The semiparametric Cox regression model is often fitted in the modeling of survival data. One of its main advantages is the ease of interpretation, as long as the hazards rates for two individuals do not vary over time. In practice the proportionality assumption of the hazards may not be true in some situations. In addition, in several survival data is common a proportion of units not susceptible to the event of interest, even if, accompanied by a sufficiently large time, which is so-called immune, “cured,” or not susceptible to the event of interest. In this context, several cure rate models are available to deal with in the long term. Here, we consider the generalized time-dependent logistic (GTDL) model with a power variance function (PVF) frailty term introduced in the hazard function to control for unobservable heterogeneity in patient populations. It allows for non-proportional hazards, as well as survival data with long-term survivors. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation was conducted to evaluate the performance of the models. Its practice relevance is illustrated in a real medical dataset from a population-based study of incident cases of melanoma diagnosed in the state of São Paulo, Brazil.

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Survival models with a frailty term are presented as an extension of Cox’s proportional hazard model, in which a random effect is introduced in the hazard function in a multiplicative form with the aim of modeling the unobserved heterogeneity in the population. Candidates for the frailty distribution are assumed to be continuous and non-negative. However, this assumption may not be true in some situations. In this paper, we consider a discretely distributed frailty model that allows units with zero frailty, that is, it can be interpreted as having long-term survivors. We propose a new discrete frailty-induced survival model with a zero-modified power series family, which can be zero-inflated or zero-deflated depending on the parameter value. Parameter estimation was obtained using the maximum likelihood method, and the performance of the proposed models was performed by Monte Carlo simulation studies. Finally, the applicability of the proposed models was illustrated with a real melanoma cancer data set.

In this paper, our proposal consists of incorporating frailty into a statistical methodology for modeling time-to-event data, based on non-proportional hazards regression model. Specifically, we use the generalized time-dependent logistic (GTDL) model with a frailty term introduced in the hazard function to control for unobservable heterogeneity among the sampling units. We also add a regression in the parameter that measures the effect of time, since it can directly reflect the influence of covariates on the effect of time-to-failure. The practical relevance of the proposed model is illustrated in a real problem based on a data set for downhole safety valves (DHSVs) used in offshore oil and gas production wells. The reliability estimation of DHSVs can be used, among others, to predict the blowout occurrence, assess the workover demand and aid decision-making actions.INDEX TERMS Downhole safety valve, frailty model, generalized time-dependent logistic, hydrogen sulfide concentration, non-proportional hazard.