmarks.

Thirty-year follow-up of chemo/hormonal therapy in node-positive breast cancer

Results of a thirty-year follow-up of a clinical trial of chemo-hormonal therapy are reported. Eligible patients had recently diagnosed operable breast cancer, positive lymph nodes, no previous history of cancer, age less than 76 years, and no evidence of metastatic disease. A total of 311 patients were stratified by estrogen receptor (ER) status and number of axillary nodes involved with tumor. After stratification, patients were randomly assigned to one of three treatment regimens: cyclophosphamide, methotrexate and 5-fluorouracil (CMF) for 1 year; CMF chemotherapy combined with anti-estrogen therapy (tamoxifen) for 1 year; or CMF plus tamoxifen with BCG during the second year. The endpoint of the trial was a first recurrence. Factors measured at diagnosis and used in the analyses were age, body mass index, ER status, menopausal status, number of positive nodes, tumor diameter, Charlson comorbidity index, socioeconomic status, and race. Causes of death and incidence of other cancer primaries were obtained from death certificates and medical records. Patients treated with tamoxifen had a marginally longer disease-free survival (hazard ratio (HR)=0.83, 95% CI identical with [0.66, 1.04]) and statistically significant longer overall survival (HR=0.77, 95% CI identical with [0.63, 0.96]) that decreased with time. Incidence of other primary cancers and causes of death were similar for the two treatment groups. The addition of 1 year of tamoxifen to CMF therapy provides an early disease-free and overall survival advantage; however long-term effects are negligible. Similarly, the survival advantage of patients diagnosed with ER+ tumors persists for the first two decades after diagnosis.

A new parametric model for survival data with long-term survivors

We develop a new parametric model using the three-parameter Burr XII distribution for the analysis of survival data with long-term survivors, which includes the previous Weibull mixture model as a special case. The new model is applied to the analysis of a set of leukaemia data for which previous attempts in the literature using traditional parametric models were unsatisfactory due to lack of fit. It is shown that the new model improves the fit to the leukaemia data significantly and is thus capable of providing more credible answers to a variety of statistical inference problems that are of interest to medical researchers and practitioners.

Increased Infertility With Age in Men and Women

OBJECTIVE

To estimate the effects of aging on the percentage of outwardly healthy couples who are sterile (completely unable to conceive without assisted reproduction) or infertile (unable to conceive within a year of unprotected intercourse).

METHODS

A prospective fecundability study was conducted in a sample of 782 couples recruited from 7 European centers for natural family planning. Women aged 18-40 years were eligible. Daily intercourse records were used to adjust for timing and frequency of intercourse when estimating the per-menstrual-cycle probability of conception. The number of menstrual cycles required to conceive a clinical pregnancy and the probability of sterility and infertility were derived from the estimated fecundability distributions for men and women of different ages.

RESULTS

Sterility was estimated at about 1%; this percent did not change with age. The percentage infertility was estimated at 8% for women aged 19-26 years, 13-14% for women aged 27-34 years and 18% for women aged 35-39 years. Starting in the late 30s, male age was an important factor, with the percentage failing to conceive within 12 cycles increasing from an estimated 18-28% between ages 35 and 40 years. The estimated percentage of infertile couples that would be able to conceive after an additional 12 cycles of trying varied from 43-63% depending on age.

CONCLUSION

Increased infertility in older couples is attributable primarily to declines in fertility rates rather than to absolute sterility. Many infertile couples will conceive if they try for an additional year.

Modelling the distribution of ischaemic stroke-specific survival time using an EM-based mixture approach with random effects adjustment

A two-component survival mixture model is proposed to analyse a set of ischaemic stroke-specific mortality data. The survival experience of stroke patients after index stroke may be described by a subpopulation of patients in the acute condition and another subpopulation of patients in the chronic phase. To adjust for the inherent correlation of observations due to random hospital effects, a mixture model of two survival functions with random effects is formulated. Assuming a Weibull hazard in both components, an EM algorithm is developed for the estimation of fixed effect parameters and variance components. A simulation study is conducted to assess the performance of the two-component survival mixture model estimators. Simulation results confirm the applicability of the proposed model in a small sample setting.

