On penalized likelihood estimation for a non-proportional hazards regression model
The fundamental assumption of proportionality of hazards in the Cox
model sometimes does not hold in practice. In this paper, a semi-parametric generalization of the Cox model that permits crossing hazard curves is described. This model allows the interaction between covariates and the baseline hazard, and has been the subject of recent investigation. It includes, for the two sample problem, the case of two Weibull distributions and two extreme value distributions differing in both scale and shape parameters. The partial likelihood approach cannot be applied here to estimate the model parameters, and flexible methods based on splines and sieves for approximating the baseline hazard have been suggested. A theoretical framework for estimation in this generalized model is developed based on penalized likelihood methods. It is shown that the optimal solution to the baseline hazard, baseline cumulative hazard and their ratio are exponential splines with knots at the unique failure times. Its relationship to prior computational approaches for this model is outlined.