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 (M1 and M2) using the formula:

HR=M1M2

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?

**Answer**

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

S1(t)=S0(t)r

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 Mr, M1 for two variables with hazard ratio r. Your statement implies

Mr=M0/r

From the definition of the median, we get

Sr(M0/r)=0.5

Then, we substitute the relationship between survival functions

S0(M0/r)r=0.5⇒S0(M0/r)=0.51/r

This holds for any r, hence

S0(t)=0.5t/M0=etlog0.5M0

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

**Attribution***Source : Link , Question Author : Adam Ryczkowski , Answer Author : Scortchi – Reinstate Monica*