smcure: an R-package for estimating semiparametric mixture cure models

CureAuxSP: An R package for estimating mixture cure models with auxiliary survival probabilities

BACKGROUND AND OBJECTIVE

There is a rising interest in exploiting aggregate information from external medical studies to enhance the statistical analysis of a modestly sized internal dataset. Currently available software packages for analyzing survival data with a cure fraction ignore the potentially available auxiliary information. This paper aims at filling this gap by developing a new R package CureAuxSP that can include subgroup survival probabilities extracted outside into an interested internal survival dataset.

METHODS

The newly developed R package CureAuxSP provides an efficient approach for information synthesis under the mixture cure models, including Cox proportional hazards mixture cure model and the accelerated failure time mixture cure model as special cases. It focuses on synthesizing subgroup survival probabilities at multiple time points and the underlying method development lies in the control variate technique. Evaluation of homogeneity assumption based on a test statistic can be automatically carried out by our package and if heterogeneity does exist, the original outputs can be further refined adaptively.

RESULTS

The R package CureAuxSP provides a main function SMC.AxuSP() that helps us adaptively incorporate external subgroup survival probabilities into the analysis of an internal survival data. We also provide another function Print.SMC.AuxSP() for printing the results with a better presentation. Detailed usages are described, and implementations are illustrated with numerical examples, including a simulated dataset with a well-designed data generating process and a real breast cancer dataset. Substantial efficiency gain can be observed by our results.

CONCLUSIONS

Our R package CureAuxSP can make the wide applications of utilizing auxiliary information possible. It is anticipated that the performance of mixture cure models can be improved for the survival data with a cure fraction, especially for those with small sample sizes.

On variable selection in a semiparametric AFT mixture cure model

In clinical studies, one often encounters time-to-event data that are subject to right censoring and for which a fraction of the patients under study never experience the event of interest. Such data can be modeled using cure models in survival analysis. In the presence of cure fraction, the mixture cure model is popular, since it allows to model probability to be cured (called the incidence) and the survival function of the uncured individuals (called the latency). In this paper, we develop a variable selection procedure for the incidence and latency parts of a mixture cure model, consisting of a logistic model for the incidence and a semiparametric accelerated failure time model for the latency. We use a penalized likelihood approach, based on adaptive LASSO penalties for each part of the model, and we consider two algorithms for optimizing the criterion function. Extensive simulations are carried out to assess the accuracy of the proposed selection procedure. Finally, we employ the proposed method to a real dataset regarding heart failure patients with left ventricular systolic dysfunction.

Estimating causal effects in observational studies for survival data with a cure fraction using propensity score adjustment

In observational studies, covariates are often confounding factors for treatment assignment. Such covariates need to be adjusted to estimate the causal treatment effect. For observational studies with survival outcomes, it is usually more challenging to adjust for the confounding covariates for causal effect estimation because of censoring. The challenge becomes even thornier when there exists a nonignorable cure fraction in the population. In this paper, we propose a causal effect estimation approach in observational studies for survival data with a cure fraction. We extend the absolute treatment effects on survival outcomes-including the restricted average causal effect and SPCE-to survival outcomes with cure fractions, and construct the corresponding causal effect estimators based on propensity score stratification. We prove the asymptotic properties of the proposed estimators and conduct simulation studies to evaluate their performances. As an illustration, the method is applied to a stomach cancer study.

A support vector machine-based cure rate model for interval censored data

The mixture cure rate model is the most commonly used cure rate model in the literature. In the context of mixture cure rate model, the standard approach to model the effect of covariates on the cured or uncured probability is to use a logistic function. This readily implies that the boundary classifying the cured and uncured subjects is linear. In this article, we propose a new mixture cure rate model based on interval censored data that uses the support vector machine to model the effect of covariates on the uncured or the cured probability (i.e. on the incidence part of the model). Our proposed model inherits the features of the support vector machine and provides flexibility to capture classification boundaries that are nonlinear and more complex. The latency part is modeled by a proportional hazards structure with an unspecified baseline hazard function. We develop an estimation procedure based on the expectation maximization algorithm to estimate the cured/uncured probability and the latency model parameters. Our simulation study results show that the proposed model performs better in capturing complex classification boundaries when compared to both logistic regression-based and spline regression-based mixture cure rate models. We also show that our model's ability to capture complex classification boundaries improve the estimation results corresponding to the latency part of the model. For illustrative purpose, we present our analysis by applying the proposed methodology to the NASA's Hypobaric Decompression Sickness Database.

Functional proportional hazards mixture cure model with applications in cancer mortality in NHANES and post ICU recovery

We develop a functional proportional hazards mixture cure model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the expectation-maximization algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness penalty. Simulation studies illustrate a satisfactory performance of the proposed method in accurately estimating the model parameters and the baseline survival function. Finally, the clinical potential of the model is demonstrated in two real data examples that incorporate rich high-dimensional biomedical signals as functional covariates measured at the baseline and constitute novel domains to apply cure survival models in contemporary medical situations. In particular, we analyze (i) minute-by-minute physical activity data from the National Health And Nutrition Examination Survey 2003-2006 to study the association between diurnal patterns of physical activity at baseline and all cancer mortality through 2019 while adjusting for other biological factors; (ii) the impact of daily functional measures of disease severity collected in the intensive care unit on post intensive care unit recovery and mortality event. Our findings provide novel epidemiological insights into the association between daily patterns of physical activity and cancer mortality. Software implementation and illustration of the proposed estimation method are provided in R.

