A nonparametric mixture model for cure rate estimation

A New Approach to Modeling the Cure Rate in the Presence of Interval Censored Data

We consider interval censored data with a cured subgroup that arises from longitudinal followup studies with a heterogeneous population where a certain proportion of subjects is not susceptible to the event of interest. We propose a two component mixture cure model, where the first component describing the probability of cure is modeled by a support vector machine-based approach and the second component describing the survival distribution of the uncured group is modeled by a proportional hazard structure. Our proposed model provides flexibility in capturing complex effects of covariates on the probability of cure unlike the traditional models that rely on modeling the cure probability using a generalized linear model with a known link function. For the estimation of model parameters, we develop an expectation maximization-based estimation algorithm. We conduct simulation studies and show that our proposed model performs better in capturing complex effects of covariates on the cure probability when compared to the traditional logit link-based two component mixture cure model. This results in more accurate (smaller bias) and precise (smaller mean square error) estimates of the cure probabilities, which in-turn improves the predictive accuracy of the latent cured status. We further show that our model's ability to capture complex covariate effects also improves the estimation results corresponding to the survival distribution of the uncured. Finally, we apply the proposed model and estimation procedure to an interval censored data on smoking cessation.

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.

A Bayesian proportional hazards mixture cure model for interval-censored data

The proportional hazards mixture cure model is a popular analysis method for survival data where a subgroup of patients are cured. When the data are interval-censored, the estimation of this model is challenging due to its complex data structure. In this article, we propose a computationally efficient semiparametric Bayesian approach, facilitated by spline approximation and Poisson data augmentation, for model estimation and inference with interval-censored data and a cure rate. The spline approximation and Poisson data augmentation greatly simplify the MCMC algorithm and enhance the convergence of the MCMC chains. The empirical properties of the proposed method are examined through extensive simulation studies and also compared with the R package "GORCure". The use of the proposed method is illustrated through analyzing a data set from the Aerobics Center Longitudinal Study.

Causal inference for time-to-event data with a cured subpopulation

When studying the treatment effect on time-to-event outcomes, it is common that some individuals never experience failure events, which suggests that they have been cured. However, the cure status may not be observed due to censoring which makes it challenging to define treatment effects. Current methods mainly focus on estimating model parameters in various cure models, ultimately leading to a lack of causal interpretations. To address this issue, we propose 2 causal estimands, the timewise risk difference and mean survival time difference, in the always-uncured based on principal stratification as a complement to the treatment effect on cure rates. These estimands allow us to study the treatment effects on failure times in the always-uncured subpopulation. We show the identifiability using a substitutional variable for the potential cure status under ignorable treatment assignment mechanism, these 2 estimands are identifiable. We also provide estimation methods using mixture cure models. We applied our approach to an observational study that compared the leukemia-free survival rates of different transplantation types to cure acute lymphoblastic leukemia. Our proposed approach yielded insightful results that can be used to inform future treatment decisions.

Partly linear single-index cure models with a nonparametric incidence link function

In cancer studies, it is commonplace that a fraction of patients participating in the study are cured, such that not all of them will experience a recurrence, or death due to cancer. Also, it is plausible that some covariates, such as the treatment assigned to the patients or demographic characteristics, could affect both the patients' survival rates and cure/incidence rates. A common approach to accommodate these features in survival analysis is to consider a mixture cure survival model with the incidence rate modeled by a logistic regression model and latency part modeled by the Cox proportional hazards model. These modeling assumptions, though typical, restrict the structure of covariate effects on both the incidence and latency components. As a plausible recourse to attain flexibility, we study a class of semiparametric mixture cure models in this article, which incorporates two single-index functions for modeling the two regression components. A hybrid nonparametric maximum likelihood estimation method is proposed, where the cumulative baseline hazard function for uncured subjects is estimated nonparametrically, and the two single-index functions are estimated via Bernstein polynomials. Parameter estimation is carried out via a curated expectation-maximization algorithm. We also conducted a large-scale simulation study to assess the finite-sample performance of the estimator. The proposed methodology is illustrated via application to two cancer datasets.

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.

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.

