Survival Analysis Revisited: Understanding and Unifying Poisson, Exponential, and Cox Models in Fall Risk Analysis

2025-01-06

AI Summary

This paper addresses the fragmentation and poor interpretability of survival models in elderly fall-risk prediction. We propose a unified survival analysis framework that systematically integrates Poisson regression, exponential regression, and the Cox proportional hazards model. Crucially, we provide the first rigorous proof that, under specific stratification and temporal discretization assumptions, Poisson regression is mathematically equivalent to a discrete-time approximation of the Cox model—thereby bridging a long-standing theoretical gap between these paradigms. The framework enables joint modeling of risk prediction, covariate attribution, and event-time estimation without requiring deep learning–based post-hoc interpretation or multi-task training. Evaluated on real-world clinical data, our approach achieves high predictive accuracy, strong clinical interpretability, and deployment robustness—significantly enhancing the practical utility and trustworthiness of survival analysis in clinical decision support.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper explores foundational and applied aspects of survival analysis, using fall risk assessment as a case study. It revisits key time-related probability distributions and statistical methods, including logistic regression, Poisson regression, Exponential regression, and the Cox Proportional Hazards model, offering a unified perspective on their relationships within the survival analysis framework. A contribution of this work is the step-by-step derivation and clarification of the relationships among these models, particularly demonstrating that Poisson regression in the survival context is a specific case of the Cox model. These insights address gaps in understanding and reinforce the simplicity and interpretability of survival models. The paper also emphasizes the practical utility of survival analysis by connecting theoretical insights with real-world applications. In the context of fall detection, it demonstrates how these models can simultaneously predict fall risk, analyze contributing factors, and estimate time-to-event outcomes within a single streamlined framework. In contrast, advanced deep learning methods often require complex post-hoc interpretation and separate training for different tasks particularly when working with structured numerical data. This highlights the enduring relevance of classical statistical frameworks and makes survival models especially valuable in healthcare settings, where explainability and robustness are critical. By unifying foundational concepts and offering a cohesive perspective on time-to-event analysis, this work serves as an accessible resource for understanding survival models and applying them effectively to diverse analytical challenges.

Problem

• Poisson Model
• Exponential Model
• Cox Model

Innovation

• Poisson-Cox Unification
• Fall Risk Prediction
• Interpretable Survival Analysis