If two 5-year survival probabilities are p1=.55 and p2=.41
the ratio is .55/.41 = 1.34 but since probabilities are in [0, 1] should I take the log first? Which is the more appropriate way to interpret the ratio?
the ratio of logs is Log(.55)/log(.41) = .671 Which is less than one although the probability .55 > .41 so taking the reciprocal I get approximately 1.49
How to interpret this...
Can I say "people in the group with higher probability have about 1.5 times the chance to survive 5 years as a person in the other group." Or do I have to qualify it and add, "on the log scale"?
Or is the former way better (without logs)? Or is there another tranformation I can use? Thanks.
If both $5$-year survival probabilities are the result of survival times having exponential distributions,
then your ratio of logs of about $0.671$ is in fact the hazard ratio (the ratio of hazard rates).
Clearly you case $1$ has a lower hazard rate than your case $2$ as its $5$-year survival probability is greater. So you should not be surprised that the ratio is less than $1$.