On the parameter estimation of Box-Cox transformation cure model

We propose an improved estimation method for the Box-Cox transformation (BCT) cure rate model parameters. Specifically, we propose a generic maximum likelihood estimation algorithm through a non-linear conjugate gradient (NCG) method with an efficient line search technique. We then apply the proposed NCG algorithm to BCT cure model. Through a detailed simulation study, we compare the model fitting results of the NCG algorithm with those obtained by the existing expectation maximization (EM) algorithm. First, we show that our proposed NCG algorithm allows simultaneous maximization of all model parameters unlike the EM algorithm when the likelihood surface is flat with respect to the BCT index parameter. Then, we show that the NCG algorithm results in smaller bias and noticeably smaller root mean square error of the estimates of the model parameters that are associated with the cure rate. This results in more accurate and precise inference on the cure rate. In addition, we show that when the sample size is large the NCG algorithm, which only needs the computation of the gradient and not the Hessian, takes less CPU time to produce the estimates. These advantages of the NCG algorithm allows us to conclude that the NCG method should be the preferred estimation method over the already existing EM algorithm in the context of BCT cure model. Finally, we apply the NCG algorithm to analyze a well-known melanoma data and show that it results in a better fit when compared to the EM algorithm.

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.

Estimation of patient flow in hospitals using up-to-date data. Application to bed demand prediction during pandemic waves

Hospital bed demand forecast is a first-order concern for public health action to avoid healthcare systems to be overwhelmed. Predictions are usually performed by estimating patients flow, that is, lengths of stay and branching probabilities. In most approaches in the literature, estimations rely on not updated published information or historical data. This may lead to unreliable estimates and biased forecasts during new or non-stationary situations. In this paper, we introduce a flexible adaptive procedure using only near-real-time information. Such method requires handling censored information from patients still in hospital. This approach allows the efficient estimation of the distributions of lengths of stay and probabilities used to represent the patient pathways. This is very relevant at the first stages of a pandemic, when there is much uncertainty and too few patients have completely observed pathways. Furthermore, the performance of the proposed method is assessed in an extensive simulation study in which the patient flow in a hospital during a pandemic wave is modelled. We further discuss the advantages and limitations of the method, as well as potential extensions.

Measures of explained variation under the mixture cure model for survival data

Explained variation is well understood under linear regression models and has been extended to models for survival data. In this article, we consider the mixture cure models. We propose two approaches to define explained variation under the mixture cure models, one based on the Kullback-Leibler information gain and the other based on residual sum of squares. We show that the proposed measures have desired properties as measures of explained variation, similar to those under other regression models. A simulation study is conducted to demonstrate the properties of the proposed measures. They are also applied to real data analyses to illustrate the use of explained variation.

A Bayesian Mixture Cure Rate Model for Estimating Short-Term and Long-Term Recidivism

Mixture cure rate models have been developed to analyze failure time data where a proportion never fails. For such data, standard survival models are usually not appropriate because they do not account for the possibility of non-failure. In this context, mixture cure rate models assume that the studied population is a mixture of susceptible subjects who may experience the event of interest and non-susceptible subjects that will never experience it. More specifically, mixture cure rate models are a class of survival time models in which the probability of an eventual failure is less than one and both the probability of eventual failure and the timing of failure depend (separately) on certain individual characteristics. In this paper, we propose a Bayesian approach to estimate parametric mixture cure rate models with covariates. The probability of eventual failure is estimated using a binary regression model, and the timing of failure is determined using a Weibull distribution. Inference for these models is attained using Markov Chain Monte Carlo methods under the proposed Bayesian framework. Finally, we illustrate the method using data on the return-to-prison time for a sample of prison releases of men convicted of sexual crimes against women in England and Wales and we use mixture cure rate models to investigate the risk factors for long-term and short-term survival of recidivism.

