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

TL;DR

This paper revisits core survival analysis methods in the context of fall risk, linking logistic, Poisson, exponential, and Cox models under a unified time-to-event framework with key quantities $S(t)$ and $h(t)$. It demonstrates that Poisson regression in survival contexts is a specific case of the Cox model with a constant baseline hazard, clarifying the connections among GLMs and time-to-event methods. It shows how these models can simultaneously predict fixed-interval risk, quantify covariate effects via hazard ratios, and estimate time-to-event within a single, interpretable framework, highlighting advantages over deep learning approaches that often lack explainability. The work emphasizes practical healthcare impact by applying the unified framework to fall detection and suggesting broader applicability to disease progression and prevention tasks.

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.