Estimation of multivariate frailty models using penalized partial likelihood

There exists a growing literature on the estimation of gamma distributed mu ltiplicative shared frailty models. There is, however, often a need to mode l more complicated frailty structures, but attempts to extend gamma frailti es run into complications. Motivated by hip replacement data with a more co mplicated dependence structure, we propose a model based on multiplicative frailties with a multivariate log-normal joint distribution. We give a just ification and an estimation procedure for this generally structured frailty model. which is a generalization of the one presented by McGilchrist (1993 , Biometrics 49, 221-225). The estimation is based on Laplace approximation of the likelihood function. This leads to estimating equations based on a penalized fixed effects partial likelihood, where the marginal distribution of the frailty terms determines the penalty term. The tuning parameters of the penalty function, i.e., the frailty variances, are estimated by maximi zing an approximate profile likelihood. The performance of the approximatio n is evaluated by simulation, and the frailty model is fitted to the hip re placement data.