There is a decent discussion of several ways to measure the extent to which data violates the PH assumption in Kleinbaum and Klein (Survival Analysis: A Self-Learning Text, 3rd ed). There is also a discussion of the extended Cox model, which relaxes the proportional hazards assumption. This reading, although not complete, should give readers a practical start on detecting and adjusting for data that violates PH. I believe examples are given in tools such as R, S-Plus and SAS.

Interestingly, it isn't the survival curves, but the log-log of the survival curves that are used to compare for parallel curves, and the reason is quite instructive.

I think hazard ratio has no real meaning and interpretation in this case. E.g. it would be clearly wrong to plug such value into a markov model if one wants to predict survival (the survivals in the model would differ from the experiment).

I think parameters like added life years and QALYs should be provided.

I never understood the fixation about Cox modelling. I believe that the reason to so many papers using Cox models is the availability of that in common statistical packages and the complete misunderstanding about other models.

Probably, researchers do not check the adjustment of the model to data...

There is a decent discussion of several ways to measure the extent to which data violates the PH assumption in Kleinbaum and Klein (Survival Analysis: A Self-Learning Text, 3rd ed). There is also a discussion of the extended Cox model, which relaxes the proportional hazards assumption. This reading, although not complete, should give readers a practical start on detecting and adjusting for data that violates PH. I believe examples are given in tools such as R, S-Plus and SAS.

Interestingly, it isn't the survival curves, but the log-log of the survival curves that are used to compare for parallel curves, and the reason is quite instructive.

Thank you all for your responses. They were all very helpful.

Please keep in mind that the Kaplan Meier curves can cross each other when there is no difference between treatment/exposure groups. However if the Cox PH model is indicating an effect of the respective exposure/treatment than the Kaplan Meier curves should not cross each other.

If KM crosses, it means there is an interaction (modification of effect) that should be reported and commented. In this case a unique HR from Cox model would not be meaningful. In any case, when I see crossing KM curves, I suspect there is a mistake in the analysis. See for example the attched paper, in which there is a well-knwon "immortal person-year bias" (the author wrongly conisdered as fixed a time-varying variable (tumor diagnosis after translplant)

Sorry, here is the attachment

In many practical cases, empirical KM curves intersect at least twice, even if the proportional hazards assumption holds and the hazard ratio is not 1. The first intersection is at time zero, when the survival fraction is 1 (i.e. zero events). This is almost always true. The second intersection when all subjects have reached time-to-event. There can be multiple crossings when the surviving sample size is small enough for random crossings to occur.

Thanks for recalling all these underlying assumptions for a correct unse and interpretation of the Cox proportional hazards model. Yes I am a physician and I do agree that in the majority of our research we do not strictly fit the mathematical conditions. In your case Ahmed I would just use stratifed Cox models.