Partial Likelihood Estimation of a Cox Model with Random Effects: an EM Algorithm based on Penalized Likelihood.

The aim of this paper is to present a general EM algorithm to estimate Mixed Proportional Hazard models including more than one random effect, through partial likelihood. We assume only that the mixing distributions admit Laplace transforms. We show how to transform inference in a single complicated model in the estimation of MPH models involving only a single frailty, which are easily manageable. We then face on gamma unobserved heterogeneity. This choice is a weak assumption as the heterogeneity distribution among survivors converges to a gamma distribution, often quickly, for many types of unobserved heterogeneity distributions. The proposed approach can thus be used to estimate a wide class of models. We describe how to use the penalized partial likelihood within the EM algorithm, to improve speed and stability. The behaviour of the estimator on different clusterings and sample sizes is assessed through a Monte Carlo study. We also provide an application on the ratiffcation of ILO conventions by developing countries over the period 1975-1995. Both the simulations and the empirical results indicate an important decrease in computing time. Furthermore, our procedure converges in settings where a standard EM algorithm does not.