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I want to model probability of an event, which is typically modelled by survival methods. However, the predicted survival function is always non-increasing. My case is however opposite: The probability of survival increases in time, i.e. a subject which already survived some time has higher probability to survive than the subject which is younger. Does it make sense to use survival analysis for this, or are there any other approaches dealing with right censored data ?

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  • $\begingroup$ What you describe doesn't seem possible to me. If $t2>t1$, then $P[X > t1]=P[X>t2]+P[t1<X]\ge P[X>t2]$. $\endgroup$ – John L Feb 25 at 14:13
  • $\begingroup$ I agree with John L on this. Survival analysis asks a question like "what is the probability of surviving to some future point t2 given that you are alive at t1?". If you have 1000 people at t1 that are alive, you cannot have more than 1000 in the future at time t2. However, it is possible that the hazard of dying/failure changes over time. For example, let's say we are dealing with newly infected COVID patients. The hazard of dying from COVID for these patients will be low initially, increase over 1-3 weeks, and then fall again. Provide a simple example and we can help sort out. $\endgroup$ – Brant Inman Feb 25 at 14:25
  • $\begingroup$ I can't edit my comment. I meant $P[X>t1]=P[X>t2]+P[t1<X\le t2] \ge P[X>t2]$. $\endgroup$ – John L Feb 25 at 14:27
  • $\begingroup$ What you describe seems to be a decreasing hazard of an event over time. The hazard is the instantaneous risk of an event given that you have already survived up to that time. A survival model can readily deal with a decreasing hazard over time. The survival function itself represents the total fraction of the initial group still alive at any particular time after the start. That necessarily is non-increasing with time. As others have commented, please edit your question to clarify just what you mean. $\endgroup$ – EdM Feb 25 at 15:24
  • $\begingroup$ I think I have messed up "hazard" with "survival". @EdM do you want to make your comment an answer to this? $\endgroup$ – pikachu Feb 26 at 16:54
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The terminology of survival and hazard functions can be confusing.*

What you describe is a decreasing hazard of an event over time. The hazard is the instantaneous risk of an event given that you have already survived up to that time. A survival model can readily deal with a decreasing hazard over time. For example, for some parameter values a Weibull distribution has a continuously decreasing hazard over time.

The survival function itself, for an event that happens only once per individual, represents the total fraction of the initial group still without the event at any particular time after the start. That necessarily is non-increasing with time.


*Earlier today, for example, I temporarily mixed up cumulative hazard with cumulative event probability, until I caught myself. Loose common-language use of such words means that special care and vigilance are needed in their technical use.

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