Is this a problem for Survival analysis? I have a dataset of individuals.  Each individual has the same start time at which we begin observing them.  There is also an end time for all individuals.  Some individuals fail before they reach the end time and some individuals never fail and reach the end time (i.e. they succeed).  This is a survival analysis problem in that I am trying to model a time to event (failure).  Where my confusion comes in is with the individuals who succeed.  I obviously cannot treat these individuals as observed failures, but I also cannot treat them as censored.  This is because censorship implies that they were not observed to fail but they will fail at some point in the future (we just don't know when).  This is not true in my case because if an individual does not fail up to the end time, it will never fail.  So how do I deal with these individuals who never fail? Is this still a survival analysis problem? 
Here is an example.
Say a group of athletes all begin training for the next olympic games.  Some of them will end up getting injured from training and will not be able to compete.  But if an athlete does not get injured until the date of the olympic games, then he/she would be able to compete (i.e. succeed).  So these athletes that don't get injured and are able to compete are not censored, because whether they get injured after the olympics is irrelevant.  All we care about is whether they get injured up to the date of the olympics
 A: A few thoughts:


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*As @PeterFlom mentioned, if you just want to find out whether or not they "make it", that's a 0/1 outcome that is amenable to logistic or binomial regression, and can be addressed outside the context of survival analysis if you don't much care about the time they took to fail.


However…you said you do care, so turning to survival analysis:


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*Yes, "time to event" questions are survival analysis questions, and this looks like a pretty clear one.

*Whether or not "Olympic censoring" is a problem really depends on the question you want to ask. Are you just talking about time until injury in a cohort of athletes training for the Olympics? If that's the case, I can see making an argument that those who do make it to the Olympics are censored - given infinite time and no other outcome they will injure themselves, we've just stopped following them.

*If, on the other hand, you are really interested in Injury while training for the Olympics then yes, treating them as censored is a problem. What you actually have now is a time until two mutually exclusive outcomes: 1) Injury before the Olympics and 2) Participation in the Olympics. You're now in the domain of competing risks survival analysis, of which the cure models suggested in the comments are a sub-set. A related set of models are "mixture models", which model, as the name suggests, a mixture of two outcomes.


This paper: http://aje.oxfordjournals.org/content/170/2/244.short by Lau, Cole and Gange in AJE is an excellent review of mixture models for survival analysis. Be warned however that, coming from a place with nicely documented and implemented packages for conventional survival analysis that the mixture model universe is…somewhat less developed from the software side.