On the integration of decision trees with mixture cure model

The mixture cure model is widely used to analyze survival data in the presence of a cured subgroup. Standard logistic regression-based approaches to model the incidence may lead to poor predictive accuracy of cure, specifically when the covariate effect is non-linear. Supervised machine learning techniques can be used as a better classifier than the logistic regression due to their ability to capture non-linear patterns in the data. However, the problem of interpret-ability hangs in the balance due to the trade-off between interpret-ability and predictive accuracy. We propose a new mixture cure model where the incidence part is modeled using a decision tree-based classifier and the proportional hazards structure for the latency part is preserved. The proposed model is very easy to interpret, closely mimics the human decision-making process, and provides flexibility to gauge both linear and non-linear covariate effects. For the estimation of model parameters, we develop an expectation maximization algorithm. A detailed simulation study shows that the proposed model outperforms the logistic regression-based and spline regression-based mixture cure models, both in terms of model fitting and evaluating predictive accuracy. An illustrative example with data from a leukemia study is presented to further support our conclusion.

Prognostic Factors Analysis of Metastatic Recurrence in Cervical Carcinoma Patients Treated with Definitive Radiotherapy: A Retrospective Study Using Mixture Cure Model

OBJECTIVES

This study aims to identify prognostic factors associated with metastatic recurrence-free survival of cervical carcinoma (CC) patients treated with radical radiotherapy and assess the cure probability of radical radiotherapy from metastatic recurrence.

METHODS

Data were from 446 cervical carcinoma patients with radical radiotherapy for an average follow up of 3.96 years. We applied a mixture cure model to investigate the association between metastatic recurrence and prognostic factors and the association between noncure probability and factors, respectively. A nonparametric test of cure probability under the framework of a mixture cure model was used to examine the significance of cure probability of the definitive radiotherapy treatment. Propensity-score-matched (PSM) pairs were generated to reduce bias in subgroup analysis.

RESULTS

Patients in advanced stages (p = 0.005) and those with worse treatment responses in the 3rd month (p = 0.004) had higher metastatic recurrence rates. Nonparametric tests of the cure probability showed that 3-year cure probability from metastatic recurrence was significantly larger than 0, and 5-year cure probability was significantly larger than 0.7 but no larger than 0.8. The empirical cure probability by mixture cure model was 79.2% (95% CI: 78.6-79.9%) for the entire study population, and the overall median metastatic recurrence time for uncured patients (patients susceptible to metastatic recurrence) was 1.60 (95% CI: 1.51-1.69) years. Locally advanced/advanced stage was a risk factor but non-significant against the cure probability (OR = 1.078, p = 0.088). The interaction of age and activity of radioactive source were statistically significant in the incidence model (OR = 0.839, p = 0.025). In subgroup analysis, compared with high activity of radioactive source (HARS), low activity of radioactive source (LARS) significantly contributed to a 16.1% higher cure probability for patients greater than 53 years old, while cure probability was 12.2% lower for the younger patients.

CONCLUSIONS

There was statistically significant evidence in the data showing the existence of a large amount of patients cured by the definitive radiotherapy treatment. HARS is a protective factor against metastatic recurrence for uncured patients, and young patients tend to benefit more than the elderly from the HARS treatment.

The estimation of long and short term survival time and associated factors of HIV patients using mixture cure rate models

BACKGROUND

HIV is one of the deadliest epidemics and one of the most critical global public health issues. Some are susceptible to die among people living with HIV and some survive longer. The aim of the present study is to use mixture cure models to estimate factors affecting short- and long-term survival of HIV patients.

METHODS

The total sample size was 2170 HIV-infected people referred to the disease counseling centers in Kermanshah Province, in the west of Iran, from 1998 to 2019. A Semiparametric PH mixture cure model and a mixture cure frailty model were fitted to the data. Also, a comparison between these two models was performed.

RESULTS

Based on the results of the mixture cure frailty model, antiretroviral therapy, tuberculosis infection, history of imprisonment, and mode of HIV transmission influenced short-term survival time (p-value < 0.05). On the other hand, prison history, antiretroviral therapy, mode of HIV transmission, age, marital status, gender, and education were significantly associated with long-term survival (p-value < 0.05). The concordance criteria (K-index) value for the mixture cure frailty model was 0.65 whereas for the semiparametric PH mixture cure model was 0.62.

CONCLUSION

This study showed that the frailty mixture cure models is more suitable in the situation where the studied population consisted of two groups, susceptible and non-susceptible to the event of death. The people with a prison history, who received ART treatment, and contracted HIV through injection drug users survive longer. Health professionals should pay more attention to these findings in HIV prevention and treatment.

Penalized maximum likelihood inference under the mixture cure model in sparse data

INTRODUCTION

When a study sample includes a large proportion of long-term survivors, mixture cure (MC) models that separately assess biomarker associations with long-term recurrence-free survival and time to disease recurrence are preferred to proportional-hazards models. However, in samples with few recurrences, standard maximum likelihood can be biased.

OBJECTIVE AND METHODS

We extend Firth-type penalized likelihood (FT-PL) developed for bias reduction in the exponential family to the Weibull-logistic MC, using the Jeffreys invariant prior. Via simulation studies based on a motivating cohort study, we compare parameter estimates of the FT-PL method to those by ML, as well as type 1 error (T1E) and power obtained using likelihood ratio statistics.

