Nonparametric proportional hazards model with differential regularization applied to spatial survival data

The proportional hazards (PH) model is one of the most widely used models in survival analysis, typically assuming a log-linear relationship between covariates and the hazard function. However, in the context of spatial survival data, where the time-to-event variable is associated with a spatial location within a given domain, this assumption is often unrealistic in capturing spatial effects. Thus, this paper proposes modeling the location effect through a nonparametric function of spatial location. The function is approximated using finite element methods on a triangulated mesh to accommodate irregular domains. Estimation is carried out within the classical partial likelihood framework, with smoothness of the spatial effect enforced through differential penalization. Using sieve methods, we establish the consistency and asymptotic normality of the parametric component. Simulations and two empirical applications demonstrate superior performance compared to existing approaches.