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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

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    $\begingroup$ You should consider "cure models". A cure model is survival model with a cure fraction, i.e. a fraction of the population that is not susceptible to the event. $\endgroup$ – ocram Jan 6 '14 at 14:26
  • $\begingroup$ You can use survival analysis, but this is a criticism of the method (that the probability of the event is zero for some cases). For example, not every person may have a heart attack. $\endgroup$ – Ellis Valentiner Jan 6 '14 at 14:30
  • $\begingroup$ (continued) One of the challenges in the cure model methodology is that it is difficult to distinguish between a censored observation and an observation from a cured subject. It appears that you won't encounter this difficulty in this particular setting. $\endgroup$ – ocram Jan 6 '14 at 14:41
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    $\begingroup$ Thanks @ocram. You are correct. In my particular setting, the individuals who make it to the end time are cured and there would be no need to keep observing them. Cure models looks like the way to go. Thanks I had never heard of them before. $\endgroup$ – user1893354 Jan 6 '14 at 14:46
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    $\begingroup$ If all you are interested in is whether they quit before the Olympics, then you could use logistic regression with the outcome being succeed/fail. If you are also interested in how long they took to fail, you will need survival models of some sort, possibly the cure models that @ocram discusses. $\endgroup$ – Peter Flom - Reinstate Monica Jan 6 '14 at 15:22
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A few thoughts:

  • 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:

  • 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.

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  • $\begingroup$ regarding the second last bullet point, the issue I have is that in the context of the problem, we are not interested in whether they get injured after the olympics. So maybe that is not the route I want to explore. It seems like your last bullet point is more appropriate. I have not studied competeing risk survival analysis - can it deal with the case when the time to one outcome is unknown (injury time) and the other is known (time of olympics)? Thanks $\endgroup$ – user1893354 Jan 6 '14 at 16:33
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    $\begingroup$ @user1893354 Yes. By modeling them as two separate processes, you could for example have no censored observations for "Time to Olympics" and a relatively sophisticated interval censoring scheme for "Time until Injury". $\endgroup$ – Fomite Jan 6 '14 at 16:37
  • $\begingroup$ +1 for "competing risks". This allows you to have actual censored values -- somehow you don't know if they made it to the Olympics or not -- as well as two possible outcomes: a) injury, b) competed in Olympics. $\endgroup$ – Wayne Oct 20 '16 at 14:41

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