An investigation of dynamic covariate effects in survival data

Survival data are often modelled by the Cox proportional hazards model, which assumes that covariate effects are constant over time. Estimation of covariate effects in such models is usually based on the partial likelihood function with the baseline hazard being estimated non-parametrically. In recent years however, several new approaches have been suggested which allow survival data to be modelled more realistically by allowing the covariate effects to vary with time. Non-proportional hazard fimctions, with covariate effects changing dynamically, can be fitted using penalised splines (P-splines). Links exist between P-spline smoothing and penalised quasi-likelihood estimation in generalised linear mixed models allowing estimation of the smoothing parameters steering the amount of smoothing. Here a hybrid form for smoothing parameter selection is suggested which combines the mixed model approach with a classical Akaike criterion. Two approaches to estimation of dynamic covariate effects in survival data are considered. One is a Poisson type approach based on the likelihood function and allows for estimation of the baseline hazard, usually treated as a nuisance parameter. The second is a numerically faster approach based on the partial likelihood function. Both approaches are evaluated with simulations and applied to data from the German Socio-Economic Panel. The partial likelihood approach is also applied to data from the West of Scotland Coronary Prevention Study