Spatial Proportional Hazards Model with Differential Regularization

TL;DR

This work develops a nonparametric spatial proportional hazards model where the hazard is $\lambda(t|\mathbf{x},\mathbf{p}) = \lambda_0(t) \exp(\mathbf{x}^\top \boldsymbol{\beta}_0 + h_0(\mathbf{p}))$, incorporating a smooth spatial effect $h_0$ on a 2D domain. Estimation combines penalized partial likelihood with a Laplacian penalty and a sieve based on $H^2$-conforming finite elements to handle irregular domains via FEM. The authors prove identifiability, consistency, and asymptotic normality of the parametric component, and demonstrate finite-sample performance through simulations and two empirical studies (London fire-response times and Campi Flegrei seismic triggering). The method yields high-resolution spatial risk maps and outperforms standard PH and GAM-based spatial approaches, with practical applicability to areal data as an extension. Overall, this framework provides robust, interpretable inference for spatially structured survival data.

Abstract

The Proportional Hazards (PH) model is one of the most common model used in survival analysis, which typically assumes a log-linear relationship between covariates and the hazard function. However, this assumption may not hold in practice. This paper introduces a nonparametric extension of the PH model, which generalizes the log-linear assumption by allowing for an unspecified, smooth function of covariates, enabling more flexible modeling. We focus on applications with spatial survival data, where the location of an event affects the risk. The proposed model captures this spatial variation using a nonparametric spatial effect. We estimate the spatial effect using finite element methods on a mesh constructed from a triangulation of the domain, which allows us to handle irregular shapes. The model remains within the classical partial likelihood framework, ensuring computational feasibility. To enforce the smoothness in the nonparametric spatial effect, we consider a differential penalization. We establish the asymptotic properties of the proposed estimator using sieve methods, demonstrating its consistency and the asymptotic normality of the parametric component. A simulation study is conducted to evaluate the model's performance, followed by two empirical applications that demonstrate its practical advantages over standard PH models, especially in settings with spatial dependence in survival data.

Figures (6)

• Figure 1: Example of a linear finite element basis function on a planar mesh. The function is locally supported, taking the value one at a specific vertex and zero at all other vertices.
• Figure 2: Spatial simulation results: $(a)$ true effect, $(b)$ sampling design, $(c)$ estimated effect, and $(d)$ pointwise error for $n = 500$ with 30% censoring.
• Figure 3: Fire incident locations coloured by fire engine arrival time (in minutes) across Greater London. Grey lines indicate the triangular finite element mesh, with the thick black outline marking the administrative boundary. The inset shows the location of Greater London within the United Kingdom.
• Figure 4: Estimated spatial effect $(a)$ and relative risk $(b)$ for fire engine response times across Greater London.
• Figure 5: Map of censored and uncensored observations/smartphones for the 18 February 2025 earthquake in Campi Flegrei (Italy). Grey lines depict the triangular mesh, while the thick black outline marks the administrative boundary. Blue and red dots represent uncensored and censored observations, respectively. The black star denotes the earthquake epicentre. The inset map shows the location of the study area, highlighted in red, within Italy.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2: Consistency
• Theorem 3: Asymptotic Distribution
• proof
• proof
• proof
• Lemma 1
• proof
• Lemma 2
• proof