A flexible-hazards cure model with application to patients with soft tissue sarcoma

In medical research, it is often of great interest to have an accurate estimation of cure rates by different treatment options and for different patient groups. If the follow-up time is sufficiently long and the sample size is large, the proportion of cured patients will make the Kaplan-Meier estimator of survival function have a flat plateau at its tail, whose value indicates the overall cure rate. However, it may be difficult to estimate and compare the cure rates for all the subsets of interest in this way, due to the limit of sample sizes and curse of dimensionality. In the current literature, most regression models for estimating cure rates assume proportional hazards (PH) between different subgroups. It turns out that the estimation of cure rates for subgroups is highly sensitive to this assumption, so more flexible models are needed, especially when this PH assumption is clearly violated. We propose a new cure model to simultaneously incorporate both PH and non-PH scenarios for different covariates. We develop a stable and easily implementable iterative procedure for parameter estimation through maximization of the nonparametric likelihood function. The covariance matrix is estimated by adding perturbation weights to the estimation procedure. In simulation studies, the proposed method provides unbiased estimation for the regression coefficients, survival curves, and cure rates given covariates, while existing models are biased. Our model is applied to a study of stage III soft tissue sarcoma and provides trustworthy estimation of cure rates for different treatment and demographic groups.

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.

Controlled variable selection in Weibull mixture cure models for high-dimensional data

Medical breakthroughs in recent years have led to cures for many diseases. The mixture cure model (MCM) is a type of survival model that is often used when a cured fraction exists. Many have sought to identify genomic features associated with a time-to-event outcome which requires variable selection strategies for high-dimensional spaces. Unfortunately, currently few variable selection methods exist for MCMs especially when there are more predictors than samples. This study develops high-dimensional penalized Weibull MCMs, which allow for identification of prognostic factors associated with both cure status and/or survival. We demonstrated how such models may be estimated using two different iterative algorithms. The model-X knockoffs method was combined with these algorithms to control the false discovery rate (FDR) in variable selection. Through extensive simulation studies, our penalized MCMs have been shown to outperform alternative methods on multiple metrics and achieve high statistical power with FDR being controlled. In an acute myeloid leukemia (AML) application with gene expression data, our proposed approach identified 14 genes associated with potential cure and 12 genes with time-to-relapse, which may help inform treatment decisions for AML patients.

Analysis of Cancer Survival Associated With Immune Checkpoint Inhibitors After Statistical Adjustment: A Systematic Review and Meta-analyses

IMPORTANCE

Appropriate clinical decision-making relies on accurate data interpretation, which in turn relies on the use of suitable statistical models. Long tails and early crossover-2 features commonly observed in immune checkpoint inhibitor (ICI) survival curves-raise questions as to the suitability of Cox proportional hazards regression for ICI survival analysis. Cox proportional hazards-Taylor expansion adjustment for long-term survival data (Cox-TEL) adjustment may provide possible solutions in this setting.

OBJECTIVE

To estimate overall survival and progression-free survival benefits of ICI therapy vs chemotherapy using Cox-TEL adjustment.

DATA SOURCES

A PubMed search was performed for all cataloged publications through May 22, 2022.

STUDY SELECTION

The search was restricted to randomized clinical trials with search terms for ICIs and lung cancer, melanoma, or urothelial carcinoma. The publications identified were further reviewed for inclusion.

DATA EXTRACTION AND SYNTHESIS

Cox proportional hazards ratios (HRs) were transformed to Cox-TEL HRs for patients with short-term treatment response (ie, short-term survivor) (ST-HR) and difference in proportions for patients with long-term survival (LT-DP) by Cox-TEL. Meta-analyses were performed using a frequentist random-effects model.

MAIN OUTCOMES AND MEASURES

Outcomes of interest were pooled overall survival (primary outcome) and progression-free survival (secondary outcome) HRs, ST-HRs, and LT-DPs. Subgroup analyses stratified by cancer type also were performed.

RESULTS

A total of 1036 publications was identified. After 3 levels of review against inclusion criteria, 13 clinical trials (7 in non-small cell lung cancer, 3 in melanoma, and 3 in urothelial carcinoma) were selected for the meta-analysis. In the primary analysis, pooled findings were 0.75 (95% CI, 0.70-0.81) for HR, 0.86 (95% CI, 0.81-0.92) for ST-HR, and 0.08 (95% CI, 0.06-0.10) for LT-DP. In the secondary analysis, the pooled values for progression-free survival were 0.77 (95% CI, 0.64-0.91) for HR, 1.02 (95% CI, 0.84-1.24) for ST-HR, and 0.10 (95% CI, 0.06-0.14) for LT-DP.

