# How do you convert hazard ratios to differences in life expectancy?

How do epidemiologists convert hazard ratios to difference in life expectancy?

For example, let's say I find that the hazard ratio of all-cause mortality in City A is 1.25 compared to City B. I want to find out how to calculate how much shorter life expectancy is predicted to be in City A compared to City B based on the findings from the hazard model.

One solution that I think works is to use life tables and calculate life expectancy with and without a 25% increase in the probability of dying each year. From these hazards, you can get the unconditional probability of dying each year, which you can multiply by age and sum to get life expectancy.

I am wondering if this is in line with the standard way of making the calculation. References to textbooks are welcome.

Hazard ratios are ratios of instantaneous rates and are relative measures. Assuming a hazard ratio for comparing two groups is a single constant (i.e, that the proportional hazards (PH) assumption holds), the log HR equals the difference in to two (parallel) log-log survival curves. HR cannot lead to differences in life expectancy by themselves. For that you may have to assume a parametric proportional hazards model. This chapter presents one of the parametric PH models—-the Weibull model. The R rms package psm and Mean functions can be used to estimate expected life lengths for this model. More generally one can fit the hazard function with a spline function to be more flexible than Weibull and still be able to extrapolate to get life expectancy and median life length.

If you have data fitting an accelerated failure time model instead, these model median life length directly, and medians and their differences are easy to estimate using these models.

From the Cox PH model you can get mean restricted survival time (RMST) which is the area under a covariate-specific survival curve from zero to some fixed time horizon that is at or before your study’s follow-up ends.

If you have a reference survival curve that goes all the way to $$S(t)=0$$ and you know a hazard ratio that will convert that curve to represent another type of study unit, and you know that the hazard ratio does not depend on time, then the survival curve for the other unit is $$S(t)^h$$ where $$h$$ is the other unit : reference unit hazard ratio. Getting the area under $$S(t)^h$$ will estimate the life expectancy for the new unit.

• This is really useful and I appreciate the reference. But you say that a parametric assumption is required…does that mean it is wrong to use the life table method I mentioned in my question? This empirical method seems more attractive to me… Commented Nov 4, 2023 at 20:56
• Estimated life expectancy is the area under a survival curve from $t=0$ to $t=z$ where $z$ is the point $S(t)$ reaches zero. You will almost never have all subjects ultimately fail in a study, so you can only estimate RMST or use parametric model extrapolation to $t=\infty$ to estimate life expectancy. Or do you have life tables that go all the way to no one surviving? If so there is another possibility which I’m adding to the end of my answer. Commented Nov 5, 2023 at 11:43
• Yes I do have the life tables! I mean, I'm using the life tables from the US's Social Security Administration linked to in my original question. I'm not trying to get life tables from the original studies--challenging as you say--just trying to get ballpark estimates of the impact on life expectancy. My feeling is that converting the hazard ratios to differences in life expectancy for a "typical person" using the SSA data would be more accurate in general than having to assume Weibull or something (curious if you disagree). Commented Nov 5, 2023 at 19:58
• Makes sense. Depends on how fine a resolution you want to describe types of persons. Commented Nov 5, 2023 at 20:31

David Spiegelhalter's "micromorts" and "microlives" give approximate conversions of hazard ratios to life expectancies for a standard (UK) population under a Gompertz model. For example, here

Very roughly, a person in their 30's facing, from some cause, a daily relative risk of death of 1.09 throughout their life is losing around 1 microlife per day.

where a microlife is half an hour of life expectancy