GOMPERTZ REGRESSION MODEL WITH GAMMA FRAILTY: A STUDY ON THE APPLICATION IN LUNG CANCER

Abstract

Survival models with frailty are used when some variables are non-available to explain the occurrence time of an event of interest. This non-availability may be considered as a random effect related to unobserved covariates, or that cannot be measured, such as environmental or genetic factors. This paper focuses on the Gamma-Gompertz (denoted by G-G) model that is one of a class of models that investigate the effects of unobservable heterogeneity. We assume that the baseline mortality rate in the G-G model is the Gompertz model, in which mortality increases exponentially with age and the frailty is a fixed property of the individual, and the distribution of frailty is a gamma distribution. The proposed methodology uses the Laplace transform to find the unconditional survival function in the individual frailty. Estimation is based on maximum likelihood methods and this distribution is compared with its particular case. A simulation study examines the bias, the mean squared errors and the coverage probabilities considering various samples sizes and censored data. A real example with lung cancer data illustrates the applicability of the methodology, where we compared the G-G and without frailty models via criteria which select thebest fitted model to the data.