# Why does the hazard ratio represent the magnitude of distance between the Kaplan-Meier plots?

From Wikipedia

The hazard ratio is simply the relationship between the instantaneous hazards in the two groups and represents, in a single number, the magnitude of distance between the Kaplan-Meier plots

As far as I know a Kaplan-Meier plot is the estimate of the survival function under a level of the covariate. The hazard ratio is the ratio between the hazard function values at two different levels of the covariate. So I was wondering why the hazard ratio represents the magnitude of distance between the Kaplan-Meier plots? Thanks!

• The very next paragraph in the article you quote proceeds to answer this question.
– whuber
Aug 6 '13 at 19:52
• @whuber: Thanks. My understanding about that paragraph is limited. My question is rather "why" than "how". My confusion comes from that one is for survival function, and the other is for hazard function.
– Tim
Aug 6 '13 at 19:56

The Cox proportional hazards model can be written in terms of the effect of predictor variables on the log relative hazard, which is also the effect on the log relative cumulative hazard scale. Log cumulative hazard is equal to the log of the -log of the cumulative survival function which Kaplan-Meier estimates. So you could say that the log hazard ratio (regression effect in the Cox model) estimates the average difference between two Kaplan-Meier estimates if you transform both of them by the log-log transformation.

• I'd rather say: the log hazard ratio estimates the difference between the two "log minus log'' survival functions (this difference is constant by the proportional hazards assumption). Do you mean that the log hazard ratio is a kind of average between the two log minus log'' Kaplan-Meier estimates ? Is it the maximum likelihood estimate in the Cox model ? Actually I don't know how it is defined. Aug 22 '13 at 19:11

In the Cox proportional hazard models for two groups we assume that the hazard rate $\lambda_1$ in group 1 is related to the hazard rate $\lambda_2$ in group 2 by $\lambda_2(t)=k\lambda_1(t)$ for a certain proportionality constant $k>0$.

Then, denoting by $S_1$ and $S_2$ the survival functions in group 1 and group 2 respectively, the proportional hazard assumption is equivalently written $\log S_2(t) = k \log S_1(t)$ (by integration), and then $\boxed{\log(-\log S_2(t))=\log(k)+\log(-\log S_1(t))}$.

I would add that this boxed formula provides a visual method to check the proportional hazards assumption: plot the two "$\log$ minus $\log$'' transformations of the Kaplan-Meier estimates $\hat S_1$ and $\hat S_2$ on the same graphic; under the proportional hazards assumption it is expected that the two curves are "parallel". With R, you get this graphic by typing:
fit <- coxph(Surv(time,status)~strata(group), mydataframe)