CONCLUSIONS AND RELEVANCE

This systematic review and meta-analysis of ICI clinical trial results noted consistently larger ST-HRs vs Cox HRs for ICI therapy, with an LT-DP of approximately 10%. These results suggest that Cox HRs may not provide a full picture of survival outcomes when the risk reduction from treatment is not constant, which may aid in the decision-making process of oncologists and patients.

Survival after active surveillance versus upfront surgery for incidental small pancreatic neuroendocrine tumours

BACKGROUND

The safety of observing small non-functioning pancreatic neuroendocrine tumours (NF-Pan-NETs) remains under debate.

METHODS

This was a multicentre retrospective study of patients with small incidental NF-Pan-NETs. Survival of patients who underwent upfront surgery versus active surveillance was compared. The risk of death was matched with that in the healthy population. The excess hazard rate and probability of a normal lifespan (NLP) were calculated. Propensity score matching (PSM) with a 1 : 1 ratio was used to minimize the risk of selection bias.

RESULTS

Some 222 patients (43.7 per cent) underwent upfront surgery and 285 (56.3 per cent) were observed. The excess hazard rate for the entire cohort was quantifiable as 0.04 (95 per cent c.i. 0 to 0.08) deaths per 1000 persons per year, and the NLP was 99.7 per cent. Patients in the active surveillance group were older (median age 65 versus 58 years; P < 0.001), and more often had co-morbidity (45.3 versus 24.8 per cent; P = 0.001), and smaller tumours (median 12 versus 13 mm; P < 0.001), less frequently located in the pancreatic body-tail (59.5 versus 69.6 per cent; P = 0.008, 59.3 versus 73.9 per cent; P = 0.001). Median follow-up was longer for patients who underwent upfront surgery (5.6 versus 2.7 years; P < 0.001). After PSM, 118 patients per group were included. The excess hazard rates were 0.2 and 0.9 deaths per 1000 persons per year (P = 0.020) for patients in the active surveillance and upfront surgery groups respectively. Corresponding NLPs were 99.9 and 99.5 per cent respectively (P = 0.011).

CONCLUSION

Active surveillance of small incidental NF-Pan-NETs is a reasonable alternative to resection.

The analysis of COVID-19 in-hospital mortality: A competing risk approach or a cure model?

Competing risk analyses have been widely used for the analysis of in-hospital mortality in which hospital discharge is considered as a competing event. The competing risk model assumes that more than one cause of failure is possible, but there is only one outcome of interest and all others serve as competing events. However, hospital discharge and in-hospital death are two outcomes resulting from the same disease process and patients whose disease conditions were stabilized so that inpatient care was no longer needed were discharged. We therefore propose to use cure models, in which hospital discharge is treated as an observed “cure” of the disease. We consider both the mixture cure model and the promotion time cure model and extend the models to allow cure status to be known for those who were discharged from the hospital. An EM algorithm is developed for the mixture cure model. We also show that the competing risk model, which treats hospital discharge as a competing event, is equivalent to a promotion time cure model. Both cure models were examined in simulation studies and were applied to a recent cohort of COVID-19 in-hospital patients with diabetes. The promotion time model shows that statin use improved the overall survival; the mixture cure model shows that while statin use reduced the in-hospital mortality rate among the susceptible, it improved the cure probability only for older but not younger patients. Both cure models show that treatment was more beneficial among older patients.

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.

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.

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.

