Using the Lambert Function to Estimate Shared Frailty Models with a Normally Distributed Random Intercept

Abstract

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, . 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.