RESULTS

In samples with relatively few events, the Firth-type penalized likelihood estimates (FT-PLEs) have mean bias closer to zero and smaller mean squared error than maximum likelihood estimates (MLEs), and can be obtained in samples where the MLEs are infinite. Under similar T1E rates, FT-PL consistently exhibits higher statistical power than ML in samples with few events. In addition, we compare FT-PL estimation with two other penalization methods (a log-F prior method and a modified Firth-type method) based on the same simulations.

DISCUSSION

Consistent with findings for logistic and Cox regressions, FT-PL under MC regression yields finite estimates under stringent conditions, and better bias-and-variance balance than the other two penalizations. The practicality and strength of FT-PL for MC analysis is illustrated in a cohort study of breast cancer prognosis with long-term follow-up for recurrence-free survival.

Fitting parametric cure models in R using the packages cuRe and rstpm2

BACKGROUND AND OBJECTIVE

Within medical research, cure models are useful for analyzing time-to-event data in the scenario where a proportion of the analyzed individuals are expected to never experience the event of interest. Cure models are also useful for modelling the relative survival in scenarios where a proportion of the individuals are expected to eventually experience a mortality rate similar to that of the general population. Here we present two R packages, cuRe and rstpm2, that provide researchers with several tools for performing statistical inference using parametric cure models.

METHODS

Cure models are commonly used to estimate 1) the proportion of individuals that are cured and 2) the event-time distribution of individuals who are not cured. This can be done using simple parametric distributions for the event-time distribution of the uncured, but our implementations also enable fitting of more flexible spline-based cure models. The parametric framework of both packages ensures that cure models for the relative survival can easily be used.

RESULTS

The cuRe package contains two main functions for estimating parametric mixture cure models; one based on simple parametric distributions (e.g. Weibull or exponential) and one utilizing a spline-based formulation of the cure model. The rstpm2 package enables estimation of spline-based latent cure models, i.e., cure models with no explicit parameters modelling the proportion of cured individuals.

CONCLUSIONS

Through the R-packages cuRe and rstpm2, a wide range of different parametric cure models can be fitted. The cuRe package also contains a number of useful post-estimation procedures for computing the time to statistical cure and conditional probability of cure, which may spread the use of cure models in medical research.

Analysis of survival data with cure fraction and variable selection: A pseudo-observations approach

In biomedical studies, survival data with a cure fraction (the proportion of subjects cured of disease) are commonly encountered. The mixture cure and bounded cumulative hazard models are two main types of cure fraction models when analyzing survival data with long-term survivors. In this article, in the framework of the Cox proportional hazards mixture cure model and bounded cumulative hazard model, we propose several estimators utilizing pseudo-observations to assess the effects of covariates on the cure rate and the risk of having the event of interest for survival data with a cure fraction. A variable selection procedure is also presented based on the pseudo-observations using penalized generalized estimating equations for proportional hazards mixture cure and bounded cumulative hazard models. Extensive simulation studies are conducted to examine the proposed methods. The proposed technique is demonstrated through applications to a melanoma study and a dental data set with high-dimensional covariates.

Short-term and long-term prognostic value of histological response and intensified chemotherapy in osteosarcoma: a retrospective reanalysis of the BO06 trial

OBJECTIVES

Cure rate models accounting for cured and uncured patients, provide additional insights into long and short-term survival. We aim to evaluate the prognostic value of histological response and chemotherapy intensification on the cure fraction and progression-free survival (PFS) for the uncured patients.

DESIGN

Retrospective analysis of a randomised controlled trial, MRC BO06 (EORTC 80931).

SETTING

Population-based study but proposed methodology can be applied to other trial designs.

PARTICIPANTS

A total of 497 patients with resectable highgrade osteosarcoma, of which 118 were excluded because chemotherapy was not started, histological response was not reported, abnormal dose was reported or had disease progression during treatment.

INTERVENTIONS

Two regimens with the same anticipated cumulative dose (doxorubicin 6×75 mg/m²/week; cisplatin 6×100 mg/m²/week) over different time schedules: every 3 weeks in regimen-C and every 2 weeks in regimen-DI.

PRIMARY AND SECONDARY OUTCOME MEASURES

The primary outcome is PFS computed from end of treatment because cure, if it occurs, may happen at any time during treatment. A mixture cure model is used to study the effect of histological response and intensified chemotherapy on the cure status and PFS for the uncured patients.

RESULTS

Histological response is a strong prognostic factor for the cure status (OR 3.00, 95% CI 1.75 to 5.17), but it has no clear effect on PFS for the uncured patients (HR 0.78, -95% CI 0.53 to 1.16). The cure fractions are 55% (46%-63%) and 29% (22%-35%), respectively, among patients with good and poor histological response (GR, PR). The intensified regimen was associated with a higher cure fraction among PR (OR 1.90, 95% CI 0.93 to 3.89), with no evidence of effect for GR (OR 0.78, 95% CI 0.38 to 1.59).

CONCLUSIONS

Accounting for cured patients is valuable in distinguishing the covariate effects on cure and PFS. Estimating cure chances based on these prognostic factors is relevant for counselling patients and can have an impact on treatment decisions.

TRIAL REGISTRATION NUMBER

ISRCTN86294690.

