Stacked Laplace-EM algorithm for duration models with time-varying and random effects

An extension of the Cox proportional hazards model for clustered survival data is proposed. This allows both general random effects (frailties) and time-varying regression coefficients, the latter being smooth functions of time. The model is fitted using a mixed-model representation of penalized spline smoothing which offers a unified framework for estimation of the baseline hazard, the smooth effects and the random effects. The estimator is computed using a stacked laplace-EM (SLaEM) algorithm. More specifically, the smoothing parameters are integrated out in the log likelihood via a Laplace approximation. The approximation itself involves an integrated log-likelihood over the random cluster effects, for which the EM algorithm is used. A marginal Akaike information criterion is developed for selection among possible candidate models. The time-varying and mixed effects model is applied to unemployment data taken from the German Socio-Economic Panel. The duration of unemployment is modeled in a flexible way including smooth covariate effects and individual random effects.