Modelo logístico generalizado dependente do tempo com fragilidade

Abstract

Several authors have preferred to model survival data in the presence of covariates through the hazard function, a fact related to its interpretation. The hazard function describes as the instantaneous average of failure changes over time. In this context, one of the most used models is the Cox s model (1972), in which the basic supposition for its use is that the ratio of the failure rates, of any two individuals, are proportional. However, experiments show that there are survival data which can not be accommodated by the Cox s model. This fact has been determinant in the developing of several types of non-proporcional hazard models. Among them we mention the accelerated failure model (Prentice, 1978), the hybrid hazard model (Etezadi-Amoli and Ciampi, 1987) and the extended hybrid hazard models (Louzada-Neto, 1997 and 1999). Mackenzie (1996) proposed a parametric family of non-proportional hazard model called generalized time-dependent logistic model - GTDL. This model is based on the generalization of the standard logistic function for the time-dependent form and is motivated in part by considering the timeeffect in its setting and, in part by the need to consider parametric structure. The frailty model (Vaupel et al., 1979, Tomazella, 2003, Tomazella et al., 2004) is characterized by the use of a random effect, ie, an unobservable random variable, which represents information that or could not or were not collected, such as, environmental and genetics factors, or yet information that, for some reason, were not considered in the planning. The frailty variable is introduced in the modeling of the hazard function, with the objective of control the unobservable heterogeneity of the units under study, including the dependence of the units that share the same hazard factors. In this work we considered an extension of the GTDL model using the frailty model as an alternative to model data which does not have a proportional hazard structure. From a classical perspective, we did a simulation study and an application with real data. We also used a Bayesian approach to a real data set.