Subgroup Identification and Regression Analysis of Clustered and Heterogeneous Interval-Censored Data

Clustered and heterogeneous interval-censored data occur in many fields such as medical studies. For example, in a migraine study with the Netherlands Twin Registry, the information including time to diagnosis of migraine and gender was collected for 3975 monozygotic and dizygotic twins. Since each study subject is observed only at discrete and periodic follow-up time points, the failure times of interest (i.e., the time when the individual first had a migraine) are known only to belong to certain intervals and hence are interval-censored. Furthermore, these twins come from different genetic backgrounds and may be associated with differential risks for developing migraines. For simultaneous subgroup identification and regression analysis of such data, we propose a latent Cox model where the number of subgroups is not assumed a priori but rather data-driven estimated. The nonparametric maximum likelihood method and an EM algorithm with monotone ascent property are also developed for estimating the model parameters. Simulation studies are conducted to assess the finite sample performance of the proposed estimation procedure. We further illustrate the proposed methodologies by an empirical analysis of migraine data.

Maximum likelihood estimation for length-biased and interval-censored data with a nonsusceptible fraction

Left-truncated data are often encountered in epidemiological cohort studies, where individuals are recruited according to a certain cross-sectional sampling criterion. Length-biased data, a special case of left-truncated data, assume that the incidence of the initial event follows a homogeneous Poisson process. In this article, we consider an analysis of length-biased and interval-censored data with a nonsusceptible fraction. We first point out the importance of a well-defined target population, which depends on the prior knowledge for the support of the failure times of susceptible individuals. Given the target population, we proceed with a length-biased sampling and draw valid inferences from a length-biased sample. When there is no covariate, we show that it suffices to consider a discrete version of the survival function for the susceptible individuals with jump points at the left endpoints of the censoring intervals when maximizing the full likelihood function, and propose an EM algorithm to obtain the nonparametric maximum likelihood estimates of nonsusceptible rate and the survival function of the susceptible individuals. We also develop a novel graphical method for assessing the stationarity assumption. When covariates are present, we consider the Cox proportional hazards model for the survival time of the susceptible individuals and the logistic regression model for the probability of being susceptible. We construct the full likelihood function and obtain the nonparametric maximum likelihood estimates of the regression parameters by employing the EM algorithm. The large sample properties of the estimates are established. The performance of the method is assessed by simulations. The proposed model and method are applied to data from an early-onset diabetes mellitus study.

Exposure assessment for Cox proportional hazards cure models with interval-censored survival data

Mixture cure models have been developed as an effective tool to analyze failure time data with a cure fraction. Used in conjunction with the logistic regression model, this model allows covariate-adjusted inference of an exposure effect on the cured probability and the hazard of failure for the uncured subjects. However, the covariate-adjusted inference for the overall exposure effect is not directly provided. In this paper, we describe a Cox proportional hazards cure model to analyze interval-censored survival data in the presence of a cured fraction and then apply a post-estimation approach by using model-predicted estimates difference to assess the overall exposure effect on the restricted mean survival time scale. For baseline hazard/survival function estimation, simple parametric models as fractional polynomials or restricted cubic splines are utilized to approximate the baseline logarithm cumulative hazard function, or, alternatively, the full likelihood is specified through a piecewise linear approximation for the cumulative baseline hazard function. Simulation studies were conducted to demonstrate the unbiasedness of both estimation methods for the overall exposure effect estimates over various baseline hazard distribution shapes. The methods are applied to analyze the interval-censored relapse time data from a smoking cessation study.

Mixture proportional hazards cure model with latent variables

A mixture proportional hazards cure model with latent variables is proposed. The proposed model assesses the effects of the observed and latent risk factors on the hazards of uncured subjects and the cure rate through a proportional hazards model and a logistic model, respectively. Factor analysis is employed to measure the latent variables through correlated multiple indicators. Maximum likelihood estimation is performed through a Gaussian quadratic technique that approximates the integration over the latent variables. A piecewise constant function is used for the unspecified baseline hazard of uncured subjects. The proposed method can be conveniently implemented by using SAS Proc NLMIXED. Simulation studies are conducted to evaluate the performance of the proposed approach. An application to a study concerning the risk factors of chronic kidney disease for type 2 diabetic patients is provided.

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.