Laplacian‐P‐splines for Bayesian inference in the mixture cure model

The mixture cure model for analyzing survival data is characterized by the assumption that the population under study is divided into a group of subjects who will experience the event of interest over some finite time horizon and another group of cured subjects who will never experience the event irrespective of the duration of follow‐up. When using the Bayesian paradigm for inference in survival models with a cure fraction, it is common practice to rely on Markov chain Monte Carlo (MCMC) methods to sample from posterior distributions. Although computationally feasible, the iterative nature of MCMC often implies long sampling times to explore the target space with chains that may suffer from slow convergence and poor mixing. Furthermore, extra efforts have to be invested in diagnostic checks to monitor the reliability of the generated posterior samples. A sampling‐free strategy for fast and flexible Bayesian inference in the mixture cure model is suggested in this article by combining Laplace approximations and penalized B‐splines. A logistic regression model is assumed for the cure proportion and a Cox proportional hazards model with a P‐spline approximated baseline hazard is used to specify the conditional survival function of susceptible subjects. Laplace approximations to the posterior conditional latent vector are based on analytical formulas for the gradient and Hessian of the log‐likelihood, resulting in a substantial speed‐up in approximating posterior distributions. The spline specification yields smooth estimates of survival curves and functions of latent variables together with their associated credible interval are estimated in seconds. A fully stochastic algorithm based on a Metropolis‐Langevin‐within‐Gibbs sampler is also suggested as an alternative to the proposed Laplacian‐P‐splines mixture cure (LPSMC) methodology. The statistical performance and computational efficiency of LPSMC is assessed in a simulation study. Results show that LPSMC is an appealing alternative to MCMC for approximate Bayesian inference in standard mixture cure models. Finally, the novel LPSMC approach is illustrated on three applications involving real survival data.

Modified score function for monotone likelihood in the semiparametric mixture cure model

The cure fraction models are intended to analyze lifetime data from populations where some individuals are immune to the event under study, and allow a joint estimation of the distribution related to the cured and susceptible subjects, as opposed to the usual approach ignoring the cure rate. In situations involving small sample sizes with many censored times, the detection of nonfinite coefficients may arise via maximum likelihood. This phenomenon is commonly known as monotone likelihood (ML), occurring in the Cox and logistic regression models when many categorical and unbalanced covariates are present. An existing solution to prevent the issue is based on the Firth correction, originally developed to reduce the estimation bias. The method ensures finite estimates by penalizing the likelihood function. In the context of mixture cure models, the ML issue is rarely discussed in the literature; therefore, this topic can be seen as the first contribution of our paper. The second major contribution, not well addressed elsewhere, is the study of the ML issue in cure mixture modeling under the flexibility of a semiparametric framework to handle the baseline hazard. We derive the modified score function based on the Firth approach and explore finite sample size properties of the estimators via a Monte Carlo scheme. The simulation results indicate that the performance of coefficients related to the binary covariates are strongly affected to the imbalance degree. A real illustration, in the melanoma dataset, is discussed using a relatively novel data set collected in a Brazilian university hospital.

Development and Evaluation of a Method to Correct Misinterpretation of Clinical Trial Results With Long-term Survival

Importance

In immune checkpoint inhibitor (ICI) trials, long tails and crossovers in survival curves-which violate the proportional hazards (PH) assumption-are commonly observed, making cure or restricted mean survival time models preferable for analysis of ICI survival data. Cox PH analysis, however, still appears in major medical journals, leading to potential misinterpretation of clinical significance.

Objective

To convert inappropriate Cox hazard ratios (HRs) to appropriate PH cure model treatment-effect estimates (HR for short-term survivors and difference in proportions [DP] for long-term survivors) for more accurate interpretation of published ICI trials.

Design and Setting

This study uses the Taylor expansion technique to demonstrate the mathematical relationship between Cox PH and PH cure models for data with long-term survival, and based on this relationship, proposes the Cox-TEL (Cox PH-Taylor expansion adjustment for long-term survival data) adjustment method. The proposed Cox-TEL method requires only 2 inputs: the reported Cox HRs and Kaplan-Meier-estimated survival probabilities.

Results

Comprehensive simulations show the strength of the proposed method in terms of power, bias, and type I error rate; these results, which are close to PH cure model estimates, were further verified in a melanoma data set (N = 285; Cox HR = 0.71; 95% CI, 0.51-0.91; Cox-TEL HR = 0.83; 95% CI, 0.60-1.07; PH cure HR = 0.86; 95% CI, 0.61-1.11; Cox-TEL DP = 0.10; 95% CI, 0.01-0.23; PH cure DP = 0.10; 95% CI, 0.00-0.21). The magnitude of potential difference between reported and adjusted HRs using real-world ICI trial results is demonstrated. For example, in the CheckMate 067 trial (nivolumab/ipilimumab combination therapy vs ipilimumab), the Cox HR was 0.54 (95% CI, 0.44-0.67), and the Cox-TEL HR was 0.90 (95% CI, 0.73-1.11).

Conclusions and Relevance

The findings of this study suggest the need to revisit published ICI survival data analysis to address potential misinterpretation. The Cox-TEL method not only is designed for this purpose, but also is user friendly and easy to implement using published clinical trial data and a freely available R software package.

Utilization of a Mixture Cure Rate Model based on the Generalized Modified Weibull Distribution for the Analysis of Leukemia Patients

OBJECTIVE

Cure rate models are survival models, commonly applied to model survival data with a cured fraction. In the existence of a cure rate, if the distribution of survival times for susceptible patients is specified, researchers usually prefer cure models to parametric models. Different distributions can be assumed for the survival times, for instance, generalized modified Weibull (GMW), exponentiated Weibull (EW), and log-beta Weibull. The purpose of this study is to select the best distribution for uncured patients' survival times by comparing the mixture cure models based on the GMW distribution and its particular cases.