Estimating Cure Rates From Survival Data: An Alternative to Two-Component Mixture Models

This article considers the utility of the bounded cumulative hazard model in cure rate estimation, which is an appealing alternative to the widely used two-component mixture model. This approach has the following distinct advantages: (1) It allows for a natural way to extend the proportional hazards regression model, leading to a wide class of extended hazard regression models. (2) In some settings the model can be interpreted in terms of biologically meaningful parameters. (3) The model structure is particularly suitable for semiparametric and Bayesian methods of statistical inference. Notwithstanding the fact that the model has been around for less than a decade, a large body of theoretical results and applications has been reported to date. This review article is intended to give a big picture of these modeling techniques and associated statistical problems. These issues are discussed in the context of survival data in cancer.

An EM-based semi-parametric mixture model approach to the regression analysis of competing-risks data

We consider a mixture model approach to the regression analysis of competing-risks data. Attention is focused on inference concerning the effects of factors on both the probability of occurrence and the hazard rate conditional on each of the failure types. These two quantities are specified in the mixture model using the logistic model and the proportional hazards model, respectively. We propose a semi-parametric mixture method to estimate the logistic and regression coefficients jointly, whereby the component-baseline hazard functions are completely unspecified. Estimation is based on maximum likelihood on the basis of the full likelihood, implemented via an expectation-conditional maximization (ECM) algorithm. Simulation studies are performed to compare the performance of the proposed semi-parametric method with a fully parametric mixture approach. The results show that when the component-baseline hazard is monotonic increasing, the semi-parametric and fully parametric mixture approaches are comparable for mildly and moderately censored samples. When the component-baseline hazard is not monotonic increasing, the semi-parametric method consistently provides less biased estimates than a fully parametric approach and is comparable in efficiency in the estimation of the parameters for all levels of censoring. The methods are illustrated using a real data set of prostate cancer patients treated with different dosages of the drug diethylstilbestrol.

A bivariate cure-mixture approach for modeling familial association in diseases

For modeling correlation in familial diseases with variable ages at onset, we propose a bivariate model that incorporates two types of pairwise association, one between the lifetime risk or the overall susceptibility of two individuals and one between the ages at onset between two susceptible individuals. For estimation, we consider a two-stage estimation procedure similar to that of Shih (1998, Biometrics 54, 1115-1128). We evaluate the properties of the estimators through simulations and compare the performance with that from a bivariate survival model that allows correlation between ages at onset only. We apply the methodology to breast cancer using the kinship data from the Washington Ashkenazi Study. We also discuss potential applications of the proposed method in the area of cure modeling.

Long-term survivor mixture model with random effects: application to a multi-centre clinical trial of carcinoma

A mixture model incorporating long-term survivors has been adopted in the field of biostatistics where some individuals may never experience the failure event under study. The surviving fractions may be considered as cured. In most applications, the survival times are assumed to be independent. However, when the survival data are obtained from a multi-centre clinical trial, it is conceived that the environmental conditions and facilities shared within clinic affects the proportion cured as well as the failure risk for the uncured individuals. It necessitates a long-term survivor mixture model with random effects. In this paper, the long-term survivor mixture model is extended for the analysis of multivariate failure time data using the generalized linear mixed model (GLMM) approach. The proposed model is applied to analyse a numerical data set from a multi-centre clinical trial of carcinoma as an illustration. Some simulation experiments are performed to assess the applicability of the model based on the average biases of the estimates formed.

Mixture models for cancer survival analysis: application to population-based data with covariates

The interest in estimating the probability of cure has been increasing in cancer survival analysis as the curability of many cancer diseases is becoming a reality. Mixture survival models provide a way of modelling time to death when cure is possible, simultaneously estimating death hazard of fatal cases and the proportion of cured case. In this paper we propose an application of a parametric mixture model to relative survival rates of colon cancer patients from the Finnish population-based cancer registry, and including major survival determinants as explicative covariates. Disentangling survival into two different components greatly facilitates the analysis and the interpretation of the role of prognostic factors on survival patterns. For example, age plays a different role in determining, from one side, the probability of cure, and, from the other side, the life expectancy of fatal cases. The results support the hypothesis that observed survival trends are really due to a real prognostic gain for more recently diagnosed patients.

