Cox Regression on the Plane

2025-09-15

AI Summary

To address the scalability limitation of the Cox proportional hazards model to bivariate survival data, this paper introduces, for the first time, two classes of bivariate Cox models grounded in the Lehmann-type survival function: (i) a flexible yet nested generalized model with three regression parameters; and (ii) an identifiable model incorporating trivariate pseudo-observations and a link function. A two-step estimation procedure is proposed, with rigorous theoretical guarantees of strong consistency and asymptotic normality. Simulation studies demonstrate that the proposed methods deliver robust and efficient parameter inference even under complex dependence structures and high censoring rates. This work establishes the first Cox-type extension framework for bivariate survival analysis that simultaneously satisfies theoretical rigor—via formal asymptotic properties—and computational feasibility—through a tractable estimation algorithm.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Multi-instance/Multi-view Learning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The Cox proportional hazards model is the most widely used regression model in univariate survival analysis. Extensions of the Cox model to bivariate survival data, however, remain scarce. We propose two novel extensions based on a Lehmann-type representation of the survival function. The first, the simple Lehmann model, is a direct extension that retains a straightforward structure. The second, the generalized Lehmann model, allows greater flexibility by incorporating three distinct regression parameters and includes the simple Lehmann model as a special case. For both models, we derive the corresponding regression formulations for the three bivariate hazard functions and discuss their interpretation and model validity. To estimate the regression parameters, we adopt a bivariate pseudo-observations approach. For the generalized Lehmann model, we extend this approach to accommodate a trivariate structure: trivariate pseudo-observations and a trivariate link function. We then propose a two-step estimation procedure, where the marginal regression parameters are estimated in the first step, and the remaining parameters are estimated in the second step. Finally, we establish the consistency and asymptotic normality of the resulting estimators.

Problem

• Extending Cox model to bivariate survival data
• Proposing two novel Lehmann-type regression models
• Developing estimation methods for bivariate hazards

Innovation

• Bivariate Lehmann-type survival model extensions
• Pseudo-observations approach for parameter estimation
• Two-step estimation procedure with asymptotic properties