Interpreting Hazard Ratios: Insights from Frailty Models

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.