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.

翻译：比例风险（PH）模型是生存分析中应用最广泛的模型之一，通常假设协变量与风险函数之间存在对数线性关系。然而，在空间生存数据背景下——其中事件发生时间变量与给定域内的空间位置相关联——这一假设在捕捉空间效应时往往不切实际。因此，本文提出通过空间位置的非参数函数来建模位置效应。该函数在三角网格上采用有限元方法进行近似，以适应不规则区域。估计在经典的部分似然框架内进行，并通过微分惩罚项确保空间效应的平滑性。利用筛法，我们建立了参数分量的一致性和渐近正态性。仿真与两项实证应用表明，相较于现有方法，本模型具有更优的性能。