"Translating" All-Cause Mortality Rate Ratios or Hazard Ratios to Age-, Longevity-, and Probability-Based Measures

Epidemiologists commonly use an adjusted hazard ratio or incidence density ratio, or a standardized mortality ratio, to measure a difference in all-cause mortality rates. They seldom translate it into an age-, time-, or probability-based measure that would be easier to communicate and to relate to. Several articles have shown how to translate from a standardized mortality ratio or hazard ratio to a longevity difference, a difference in actuarial ages, or a probability of being outlived. In this paper, we describe the settings where these translations are and are not appropriate and provide some of the heuristics behind the formulae. The tools that yield differences in "effective age" and in longevity are applicable when both 1) the mortality rate ratio (hazard ratio) is constant over age and 2) the rates themselves are log-linear in age. The "probability/odds of being outlived" metric is applicable whenever the first condition holds, and thus it provides no direct information on the magnitude of the effective age/longevity difference.