I have two questions:

1) Can anyone provide me with a reference regarding calculation of Hazard
ratio for two groups of data. How is it being manually calculated with an
example. Unlike median time ratio which is the ratio of median times in two
groups, at what time is the hazard ratio calculation done?

2) In kaplan-meier statistics of stratifying into two groups, a p-value is
often calculated (e.g. log-rank p-value). p-value is the test is how
significant the separation is between two groups compared to random. What is
"random" in kaplan-meier statistics. How is p-value calculated?

Thanks for everyone's time to read and hopefully respond as well.

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