A functional proportional hazard cure rate model for interval-censored data

Existing survival models involving functional covariates typically rely on the Cox proportional hazards structure and the assumption of right censorship. Motivated by the aim of predicting the time of conversion to Alzheimer's disease from sparse biomarker trajectories in patients with mild cognitive impairment, we propose a functional mixture cure rate model with both functional and scalar covariates for interval censoring and sparsely sampled functional data. To estimate the nonparametric coefficient function that depicts the effect of the shape of the trajectories on the survival outcome and cure probability, we utilize the functional principal component analysis to extract the functional features from the sparsely and irregularly sampled trajectories. To obtain parameter estimates from the mixture cure rate model with interval censoring, we apply the expectation-maximization algorithm based on Poisson data augmentation. The estimation accuracy of our method is assessed via a simulation study and we apply our model on Alzheimer's disease Neuroimaging Initiative data set.

An improved method for the effect estimation of the intermediate event on the outcome based on the susceptible pre-identification

Background

In follow-up studies, the occurrence of the intermediate event may influence the risk of the outcome of interest. Existing methods estimate the effect of the intermediate event by including a time-varying covariate in the outcome model. However, the insusceptible fraction to the intermediate event in the study population has not been considered in the literature, leading to effect estimation bias due to the inaccurate dataset.

Methods

In this paper, we propose a new effect estimation method, in which the susceptible subpopulation is identified firstly so that the estimation could be conducted in the right population. Then, the effect is estimated via the extended Cox regression and landmark methods in the identified susceptible subpopulation. For susceptibility identification, patients with observed intermediate event time are classified as susceptible. Based on the mixture cure model fitted the incidence and time of the intermediate event, the susceptibility of the patient with censored intermediate event time is predicted by the residual intermediate event time imputation. The effect estimation performance of the new method was investigated in various scenarios via Monte-Carlo simulations with the performance of existing methods serving as the comparison. The application of the proposed method to mycosis fungoides data has been reported as an example.

Results

The simulation results show that the estimation bias of the proposed method is smaller than that of the existing methods, especially in the case of a large insusceptible fraction. The results hold for small sample sizes. Besides, the estimation bias of the new method decreases with the increase of the covariates, especially continuous covariates, in the mixture cure model. The heterogeneity of the effect of covariates on the outcome in the insusceptible and susceptible subpopulation, as well as the landmark time, does not affect the estimation performance of the new method.

Conclusions

Based on the pre-identification of the susceptible, the proposed new method could improve the effect estimation accuracy of the intermediate event on the outcome when there is an insusceptible fraction to the intermediate event in the study population.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-021-01378-8.

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.

Sequence kernel association test for survival outcomes in the presence of a non-susceptible fraction

In this work, we propose a single nucleotide polymorphism set association test for survival phenotypes in the presence of a non-susceptible fraction. We consider a mixture model with a logistic regression for the susceptibility indicator and a proportional hazards regression to model survival in the susceptible group. We propose a joint test to assess the significance of the genetic variant in both logistic and survival regressions simultaneously. We adopt the spirit of SKAT and conduct a variance-component test treating the genetic effects of multiple variants as random. We derive score-type test statistics, and we investigate several approaches to compute their $p$-values. The finite-sample properties of the proposed tests are assessed and compared to existing approaches by simulations and their use is illustrated through an application to ovarian cancer data from the Consortium of Investigators of Modifiers of BRCA1 and BRCA2.

Promotion time cure rate model with a neural network estimated nonparametric component

Promotion time cure rate models (PCM) are often used to model the survival data with a cure fraction. Medical images or biomarkers derived from medical images can be the key predictors in survival models. However, incorporating images in the PCM is challenging using traditional nonparametric methods such as splines. We propose to use neural network to model the nonparametric or unstructured predictors' effect in the PCM context. Expectation-maximization algorithm with neural network for the M-step is used for parameter estimation. Asymptotic properties of the proposed estimates are derived. Simulation studies show good performance in terms of both prediction and estimation. We finally apply our methods to analyze the brain images from open access series of imaging studies data.

