Long term survival model with frailt terms based on the power variance function family

Abstract We present a new class of survival models that combine unobserved population heterogeneity and cured individuals by incorporating frailty terms from the Power Variance Function (PVF) family, referred to as PVF-Induced Defective (PVF-ID) models. This structure broadens classic frailty theory and defective lifetimes under a proportional-hazards lens, letting researchers flexibly track long-term survivors in mixed cohorts. Inference follows a full-Bayesian routine run via Hamiltonian Monte Carlo, and key frequentist properties are cross-checked using simulation studies. When applied to HIV/AIDS records, the model reliably pinpoints important covariates and quantifies the fraction of cured patients. In this case study, among several baseline hazards, the simple exponential form wins by both leave-plus-one marginal likelihood (LPML) and raw marginal likelihood. Together, these findings position the PVF-ID framework as a robust, versatile toolbox for studying survival data with immune subpopulations.