Frailty models are generally used to model heterogeneity between the individuals. The distribution of the frailty variable is often assumed to be continuous. However, there are situations where a discretely-distributed frailty may be appropriate. In this paper, we propose extending the proportional hazards frailty models to allow a discrete distribution for the frailty variable. Having zero frailty can be interpreted as being immune or cured (long-term survivors). Thus, we develop a new survival model induced by discrete frailty with zero-inflated power series distribution, which can account for overdispersion. A numerical study is carried out under the scenario that the baseline distribution follows an exponential distribution, however this assumption can be easily relaxed and some other distributions can be considered. Moreover, this proposal allows for a more realistic description of the non-risk individuals, since individuals cured due to intrinsic factors (immune) are modeled by a deterministic fraction of zero-risk while those cured due to an intervention are modeled by a random fraction. Inference is developed by the maximum likelihood method for the estimation of the model parameters. A simulation study is performed in order to evaluate the performance of the proposed inferential method. Finally, the proposed model is applied to a data set on malignant cutaneous melanoma to illustrate the methodology.

中文翻译：

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一个新的长期生存模型，其离散脆弱性引起了色散。

脆弱模型通常用于建模个体之间的异质性。通常认为脆弱变量的分布是连续的。但是，在某些情况下，散布的脆弱可能是适当的。在本文中，我们建议扩展比例风险脆弱性模型，以允许脆弱性变量的离散分布。零脆弱可以被解释为是免疫的或治愈的（长期幸存者）。因此，我们开发了一种由零散脆弱性导致的零膨胀幂级数分布诱导的新生存模型，该模型可以解决过度分散问题。在基线分布遵循指数分布的情况下进行了数值研究，但是可以轻松地放宽此假设，并可以考虑其他一些分布。此外，该提议允许对非风险个体进行更现实的描述，因为由于内在因素（免疫）治愈的个体是通过确定的零风险分数建模的，而由于干预而治愈的个体是通过随机分数建模的。通过最大似然法进行推论，以估计模型参数。为了评估所提出的推理方法的性能，进行了仿真研究。最后，将所提出的模型应用于恶性皮肤黑色素瘤的数据集，以说明该方法。通过最大似然法进行推论，以估计模型参数。为了评估所提出的推理方法的性能，进行了仿真研究。最后，将所提出的模型应用于恶性皮肤黑色素瘤的数据集，以说明该方法。通过最大似然法进行推论，以估计模型参数。为了评估所提出的推理方法的性能，进行了仿真研究。最后，将所提出的模型应用于恶性皮肤黑色素瘤的数据集，以说明该方法。