Frailty models are essential tools in survival analysis for addressing unobserved heterogeneity and random effects in the data. These models incorporate a random effect, the frailty, which is assumed to impact the hazard rate multiplicatively. In this paper, we introduce a novel class of frailty models in both univariate and multivariate settings, using phase-type distributions as the underlying frailty specification. We investigate the properties of these phase-type frailty models and develop expectation-maximization algorithms for their maximum-likelihood estimation. In particular, we show that the resulting model shares similarities with the Gamma frailty model, has closed-form expressions for its functionals, and can approximate any other frailty model. Through a series of simulated and real-life numerical examples, we demonstrate the effectiveness and versatility of the proposed models in addressing unobserved heterogeneity in survival analysis.

翻译：脆弱模型是生存分析中处理未观测异质性和数据随机效应的关键工具。这些模型引入了一个随机效应——脆弱因子，假定其以乘积形式影响风险率。本文提出了一类新颖的脆弱模型，适用于单变量与多变量场景，其核心采用相位型分布作为脆弱因子的设定框架。我们系统研究了这类相位型脆弱模型的性质，并开发了基于期望最大化算法的极大似然估计方法。特别地，我们证明该模型与Gamma脆弱模型具有相似特性，其泛函表达式具有闭合形式，且能逼近任意其他脆弱模型。通过系列仿真与实际数据算例，我们验证了所提模型在解决生存分析中未观测异质性问题的有效性与普适性。