# Can Hazard Ratio be translated into ratio of medians of survival time?

In one paper describing results of survival analysis I have read a statement that implies that one can translate Hazard ratio (HR) into ratio of median survival times ($M_1$ and $M_2$) using the formula:

$HR = \frac{M_1}{M_2}$

I’m sure it doesn’t hold when one cannot assume proportional hazard model (as nothing works if HR is not well-defined). But I suspect, that even then it wouldn’t work for any survival distribution except exponential. Is my intuition right?

Your intuition is correct. The following relationship between survival functions holds:

where $r$ is the hazard ratio (see, e.g. the Wikipedia article Hazard ratio). From this we may show that your statement implies an exponential survival function.

Let us denote the medians by $M_r$, $M_1$ for two variables with hazard ratio $r$. Your statement implies

From the definition of the median, we get

Then, we substitute the relationship between survival functions

This holds for any $r$, hence

Hence, if the statement in your question holds for arbitrary HR, the survival distribution must be exponential.