Modelos de sobrevivência induzida por fragilidade da família PVF: uma abordagem bayesiana

Abstract

In survival analysis, it is common to incorporate covariates such as age, sex, disease severity, among others. However, there are several unobserved factors that may also influence survival, including overall health status, lifestyle, occupation, or genetic predisposition. Such factors are often referred to as frailty and represent unobserved heterogeneity among individuals in the study. The objective of this work is to develop frailty models capable of capturing this unobserved heterogeneity in survival data. We introduce a new class of survival models that combines population unobserved heterogeneity and the presence of cured individuals through the incorporation of frailty terms from the PVF family (Power Variance Functions), referred to as PVF-Induced Defective (PVF-ID) models. This framework extends the classical theory of frailty models and defective lifetime models, allowing researchers to flexibly model the presence of long-term survivors. Inference is performed through a fully Bayesian procedure implemented via Hamiltonian Monte Carlo, while relevant frequentist properties are assessed through simulation studies. When applied to survival data related to HIV/AIDS, the model consistently identifies the effects of the covariates. Additionally, a bivariate survival model with cure fraction and frailty is proposed, based on Poisson mixtures and the PVF family. The model is extended to a regression framework, allowing the assessment of covariate effects on survival times. Inference is conducted under a Bayesian approach using Hamiltonian Monte Carlo implemented in RStan, and simulation studies are carried out to evaluate the performance of the estimators. The flexibility of the proposed structure allows the model to capture both dependence between failure times and unobserved heterogeneity. The practical utility of the model is illustrated using a real dataset.