Correlated destructive generalized power series cure rate models and associated inference with an application to a cutaneous melanoma data

In this paper, we propose a new cure rate survival model, which extends the model of Rodrigues et al. (2011) by incorporating a structure of dependence between the initiated cells. To create the structure of the correlation between the initiated cells, we use an extension of the generalized power series distribution by including an additional parameter @r (the inflated-parameter generalized power series (IGPS) distribution, studied by Kolev and Minkova (2000)). It has a natural interpretation in terms of both a ''zero-inflated'' proportion and a correlation coefficient. In our approach, the number of initiated cells is assumed to follow the IGPS distribution. The IGPS distribution is a natural choice for modeling correlated count data that exhibit overdispersion. The primary advantage of this distributional assumption is that the correlation structure induced by the additional parameter @r results in a natural characterization of the association between the initiated cells. Moreover, it provides a simple and realistic interpretation for the biological mechanism of the occurrence of the event of interest as it includes a process of destruction of tumor cells after an initial treatment or the capacity of an individual exposed to irradiation to repair initiated cells that result in cancer being induced. This means that what is recorded is only the undamaged portion of the original number of initiated cells not eliminated by the treatment or repaired by the repair system of an individual. Parameter estimation of the proposed model is then discussed through the maximum likelihood estimation procedure. Finally, we illustrate the usefulness of the proposed model by applying it to real cutaneous melanoma data.