I recently study survival analysis since couple of months ago, and there is one question which makes me curious. a simple and (maybe) a stupid question.
Why do we prefer to use ln-ln survival curve rather than survival curve in proportional hazard test? As you know, the $\hat{S}$ we use in ln-ln formula is survival function based on Kaplan-Meier, so why we don't use that 'original' KM survival function?