MATERIALS AND METHODS

A data set of 91 patients with high-risk acute lymphoblastic leukemia (ALL) followed for five years from 1982 to 1987 was chosen for fitting the mixture cure model. We used the maximum likelihood estimation technique via R software 3.6.2 to obtain the estimates for parameters of the proposed model in the existence of cure rate, censored data, and covariates. For the best model choice, the Akaike information criterion (AIC) was implemented.

RESULTS

After comparing different parametric models fitted to the data, including or excluding cure fraction, without covariates, the smallest AIC values were obtained by the EW and the GMW distributions, (953.31/969.35) and (955.84/975.99), respectively. Besides, assuming a mixture cure model based on GMW with covariates, an estimated ratio between cure fractions for allogeneic and autologous bone marrow transplant groups (and its 95% confidence intervals) were 1.42972 (95% CI: 1.18614 - 1.72955).

CONCLUSION

The results of this study reveal that the EW and the GMW distributions are the best choices for the survival times of Leukemia patients.
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An additive hazards cure model with informative interval censoring

The existence of a cured subgroup happens quite often in survival studies and many authors considered this under various situations (Farewell in Biometrics 38:1041-1046, 1982; Kuk and Chen in Biometrika 79:531-541, 1992; Lam and Xue in Biometrika 92:573-586, 2005; Zhou et al. in J Comput Graph Stat 27:48-58, 2018). In this paper, we discuss the situation where only interval-censored data are available and furthermore, the censoring may be informative, for which there does not seem to exist an established estimation procedure. For the analysis, we present a three component model consisting of a logistic model for describing the cure rate, an additive hazards model for the failure time of interest and a nonhomogeneous Poisson model for the observation process. For estimation, we propose a sieve maximum likelihood estimation procedure and the asymptotic properties of the resulting estimators are established. Furthermore, an EM algorithm is developed for the implementation of the proposed estimation approach, and extensive simulation studies are conducted and suggest that the proposed method works well for practical situations. Also the approach is applied to a cardiac allograft vasculopathy study that motivated this investigation.

Firth adjusted score function for monotone likelihood in the mixture cure fraction model

Models for situations where some individuals are long-term survivors, immune or non-susceptible to the event of interest, are extensively studied in biomedical research. Fitting a regression can be problematic in situations involving small sample sizes with high censoring rate, since the maximum likelihood estimates of some coefficients may be infinity. This phenomenon is called monotone likelihood, and it occurs in the presence of many categorical covariates, especially when one covariate level is not associated with any failure (in survival analysis) or when a categorical covariate perfectly predicts a binary response (in the logistic regression). A well known solution is an adaptation of the Firth method, originally created to reduce the estimation bias. The method provides a finite estimate by penalizing the likelihood function. Bias correction in the mixture cure model is a topic rarely discussed in the literature and it configures a central contribution of this work. In order to handle this point in such context, we propose to derive the adjusted score function based on the Firth method. An extensive Monte Carlo simulation study indicates good inference performance for the penalized maximum likelihood estimates. The analysis is illustrated through a real application involving patients with melanoma assisted at the Hospital das Clínicas/UFMG in Brazil. This is a relatively novel data set affected by the monotone likelihood issue and containing cured individuals.

Semiparametric estimation of the cure fraction in population-based cancer survival analysis

With rapid development in medical research, the treatment of diseases including cancer has progressed dramatically and those survivors may die from causes other than the one under study, especially among elderly patients. Motivated by the Surveillance, Epidemiology, and End Results (SEER) female breast cancer study, background mortality is incorporated into the mixture cure proportional hazards (MCPH) model to improve the cure fraction estimation in population-based cancer studies. Here, that patients are "cured" is defined as when the mortality rate of the individuals in diseased group returns to the same level as that expected in the general population, where the population level mortality is presented by the mortality table of the United States. The semiparametric estimation method based on the EM algorithm for the MCPH model with background mortality (MCPH+BM) is further developed and validated via comprehensive simulation studies. Real data analysis shows that the proposed semiparametric MCPH+BM model may provide more accurate estimation in population-level cancer study.

Oral microbiome and onset of oral mucositis in patients with squamous cell carcinoma of the head and neck

BACKGROUND

Oral mucositis (OM) is a debilitating sequela for patients treated for squamous cell carcinoma of the head and neck (HNSCC). This study investigated whether oral microbial features before treatment or during treatment are associated with the time to onset of severe OM in patients with HNSCC.

METHODS

This was a cohort study of newly diagnosed patients with locoregional HNSCC who received chemotherapy with or without radiotherapy from April 2016 to September 2017. OM was based on the National Cancer Institute's Common Terminology Criteria for Adverse Events, version 4.0. The oral microbiome was characterized on the basis of the 16S ribosomal RNA V4 region with the Illumina platform. A mixture cure model was used to generate hazard ratios for the onset of severe OM.

