Multi‐dimensional penalized hazard model with continuous covariates: applications for studying trends and social inequalities in cancer survival

Describing the dynamics of patient mortality hazard is a major concern for cancer epidemiologists. In addition to time and age, other continuous covariates have often to be included in the model. For example, survival trend analyses and socio‐economic studies deal respectively with the year of diagnosis and a deprivation index. Taking advantage of a recent theoretical framework for general smooth models, the paper proposes a penalized approach to hazard and excess hazard models in time‐to‐event analyses. The baseline hazard and the functional forms of the covariates were specified by using penalized natural cubic regression splines with associated quadratic penalties. Interactions between continuous covariates and time‐dependent effects were dealt with by forming a tensor product smooth. The smoothing parameters were estimated by optimizing either the Laplace approximate marginal likelihood criterion or the likelihood cross‐validation criterion. The regression parameters were estimated by direct maximization of the penalized likelihood of the survival model, which avoids data augmentation and the Poisson likelihood approach. The implementation proposed was evaluated on simulations and applied to real data. It was found to be numerically stable, efficient and useful for choosing the appropriate degree of complexity in overall survival and net survival contexts; moreover, it simplified the model building process.