Two useful reformulations of the hazard ratio

The hazard ratio is a standard summary for comparing survival curves yet hazard ratios are often difficult for scientists and clinicians to interpret. Insight into the interpretation of hazard ratios is obtained by relating hazard ratios to the maximum difference and an average difference between survival probabilities. These reformulations of the hazard ratio are useful in classroom discussions of survival analysis and when discussing analyses with scientists and clinicians. Large-sample distribution theory is provided for these reformulations of the hazard ratio. Two examples are used to illustrate the ideas.