RESULTS

Eighty-six percent of the patients developed OM (n = 57 [33 nonsevere cases and 24 severe cases]) with a median time to onset of OM of 21 days. With adjustments for age, sex, and smoking status, genera abundance was associated with the hazard for the onset of severe OM as follows: 1) at the baseline (n = 66), Cardiobacterium (P = .03) and Granulicatella (P = .04); 2) immediately before the development of OM (n = 57), Prevotella (P = .03), Fusobacterium (P = .03), and Streptococcus (P = .01); and 3) immediately before the development of severe OM (n = 24), Megasphaera (P = .0001) and Cardiobacterium (P = .03). There were no differences in α-diversity between the baseline samples and Human Microbiome Project data.

CONCLUSIONS

Changes in the abundance of genera over the course of treatment were associated with the onset of severe OM. The mechanism and therapeutic implications of these findings need to be investigated in future studies.

The cure model in perinatal epidemiology

In the statistical literature, the class of survival analysis models known as cure models has received much attention in recent years. Cure models seem not, however, to be part of the statistical toolbox of perinatal epidemiologists. In this paper, we demonstrate that in perinatal epidemiological studies where one investigates the relation between a gestational exposure and a condition that can only be ascertained after several years, cure models may provide the correct statistical framework. The reason for this is that the hypotheses being tested often concern an unobservable outcome that, in view of the hypothesis, should be thought of as occurring at birth, even though it is only detectable much later in life. The outcome of interest can therefore be viewed as a censored binary variable. We illustrate our argument with a simple cure model analysis of the possible relation between gestational exposure to paracetamol and attention-deficit hyperactivity disorder, using data from the Norwegian Mother, Father and Child Cohort Study conducted by the Norwegian Institute of Public Health, and information about the attention-deficit hyperactivity disorder diagnoses obtained from the Norwegian Patient Registry.

EM algorithms for fitting multistate cure models

Multistate cure models are multistate models in which transitions into one or more of the states cannot occur for a fraction of the population. In the study of cancer, multistate cure models can be used to identify factors related to the rate of cancer recurrence, the rate of death before and after recurrence, and the probability of being cured by initial treatment. However, the previous method for fitting multistate cure models requires substantial custom programming, making these valuable models less accessible to analysts. In this article, we present an Expectation-Maximization (EM) algorithm for fitting the multistate cure model using maximum likelihood. The proposed algorithm makes use of a weighted likelihood representation allowing it to be easily implemented with standard software and can incorporate either parametric or non-parametric baseline hazards for the state transition rates. A common complicating feature in cancer studies is that the follow-up times for recurrence and death may differ. Additionally, we may have missingness in the covariates. We propose a Monte Carlo EM (MCEM) algorithm for fitting the multistate cure model in the presence of covariate missingness and/or unequal follow-up of the two outcomes, we describe a novel approach for obtaining standard errors, and we provide some software. Simulations demonstrate good algorithmic performance as long as the modeling assumptions are sufficiently restrictive. We apply the proposed algorithm to a study of recurrence and death in patients with head and neck cancer.

Study design of single-arm phase II immunotherapy trials with long-term survivors and random delayed treatment effect

In the traditional study design of a single-arm phase II cancer clinical trial, the one-sample log-rank test has been frequently used. A common practice in sample size calculation is to assume that the event time in the new treatment follows exponential distribution. Such a study design may not be suitable for immunotherapy cancer trials, when both long-term survivors (or even cured patients from the disease) and delayed treatment effect are present, because exponential distribution is not appropriate to describe such data and consequently could lead to severely underpowered trial. In this research, we proposed a piecewise proportional hazards cure rate model with random delayed treatment effect to design single-arm phase II immunotherapy cancer trials. To improve test power, we proposed a new weighted one-sample log-rank test and provided a sample size calculation formula for designing trials. Our simulation study showed that the proposed log-rank test performs well and is robust of misspecified weight and the sample size calculation formula also performs well.

Validation of the Eighth Edition Clinical T Categorization System for Clinical Stage IA, Resected Lung Adenocarcinomas: Prognostic Implications of the Ground-Glass Opacity Component

INTRODUCTION

There is controversy regarding the clinical T (cT) category of lung adenocarcinomas that manifest as part-solid nodules (PSNs). We aimed to validate the cT category and to evaluate the independent prognostic role of the nodule type (i.e., part-solid versus solid).

METHODS

We retrospectively evaluated the prognostic value of clinico-radiologic factors regarding the overall survival of patients with clinical stage IA lung adenocarcinomas that were resected between 2008 and 2014. cT Category, nodule type, and their interaction term were included in the multivariable Cox regression analysis with other variables. In addition, a mixture cure model analysis was performed to investigate the association between the covariates and long-term survival.

RESULTS

A total of 744 patients (420 women; 362 PSNs; median age, 63 y) were included. The multivariable-adjusted hazard ratio (HR) of the nodule type was not significant (1.30, 95% confidence interval [CI]: 0.80-2.10, p = 0.291). However, the cT categories were significantly associated with overall survival (HR of cT1b, 2.33 [95% CI: 1.07-5.06, p = 0.033]; HR of cT1c, 5.74 [95% CI: 2.51-13.12, p < 0.001]). There were no interactions between the nodule type and the cT categories (all p > 0.05). The multivariable mixture cure model revealed that solid nodules were associated with a decreased probability of long-term survival (OR = 0.40, 95% CI: 0.18-0.92, p = 0.030). In addition, cT1c was a negative predictor of long-term survival (OR = 0.26, 95% CI: 0.07-0.94, p = 0.040).

CONCLUSIONS

The cT categorization system is valid for PSNs and solid nodules. Nevertheless, PSNs are a prognostic factor associated with long-term survival.

