# How should I deal with the consequences of proportional hazards violations in log-rank (and related) tests?

I have (censored) time-to event data for subjects in four groups. I would like to do something like a logrank test, but the survival curves do not satisfy the proportional hazards assumption. I think I have heard that the consequence of a prop. hazards violation is loss of power to identify differences between survival curves, however, in the case of my study, the logrank tests do show significant differences between the groups.

The proportional hazards violation takes the following form: one group has a relatively larger probability of an event early in the observation period, and another group has a relatively larger probability of an event late in the observation period. I believe that the G-$\rho$ family of tests (for example in the survdiff function in R's survival package) can be parameterized such that the earlier or later portion of the observation period is more heavily weighted. However, in this case, the different groups would "do better" (the event being studied is a good thing) in different specifications for the test.

I would like to know several things:

1. if the logrank function does find significant difference despite the presence of a proportional hazards violation, can we interpret this as indicating a true (overall or average) difference between the curves? or does this violation mean the test's results are totally meaningless?
2. is there a principled way of describing survival times in cases like mine? I would ideally like to be able to report an overall hazard ratio (I know -- this would lack external validity with the non-proportional hazards and censored observations, but would be useful in describing the experiment), as well as give information about which groups were more likely to have events at which times. I could choose a break point in the middle of the observation period and just do separate tests before and after (assuming these subsets of the data did satisfy the proportional hazards assumption), but the choice of such a point feels somewhat ad hoc.

Related discussion:

This thread discusses alternatives to the logrank test, but doesn't consider my issue: What are the pros and cons of using the logrank vs. the Mantel-Haenszel method for computing the Hazard Ratio in survival analysis?