A new survival model with surviving fraction: An application to colorectal cancer data

Abstract

We propose a new survival model for lifetime data in the presence of surviving fraction and obtain some of its properties. Its genesis is based on extensions of the promotion time cure model, where an extra parameter controls the heterogeneity or dependence of an unobserved number of lifetimes. We construct a regression model to evaluate the effects of covariates in the cured fraction. We discuss inference aspects for the proposed model in a classical approach, where some maximum likelihood tools are explored. Further, an expectation maximization algorithm is developed to calculate the maximum likelihood estimates of the model parameters. We also perform an empirical study of the likelihood ratio test in order to compare the promotion time cure and the proposed models. We illustrate the usefulness of the new model by means of a colorectal cancer data set.

Bayesian survival model induced by frailty for lifetime with long‐term survivors

“…For this reason, there is a vast literature on "cure rate models" for survival data, also called "survival models with a surviving fraction" or "long-term survival models." Most of these models were investigated in a competing risks scenario (Cancho, Rodrigues, & de Castro, 2011;Rodrigues, de Castro, Cancho, & Balakrishnan, 2009;Tsodikov, Ibrahim, & Yakovlev, 2003), but they can also be obtained from the proportional hazard models with discrete frailty distributions (Barriga, Cancho, Garibay, Cordeiro, & Ortega, 2018;Caroni, Crowder, & Kimber, 2010;De Angelis, Capocaccia, Hakulinen, Soderman, & Verdecchia, 1999;de Souza, Cancho, Rodrigues, & Balakrishnan, 2017;Dunson & Zhou, 2000;Economou & Stehlik, 2015;Leão, Leiva, Saulo, & Tomazella, 2017;Mazroui, Mathoulin-Pelissier, MacGrogan, Brouste, & Rondeau, 2013;Wienke, 2011). Wienke (2011) introduced the frailty model in a context of univariate survival data analysis to model the unobserved heterogeneity of individuals.…”

“…The PVF family includes as special cases the gamma, inverse Gaussian, and stable positive distributions. For this formulation, the heterogeneity of the random variables Z ′ s is equivalent to the model introduced by Barriga et al (2018), which defines a random component in the expected value of the latent causes in the BCH model by Yakovlev and Tsodikov (1996). It is different from the models proposed by de Souza et al (2017) and Cancho et al (2018) since it does not include a random component in the expected value of the number of latent risks.…”

“…This class of models includes the BCH model and the model defined by Barriga et al (2018), among other models. Further, a new regression model is constructed to evaluate the covariate effects on the disease-free proportion.…”

“…Further, a new regression model is constructed to evaluate the covariate effects on the disease-free proportion. Note that Barriga et al (2018) used a classic analysis and the inferential part was carried out under the asymptotic distribution of the maximum likelihood estimators, which in situations when the sample is small, may present difficult results to be justified. In this research, we develop the Bayesian inference procedure based on the Monte Carlo Markov Chain (MCMC), specifically the collapsed Gibbs of Liu (1994) for efficient sampling from posterior distribution of the model parameters under noninformative improper priors.…”

“…The paper is organized as follows. In Section 2, we formulate a model called the BCH-Power Variance Function (BCH-PVF) cure rate model, a generalization of the model introduced by Barriga et al (2018). In Section 3, we obtain some of its mathematical properties directly from those of the exponentiated distribution defined from the baseline F 0 .…”