Analysis of clustered failure time data with cure fraction using copula

Clustered survival data in the presence of cure has received increasing attention. In this paper, we consider a semiparametric mixture cure model which incorporates a logistic regression model for the cure fraction and a semiparametric regression model for the failure time. We utilize Archimedean copula (AC) models to assess the strength of association for both susceptibility and failure times between susceptible individuals in the same cluster. Instead of using the full likelihood approach, we consider a composite likelihood function and a two-stage estimation procedure for both marginal and association parameters. A Jackknife procedure that takes out one cluster at a time is proposed for the variance estimation of the estimators. Akaike information criterion is applied to select the best model among ACs. Simulation studies are performed to validate our estimating procedures, and two real data sets are analyzed to demonstrate the practical use of our proposed method.

Exposure, hazard, and survival analysis of diffusion on social networks

Sociologists, economists, epidemiologists, and others recognize the importance of social networks in the diffusion of ideas and behaviors through human societies. To measure the flow of information on real-world networks, researchers often conduct comprehensive sociometric mapping of social links between individuals and then follow the spread of an "innovation" from reports of adoption or change in behavior over time. The innovation is introduced to a small number of individuals who may also be encouraged to spread it to their network contacts. In conjunction with the known social network, the pattern of adoptions gives researchers insight into the spread of the innovation in the population and factors associated with successful diffusion. Researchers have used widely varying statistical tools to estimate these quantities, and there is disagreement about how to analyze diffusion on fully observed networks. Here, we describe a framework for measuring features of diffusion processes on social networks using the epidemiological concepts of exposure and competing risks. Given a realization of a diffusion process on a fully observed network, we show that classical survival regression models can be adapted to estimate the rate of diffusion, and actor/edge attributes associated with successful transmission or adoption, while accounting for the topology of the social network. We illustrate these tools by applying them to a randomized network intervention trial conducted in Honduras to estimate the rate of adoption of 2 health-related interventions-multivitamins and chlorine bleach for water purification-and determine factors associated with successful social transmission.

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

BACKGROUND AND OBJECTIVE

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

METHODS

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

RESULTS

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

CONCLUSIONS

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

Actual 10-year survival after hepatic resection of colorectal liver metastases: what factors preclude cure?

BACKGROUND

Hepatic resection of colorectal liver metastases is associated with long-term survival. This study analyzes actual 10-year survivors after resection of colorectal liver metastases, reports the observed rate of cure, and identifies factors that preclude cure.

METHODS

A single-institution, prospectively maintained database was queried for all initial resections for colorectal liver metastases for the years 1992-2004. Observed cure was defined as actual 10-year survival with either no recurrence or resected recurrence with at least 3 years of disease-free follow-up. Clinical risk score was dichotomized into low (0-2) and high (3-5). Semiparametric proportional hazards mixture cure model was utilized to estimate probability of cure.

RESULTS

We included 1,211 patients with a median follow-up for survivors of 11 years. Median disease-specific survival was 4.9 years (95% CI: 4.4-5.3). 295 patients (24.4%) were actual 10-year survivors. The observed cure rate was 20.6% (n = 250). Among 250 cured patients, 192 (76.8%) had no recurrence and 58 (23.2%) had a resected recurrence with at least 3 years of disease-free follow-up. Extrahepatic disease (n = 88), carcinoembryonic antigen >200 ng/mL (n = 119), positive margin (n = 109), and >10 tumors (n = 31) had observed cure rates less than 10%. In cure model analysis, patients with both extrahepatic disease and high clinical risk score (n = 31) had an estimated probability of cure of 3.5%.

CONCLUSION

Actual 10-year survival after resection of colorectal liver metastases is 24% with an observed 20% cure rate. Patients with both high clinical risk score and extrahepatic disease have an estimated probability of cure less than 5%. When such factors are identified, strong consideration may be given to preoperative strategies, such as neoadjuvant chemotherapy, to help select patients for surgical therapy.

Defining the Chance of Statistical Cure Among Patients with Extrahepatic Biliary Tract Cancer

BACKGROUND

While surgery offers the best curative-intent treatment, many patients with biliary tract malignancies have poor long-term outcomes. We sought to apply a non-mixture cure model to calculate the cure fraction and the time to cure after surgery of patients with peri-hilar cholangiocarcinoma (PHCC) or gallbladder cancer (GBC).

METHODS

Using the Extrahepatic Biliary Malignancy Consortium, 576 patients who underwent curative-intent surgery for gallbladder carcinoma or peri-hilar cholangiocarcinoma between 1998 and 2014 at 10 major hepatobiliary institutions were identified and included in the analysis. A non-mixture cure model was adopted to compare mortality after surgery to the mortality expected for the general population matched by sex and age.

RESULTS

The median and 5-year overall survival (OS) were 1.9 years (IQR, 0.9-4.9) and 23.9 % (95 % CI, 19.6-28.6). Among all patients with PHCC or GBC, the probability of being cured after surgery was 14.5 % (95 % CI, 8.7-23.2); the time to cure was 9.7 years and the median survival of uncured patients was 1.8 years. Determinants of cure probabilities included lymph node metastasis and CA 19.9 level (p ≤ 0.05). The cure fraction for patients with a CA 19.9 < 50 U/ml and no lymph nodes metastases were 39.0 % versus only 5.1 % among patients with a CA 19.9 ≥ 50 who also had lymph node metastasis.

