# Interpreting Hazard Ratios: Insights from Frailty Models

## TL;DR

This paper examines the interpretation of hazard ratios from Cox models, highlighting issues with conditional estimates and proposing a method to derive more individual-relevant hazard ratios using twin study data and sensitivity analysis.

## Contribution

It introduces a two-step approach to estimate hazard ratios with causal relevance, accounting for unmeasured heterogeneity and providing a framework for sensitivity analysis.

## Key findings

- Hazard ratios from conventional models may not reflect causal effects.
- A proposed method uses twin data to assess unmeasured heterogeneity.
- The approach offers more interpretable hazard ratios under certain assumptions.

## Abstract

Hazard ratios are often used to evaluate time to event outcomes, but they may be hard to interpret. A particular issue arise because hazards are typically estimated conditional on survival, i.e.\ on left truncated samples. Then, hazard ratios from conventional models cannot be interpreted as counterfactual hazard ratios that are immediately relevant to individual patients. This article explores how the hazard ratios from Cox models may differ from hazard ratios with a causal interpretation. Using summary data from twin studies, I suggest an approach to learn about the unmeasured heterogeneity in risk of an outcome, and this information allows us to explore the interpretation and magnitude of hazard ratios. Under explicit parametric assumptions, I present a two-step method to obtain hazard ratios that are more relevant to individual subjects. The strategy relies on untestable assumptions, but may nevertheless be useful for sensitivity analyses that are relatively easy to interpret.