Department of Statistics, Universidade de São Paulo, São Carlos, SP, Brazil.
Department of Statistics, Universidade Federal de São Carlos, São Carlos, SP, Brazil.
Lifetime Data Anal. 2020 Apr;26(2):221-244. doi: 10.1007/s10985-019-09472-2. Epub 2019 Apr 9.
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.
脆弱性模型通常用于对个体之间的异质性进行建模。脆弱性变量的分布通常假定为连续的。然而，在某些情况下，离散分布的脆弱性可能是合适的。在本文中，我们提出将比例风险脆弱性模型扩展到允许脆弱性变量具有离散分布。零脆弱性可以解释为免疫或治愈（长期幸存者）。因此，我们开发了一种新的生存模型，由具有零膨胀幂级数分布的离散脆弱性引起，该模型可以考虑过度分散。在基线分布遵循指数分布的情况下进行了数值研究，但是可以很容易地放宽此假设，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）而治愈的个体通过零风险的确定性分数进行建模，而由于干预而治愈的个体则通过随机分数进行建模。通过最大似然法对模型参数进行了推断。进行了模拟研究以评估所提出的推断方法的性能。最后，将所提出的模型应用于恶性皮肤黑素瘤数据集，以说明该方法。
