This question is complementary to question 1 asked by @Jesse here. It is about the case when we don't find a significant result when performing the log-rank test and when the proportional hazard assumption does not hold. In his/her answer to Jesse, @Aniko has started to answer my present question as follows: If it does not reject, then you have to worry about the proportionality of hazards and power. I totally understand we should worry, but I'd like to know how we should handle our anxiety then.
The log-rank test is always "valid" even if the proportional hazard assumption does not hold, in the following sense: under the null hypothesis $H_0$ (equality of hazard rates), the probability to reject achieves (approximately or asymptotically) the nominal significance level, that is, the power is low under $H_0$. But when the proportional hazard assumption holds true, then the behavior of the power under $H_1$ is clear because it should be an increasing function of the proportionality constant $k$.
But when this proportional hazard assumption does not hold, are there some typical "$H_1$ situations" for which it is known that the power is still low ? And for a given dataset, are there some visual methods to check whether we likely are in such a situation ?
Best Answer
The log-rank test compares the observed and expected (under $H_0$) number of events in the two groups. It will have low power, when the observed number of events matches the expected one even though the survival functions are not the same. A typical scenario for this is when the survival curves cross: one group has more events early on, the other has more events later, and the overall number of events balances out. You might be interested in the following paper by Logan et al, 2008.