Statistical Inference Of The Mixture Cure Model For Survival Data

Survival analysis is a statistical method of studying the time-to-event of interest,and a major assumption in using traditional survival models is that all subjects will eventually experience the event of interest when the follow-up is long enough.However,with rapid developments in medical technology and significant progress in health science,it is often observed that a portion of subjects will never experience the event of interest and thus can be treated as cured or longterm survivors.Using traditional survival models that do not take the cure fraction into account to analyze the survival data from such trials will result in biased estimates,leading to the waste of available information in the data.Taking this feature into account,the traditional survival model has been extended to what is commonly referred to as the cure model.In this context,this dissertation is based on the mixture cure model to investigate survival data with a cure fraction in a gradual process with three steps from model estimation,predictive accuracy evaluation to causal effects analysis.We have conducted extensive simulation studies on the proposed statistical methods and computational algorithms,and applied them to real data,which have potential contributions to the development of precision medicine in the future.The organization of this dissertation is as follows.In the first part,we conduct a statistical analysis of interval-censored survival data with a cure fraction based on a mixture cure model,where the survival function of susceptible subjects is assumed to be a semi-parametric additive risk model.We propose a sieve maximum likelihood estimation method based on Bernstein polynomials.However,due to the complexity of the likelihood structure,it is time-consuming to directly maximize the likelihood and obtain the sieve maximum likelihood estimator.As a result,we propose an EM algorithm by exploiting a Poisson data augmentation for finding the sieve maximum likelihood estimator based on the additive risk mixture cure model.Under some mild conditions,the asymptotic properties of the proposed estimator are established,including consistency,convergence rate and asymptotic normality.In the second part,we continue to explore the statistical issues of assessing the predictive accuracy for survival outcomes with a cure fraction,based on the mixture cure model.However,developing valid statistical metrics to measure the predictive accuracy of risk scores based on survival probabilities has been challenging statistically,since both the disease status and cure status are unknown among subjects who are censored.Based on the sieve maximum likelihood estimator under the mixture cure model given in the first part,we propose a time-dependent receiver operating characteristic(ROC)curve semi-parametric estimator.Next,we further provide a Bernstein-based smoothing method in the estimation procedure,which can lead to a substantial gain in efficiency.In addition,we provide an estimator of the area under the time-dependent ROC curve,which allows for an intuitive global assessment of the discriminative power of the risk score.In the third part,we further investigate statistical issues regarding causal effect estimation in observational studies for survival data with a cure fraction,based on the mixture cure model.In observational studies,covariates may contain confounders of treatment assignment.Therefore,our goal is to adjust these covariates to correctly estimate a causal effect of treatment or exposure on survival time with a cure fraction.We extend the absolute effect measures under traditional survival data,including restricted average causal effect and survival probability causal effect,to survival data with a cure fraction.Then,we construct corresponding causal effect estimators based on propensity score stratification,where the mixture cure model for each stratum is based on a more flexible generalized product-limit nonparametric estimator.Under some mild conditions,we establish the consistency and asymptotic normality of the proposed estimators.