Abstract

Shared frailty models, that is, hazard regression models for censored data including random effects acting multiplicatively on the hazard, are commonly used to analyze time-to-event data possessing a hierarchical structure. When the random effects are assumed to be normally distributed, the cluster-specific marginal likelihood has no closed-form expression. A powerful method for approximating such integrals is the adaptive Gauss-Hermite quadrature (AGHQ). However, this method requires the estimation of the mode of the integrand in the expression defining the cluster-specific marginal likelihood: it is generally obtained through a nested optimization at the cluster level for each evaluation of the likelihood function. In this work, we show that in the case of a parametric shared frailty model including a normal random intercept, the cluster-specific modes can be determined analytically by using the principal branch of the Lambert function, $W_0$. Besides removing the need for the nested optimization procedure, it provides closed-form formulas for the gradient and Hessian of the approximated likelihood making its maximization by Newton-type algorithms convenient and efficient. The Lambert-based AGHQ (LAGHQ) might be applied to other problems involving similar integrals, such as the normally distributed random intercept Poisson model and the computation of probabilities from a Poisson lognormal distribution.

摘要

共享脆弱性模型，即包含对危害进行乘法作用的随机效应的审查数据的危害回归模型，通常用于分析具有层次结构的事件发生时间数据。当假设随机效应服从正态分布时，特定于簇的边际似然没有封闭形式的表达式。一种用于逼近此类积分的强大方法是自适应高斯-埃尔米特积分 (AGHQ)。然而，这种方法需要在定义集群特定边际似然的表达式中估计被积函数的模式：它通常是通过对似然函数的每次评估在集群级别进行嵌套优化获得的。在这项工作中，我们展示了在包含正常随机截距的参数共享脆弱性模型的情况下，$W_0$. 除了消除对嵌套优化过程的需要外，它还为近似似然的梯度和 Hessian 提供了封闭形式的公式，使其通过牛顿型算法的最大化变得方便和高效。基于 Lambert 的 AGHQ (LAGHQ) 可能适用于涉及类似积分的其他问题，例如正态分布随机截距泊松模型和泊松对数正态分布的概率计算。