  Survival models incorporating cure fractions, commonly known as cure fraction
models or long-term survival models, are widely employed in epidemiological
studies to account for both immune and susceptible patients in relation to the
failure event of interest under investigation. In such studies, there is also a
need to estimate the unobservable heterogeneity caused by prognostic factors
that cannot be observed. Moreover, the hazard function may exhibit a
non-monotonic form, specifically, an unimodal hazard function. In this article,
we propose a long-term survival model based on the defective version of the
Dagum distribution, with a power variance function (PVF) frailty term
introduced in the hazard function to control for unobservable heterogeneity in
patient populations, which is useful for accommodating survival data in the
presence of a cure fraction and with a non-monotone hazard function. The
distribution is conveniently reparameterized in terms of the cure fraction, and
then associated with the covariates via a logit link function, enabling direct
interpretation of the covariate effects on the cure fraction, which is not
usual in the defective approach. It is also proven a result that generates
defective models induced by PVF frailty distribution. We discuss maximum
likelihood estimation for model parameters and evaluate its performance through
Monte Carlo simulation studies. Finally, the practicality and benefits of our
model are demonstrated through two health-related datasets, focusing on severe
cases of COVID-19 in pregnant and postpartum women and on patients with
malignant skin neoplasms.
