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!
Best Answer
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.