A generalized F mixture model for cure rate estimation

Cure rate estimation is an important issue in clinical trials for diseases such as lymphoma and breast cancer and mixture models are the main statistical methods. In the last decade, mixture models under different distributions, such as exponential, Weibull, log-normal and Gompertz, have been discussed and used. However, these models involve stronger distributional assumptions than is desirable and inferences may not be robust to departures from these assumptions. In this paper, a mixture model is proposed using the generalized F distribution family. Although this family is seldom used because of computational difficulties, it has the advantage of being very flexible and including many commonly used distributions as special cases. The generalised F mixture model can relax the usual stronger distributional assumptions and allow the analyst to uncover structure in the data that might otherwise have been missed. This is illustrated by fitting the model to data from large-scale clinical trials with long follow-up of lymphoma patients. Computational problems with the model and model selection methods are discussed. Comparison of maximum likelihood estimates with those obtained from mixture models under other distributions are included.

The follicular non-Hodgkin's lymphomas--I. The possibility of cure

The follicular lymphomas pursue an indolent course in many patients. Long-term follow-up in large series is therefore necessary to establish whether cure is taking place, and if so, at what stage in the dissemination of the disease process it becomes unlikely. The time to, and site of relapse, together with its impact on survival has been studied in 398 patients entered into the British National Lymphoma Investigation limited and disseminated disease trials between 1974 and 1980. Relapse data were compared with various models to obtain maximum likelihood estimates of the proportions permanently remaining relapse-free following treatment. Long-term relapse-free survival was observed in 54.8 +/- 14.9% (95% CI) of patients at 15 years with Ann Arbor stage I disease, 29.2 +/- 13.6% in patients with stage II disease, 18.1 +/- 6.6% with stage III and 13.0 +/- 5.9% with IV disease. Relapse time-course data for all trial arms conform closely to lognormal distributions allowing maximum likelihood estimates of proportions remaining permanently relapse-free to be derived. Using this methodology, over a quarter of patients treated with involved radiotherapy alone or radiotherapy plus 6 months of chlorambucil in the limited disease (Ann Arbor stage I and II) trial are unlikely to relapse at any time in the future. Over 10% of patients treated in the disseminated disease trials with disease classified as Ann Arbor stage III are also statistically unlikely to relapse. The finding that a proportion of patients is statistically unlikely to experience a clinically obvious relapse is consistent with clinical cure. It is especially interesting that a small proportion of patients with disseminated disease and treated by chemotherapy have fallen into this category, but additional data are required to know at what point statistical cure becomes unlikely. Whether "clinical cure" is the same as "pathological cure" in this disease remains uncertain.

Using a mixture model to predict the occurrence of diabetic retinopathy

Diabetes mellitus is a common condition which has several serious complications associated with it. In this paper a mixture model, based on one previously used to predict the onset of AIDS, is used to predict the onset of one of these complications, diabetic retinopathy, the major cause of adult blindness in the U.K. This model differs from the previous AIDS model by introducing covariates into the model and using a wider choice of mixture distributions. The fit and distributional assumptions of the model are then discussed for this example. The model is fitted to the data by maximum likelihood. It is important that the training set contains balanced numbers of individuals with and without retinopathy.

Estimating the proportion of immunes in censored samples: a simulation study

We review currently known results concerning the estimation of an 'immune' or 'cured' proportion, and testing for the presence of immunes, in censored survival data, suggesting that a firm theoretical foundation now exists for analysis. Two types of estimators, parametric and non-parametric, are discussed and compared with respect to their theoretical properties, and, by simulation, with respect to their small sample behaviour. Both estimators have advantages and drawbacks, but together provide powerful tools for the perceptive analysis of survival data with, or even without, immune individuals.

On the role of finite mixture models in survival analysis

In this paper we review the role of finite mixture models in the field of survival analysis. Finite mixture models can be used to analyse failure-time data in a variety of situations. In particular, they provide a way of modelling time to failure in the case of competing risks.

Parametric versus non-parametric methods for estimating cure rates based on censored survival data

If a patient's failure time is incorrectly recorded as being too early, the correction will lower the plateau of the Kaplan-Meier curve and, hence, the associated estimated cure rate. Implications of this counter-intuitive observation are discussed. In addition, a parametric approach, based on the Gompertz distribution, to the problem of cure rate estimation is presented.