Defective regression models for cure rate modeling in Marshall-Olkin family

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.