CONCLUSIONS

Examining an "all comer" cohort, <15 % of patients with PHCC or GBC could be considered cured after surgery. Factors such CA 19.9 level and lymph node metastasis independently predicted long-term outcome. Estimating the odds of statistical cure following surgery for biliary tract cancer can assist in decision-making as well as inform discussions around survivorship.

Extinct or still out there? Disentangling influences on extinction and rediscovery helps to clarify the fate of species on the edge

Each year, two or three species that had been considered to be extinct are rediscovered. Uncertainty about whether or not a species is extinct is common, because rare and highly threatened species are difficult to detect. Biological traits such as body size and range size are expected to be associated with extinction. However, these traits, together with the intensity of search effort, might influence the probability of detection and extinction differently. This makes statistical analysis of extinction and rediscovery challenging. Here, we use a variant of survival analysis known as cure rate modelling to differentiate factors that influence rediscovery from those that influence extinction. We analyse a global data set of 99 mammals that have been categorized as extinct or possibly extinct. We estimate the probability that each of these mammals is still extant and thus estimate the proportion of missing (presumed extinct) mammals that are incorrectly assigned extinction. We find that body mass and population density are predictors of extinction, and body mass and search effort predict rediscovery. In mammals, extinction rate increases with body mass and population density, and these traits act synergistically to greatly elevate extinction rate in large species that also occurred in formerly dense populations. However, when they remain extant, larger-bodied missing species are rediscovered sooner than smaller species. Greater search effort increases the probability of rediscovery in larger species of missing mammals, but has a minimal effect on small species, which take longer to be rediscovered, if extant. By separating the effects of species characteristics on extinction and detection, and using models with the assumption that a proportion of missing species will never be rediscovered, our new approach provides estimates of extinction probability in species with few observation records and scant ecological information.

What Cure Models Can Teach us About Genome-Wide Survival Analysis

The aim of logistic regression is to estimate genetic effects on disease risk, while survival analysis aims to determine effects on age of onset. In practice, genetic variants may affect both types of outcomes. A cure survival model analyzes logistic and survival effects simultaneously. The aim of this simulation study is to assess the performance of logistic regression and traditional survival analysis under a cure model and to investigate the benefits of cure survival analysis. We simulated data under a cure model and varied the percentage of subjects at risk for disease (cure fraction), the logistic and survival effect sizes, and the contribution of genetic background risk factors. We then computed the error rates and estimation bias of logistic, Cox proportional hazards (PH), and cure PH analysis, respectively. The power of logistic and Cox PH analysis is sensitive to the cure fraction and background heritability. Our results show that traditional Cox PH analysis may erroneously detect age of onset effects if no such effects are present in the data. In the presence of genetic background risk even the cure model results in biased estimates of both the odds ratio and the hazard ratio. Cure survival analysis takes cure fractions into account and can be used to simultaneously estimate the effect of genetic variants on disease risk and age of onset. Since genome-wide cure survival analysis is not computationally feasible, we recommend this analysis for genetic variants that are significant in a traditional survival analysis.

Can hepatic resection provide a long-term cure for patients with intrahepatic cholangiocarcinoma?

BACKGROUND

A patient can be considered statistically cured from a specific disease when their mortality rate returns to the same level as that of the general population. In the current study, the authors sought to assess the probability of being statistically cured from intrahepatic cholangiocarcinoma (ICC) by hepatic resection.

METHODS

A total of 584 patients who underwent surgery with curative intent for ICC between 1990 and 2013 at 1 of 12 participating institutions were identified. A nonmixture cure model was adopted to compare mortality after hepatic resection with the mortality expected for the general population matched by sex and age.

RESULTS

The median, 1-year, 3-year, and 5-year disease-free survival was 10 months, 44%, 18%, and 11%, respectively; the corresponding overall survival was 27 months, 75%, 37%, and 22%, respectively. The probability of being cured of ICC was 9.7% (95% confidence interval, 6.1%-13.4%). The mortality of patients undergoing surgery for ICC was higher than that of the general population until year 10, at which time patients alive without tumor recurrence can be considered cured with 99% certainty. Multivariate analysis demonstrated that cure probabilities ranged from 25.8% (time to cure, 9.8 years) in patients with a single, well-differentiated ICC measuring ≤5 cm that was without vascular/periductal invasion and lymph nodes metastases versus <0.1% (time to cure, 12.6 years) among patients with all 6 of these risk factors. A model with which to calculate cure fraction and time to cure was developed.

CONCLUSIONS

The cure model indicated that statistical cure was possible in patients undergoing hepatic resection for ICC. The overall probability of cure was approximately 10% and varied based on several tumor-specific factors. Cancer 2015;121:3998-4006. © 2015 American Cancer Society.

NPHMC: an R-package for estimating sample size of proportional hazards mixture cure model

Due to advances in medical research, more and more diseases can be cured nowadays, which largely increases the need for an easy-to-use software in calculating sample size of clinical trials with cure fractions. Current available sample size software, such as PROC POWER in SAS, Survival Analysis module in PASS, powerSurvEpi package in R are all based on the standard proportional hazards (PH) model which is not appropriate to design a clinical trial with cure fractions. Instead of the standard PH model, the PH mixture cure model is an important tool in handling the survival data with possible cure fractions. However, there are no tools available that can help design a trial with cure fractions. Therefore, we develop an R package NPHMC to determine the sample size needed for such study design.