Regression model have a substantial impact on interpretation of treatments, genetic characteristics and other covariates in survival analysis. In many datasets, the description of censoring and survival curve reveals the presence of cure fraction on data, which leads to alternative modelling. The most common approach to introduce covariates under a parametric estimation are the cure rate models and their variations, although the use of defective distributions have introduced a more parsimonious and integrated approach. Defective distributions is given by a density function whose integration is not one after changing the domain of one the parameters. In this work, we introduce two new defective regression models for long-term survival data in the Marshall-Olkin family: the Marshall-Olkin Gompertz and the Marshall-Olkin inverse Gaussian. The estimation process is conducted using the maximum likelihood estimation and Bayesian inference. We evaluate the asymptotic properties of the classical approach in Monte Carlo studies as well as the behavior of Bayes estimates with vague information. The application of both models under classical and Bayesian inferences is provided in an experiment of time until death from colon cancer with a dichotomous covariate. The Marshall-Olkin Gompertz regression presented the best adjustment and we present some global diagnostic and residual analysis for this proposal.

翻译：回归模型在生存分析中对治疗、遗传特征及其他协变量的解释具有重要影响。在许多数据集中，对删失和生存曲线的描述揭示了数据中存在治愈部分，这催生了替代建模方法。在参数估计下引入协变量的最常见方法是治愈率模型及其变体，尽管缺陷分布的使用引入了一种更为简约和综合的建模途径。缺陷分布由其密度函数定义，当改变其中一个参数的定义域时，其积分不等于一。本文在Marshall-Olkin族中针对长期生存数据提出了两种新的缺陷回归模型：Marshall-Olkin Gompertz模型和Marshall-Olkin逆高斯模型。估计过程采用最大似然估计和贝叶斯推断进行。我们通过蒙特卡洛研究评估了经典方法的渐近性质，以及贝叶斯估计在模糊先验信息下的表现。两种模型在经典与贝叶斯推断框架下的应用通过一个结肠癌死亡时间的二元协变量实验进行展示。Marshall-Olkin Gompertz回归模型表现出最佳的拟合效果，并针对该模型提供了全局诊断和残差分析。