Convergent stochastic algorithm for estimation in general multivariate correlated frailty models using integrated partial likelihood

The Cox model with unspecified baseline hazard is often used to model survival data. In the case of correlated event times, this model can be extended by introducing random effects, also called frailty terms, leading to the frailty model. Few methods have been put forward to estimate parameters of such frailty models, and they often consider only a particular distribution for the frailty terms and specific correlation structures. In this paper, a new efficient method is introduced to perform parameter estimation by maximizing the integrated partial likelihood. The proposed stochastic estimation procedure can deal with frailty models with a broad choice of distributions for the frailty terms and with any kind of correlation structure between the frailty components, also allowing random interaction terms between the covariates and the frailty components. The almost sure convergence of the stochastic estimation algorithm towards a critical point of the integrated partial likelihood is proved. Numerical convergence properties are evaluated through simulation studies and comparison with existing methods is performed. In particular, the robustness of the proposed method with respect to different parametric baseline hazards and misspecified frailty distributions is demonstrated through simulation. Finally, the method is applied to a mastitis and a bladder cancer dataset.