Title: Survival models induced by zero-modified power series discrete frailty: Application with a melanoma dataset

Abstract

Survival models with a frailty term are presented as an extension of Cox’s proportional hazard model, in which a random effect is introduced in the hazard function in a multiplicative form with the aim of modeling the unobserved heterogeneity in the population. The frailty term is a non-negative variable that indicates the frailty of the unit and it represents the information that can not be or has not been observed as environmental factor, genetic factor, or other information that for some reason was not considered in the planning of the study. Candidates for the frailty distribution are assumed to be continuous and non-negative. However, this assumption may not be true in some situations, it is appropriate to consider discretely-distributed frailty, for example, when heterogeneity in lifetimes arises because of the presence of a random number of flaws in a unit or because of exposure to damage on a random number of occasions. In this paper, we consider a discretely-distributed frailty model that allows units with zero frailty, that is, it can be interpreted as having long-term survivors. We propose a new discrete frailty-induced survival model with a Zero-Modified Power Series family, which can be zero-inflated or zero-deflated depending on the parameter value. Parameter estimation was obtained using the maximum likelihood method, and the performance of the proposed models was performed by different Monte Carlo simulation studies in which different scenarios were analyzed for the parameter p. Finally, the applicability of the proposed models was illustrated with a real melanoma cancer dataset.