The ROC of Cox proportional hazards cure models with application in cancer studies

With recent advancement in cancer screening and treatment, many patients with cancers are identified at early stage and clinically cured. Importantly, uncured patients should be treated timely before the cancer progresses to advanced stages for which therapeutic options are rather limited. It is also crucial to identify uncured subjects among patients with early-stage cancers for clinical trials to develop effective adjuvant therapies. Thus, it is of interest to develop statistical predictive models with as high accuracy as possible in predicting the latent cure status. The receiver operating characteristic curve (ROC) and the area under the ROC curve (AUC) are among the most widely used statistical metrics for assessing predictive accuracy or discriminatory power for a dichotomous outcome (cured/uncured). Yet the conventional AUC cannot be directly used due to incompletely observed cure status. In this article, we proposed new estimates of the ROC curve and its AUC for predicting latent cure status in Cox proportional hazards (PH) cure models and transformation cure models. We developed explicit formulas to estimate sensitivity, specificity, the ROC and its AUC without requiring to know the patient cure status. We also developed EM type estimates to approximate sensitivity, specificity, ROC and AUC conditional on observed data. Numerical studies were used to assess their finite-sample performance of the proposed methods. Both methods are consistent and have similar efficiency as shown in our numerical studies. A melanoma dataset was used to demonstrate the utility of the proposed estimates of the ROC curve for the latent cure status. We also have developed an [Formula: see text] package called [Formula: see text] to efficiently compute the proposed estimates.

Sporadic non-functioning pancreatic neuroendocrine tumours: multicentre analysis

BACKGROUND

Outcomes after surgery for sporadic pancreatic neuroendocrine neoplasms (Pan-NENs) were evaluated.

METHODS

This multicentre study included patients who underwent radical pancreatic resection for sporadic non-functioning Pan-NENs. In survival analysis, the risk of mortality in this cohort was analysed in relation to that of the matched healthy Italian population. Relative survival (RS) was calculated as the rate between observed and expected survival. Factors related to RS were investigated using multivariable modelling.

RESULTS

Among 964 patients who had pancreatic resection for sporadic non-functioning Pan-NENs, the overall RS rate was 91.8 (95 per cent c.i. 81.5 to 96.5) per cent. 2019 WHO grade (hazard ratio (HR) 5.75 (s.e. 4.63); P = 0.030) and European Neuroendocrine Tumour Society (ENETS) TNM stage (6.73 (3.61); P < 0.001) were independent predictors of RS. The probability of a normal lifespan for patients with G1, G2, G3 Pan-NENS, and pancreatic neuroendocrine carcinomas (Pan-NECs) was 96.7, 54.8, 0, and 0 per cent respectively. The probability of a normal lifespan was 99.8, 99.3, 79.8, and 46.8 per cent for those with stage I, II, III, and IV disease respectively. The overall disease-free RS rate was 73.6 (65.2 to 79.5) per cent. 2019 WHO grade (HR 2.10 (0.19); P < 0.001) and ENETS TNM stage (HR 2.50 (0.24); P < 0.001) significantly influenced disease-free RS. The probability of disease-free survival was 93.2, 84.9, 45.2, and 6.8 per cent for patients with stage I, II, III, and IV disease, and 91.9, 45.2, 9.4, and 0.7 per cent for those with G1, G2, G3 Pan-NENS, and Pan-NECs, respectively.

CONCLUSION

A surgical approach seems without benefit for Pan-NECs, and unnecessary for small G1 sporadic Pan-NENs. Surgery alone may be insufficient for stage III-IV and G3 Pan-NENs.

Marginal analysis of current status data with informative cluster size using a class of semiparametric transformation cure models

This research is motivated by a periodontal disease dataset that possesses certain special features. The dataset consists of clustered current status time-to-event observations with large and varying cluster sizes, where the cluster size is associated with the disease outcome. Also, heavy censoring is present in the data even with long follow-up time, suggesting the presence of a cured subpopulation. In this paper, we propose a computationally efficient marginal approach, namely the cluster-weighted generalized estimating equation approach, to analyze the data based on a class of semiparametric transformation cure models. The parametric and nonparametric components of the model are estimated using a Bernstein-polynomial based sieve maximum pseudo-likelihood approach. The asymptotic properties of the proposed estimators are studied. Simulation studies are conducted to evaluate the performance of the proposed estimators in scenarios with different degree of informative clustering and within-cluster dependence. The proposed method is applied to the motivating periodontal disease data for illustration.