Is it possible to compare survival curves from different models (Cox and Fine-Gray)? Ideally in R I would like to perform a time to event analysis on some biological example with animals. The event of interest is a terminal event X. But the animal may die from other reason before it happens. So I would like to compare three curves:


*

*Death and the terminal event X are equated. I can assign "1" to an animal regardless of the cause. I can use Kaplan-Meier.

*Death is censored. I am interested only in the event X. I can assign "0" to death. I can use Kaplan-Meier.

*Death is a competing risk. So I model the time to event X "corrected" for deaths. I am asked to use the Fine-Gray subdistribution hazard (I know it's under criticism, but I have to).
First question - do these models have any special name? I heard about crude, cause specific, and net survival, but still confuse them. 
Each of the models can "generate" a survival curve to me. In R it's "survfit" I believe. Now, having three curves, telling me the probability of the event, I would like to compare them with a test, say, LogRank.
How to do that, knowing these are 3 separate models?
Or maybe it would be easier to compare median survival times? But it may happen the curve  won't reach 50%. And I don't know any test that compares survival medians.
 A: What you described are different outcomes. You shouldn't do a hypothesis test to see if the survival curve of event X = the survival curve of (event X or death). They are different by definition. If there are not many deaths, the curves may look similar. The uncertainty in your curves may be much greater than the difference between the curves, which is what a hypothesis test would be looking at. But the curves are still different. Your null hypothesis is rejected by definition without ever doing a hypothesis test.
I focused on comparing outcomes 1 vs 2 above, but a similar argument holds for, say, comparing 2 and 3. Suppose you use the Fine and Gray model to estimate a cumulative incidence curve and compare it to 1 minus the Kaplan-Meier plot. You would be trying to estimate the cumulative incidence of event X in both cases (potentially badly with the 1-KM plot if the competing risk impactful). The null is true by the definition of what you are trying to estimate. If the curves looked different, you would simply focus on the method known to do a better job in your setting.
Rather than looking for statistical tests, in this case you are better off just plotting your curves with confidence bands so the difference between the curves can be seen in the light of the uncertainty of the estimates.
