The defective generalized Gompertz distribution and its use in the analysis of lifetime data in presence of cure fraction, censored data and covariates

Survival analysis methods are widely used in studies where the variable of interest is related to the time until the occurrence of an event. The usual methods assume that all individuals under study are subject to this event, but there are practical situations where this assumption is unrealistic. In some cases it is possible that a percentage of individuals are immune to the event of interest or, especially in cancer clinical trials, they were cured from their disease after a given treatment. In the literature, this percentage is usually referred as "cure fraction". In the present paper, we have proposed a model based on a modification of the generalized Gompertz distribution introduced by El-Gohary et al. (2013) to account for the presence of a cure fraction. We also considered the presence of censored data and covariates. Maximum likelihood and Bayesian methods for estimation of the model parameters are presented. A simulation study is provided to evaluate the performance of the maximum likelihood method in estimating parameters. In the Bayesian analysis, posterior distributions of the parameters are estimated using the Markov chain Monte Carlo (MCMC) method. An example involving a real data set is presented.

生存分析（Survival Analysis）方法广泛应用于关注变量与某事件发生前时长相关的研究中。传统生存分析方法通常假设研究中的所有个体均会发生所关注的事件，但在实际场景中该假设往往并不符合现实。部分情况下，一定比例的个体可能对所关注的事件具有免疫性；尤其是在癌症临床试验中，部分受试者在接受特定治疗后可被治愈。在现有学术文献中，该比例通常被称为“治愈比例（cure fraction）”。本文提出了一种基于El-Gohary等人（2013年）提出的广义Gompertz分布（Generalized Gompertz Distribution）改进的模型，以适配存在治愈比例的研究场景。同时，本文还考虑了删失数据（Censored Data）与协变量（Covariates）的存在。本文给出了用于估计模型参数的极大似然估计法与贝叶斯估计法。通过模拟研究评估了极大似然估计法在参数估计中的性能表现。在贝叶斯分析环节，本文采用马尔可夫链蒙特卡洛（Markov chain Monte Carlo, MCMC）方法估计参数的后验分布。最后，本文给出了一个基于真实数据集的应用实例。