# Survival analysis where P(event) < 1

Suppose you are interested in analyzing time to event data for a sample of patients. You are interested in the time elapsed until a patient contracts an illness. However, a majority of patients will never contract the illness in question, i.e they are structural zeros. Also, your data is right censored; you do not observe when all patients contract an illness.

Most survival analysis techniques deal with phenomena where the lifetime probability of an event is equal to one. What are some techniques to deal with time to event data where lifetime probability of an event is less than one?

Edit

To put another way, we normally assume that as t approaches infinity, S(t) approaches 0. Under this assumption, integrating over S(t) gives us expected lifetime. But what if S(t) asymptotically approaches, say, .9 as t approaches infinity? How do you even integrate S(t)?

• I don't think your assumption that most methods assume lifetime probability is one is correct, a lot of survival analysis is about finding how covariables modify survival (e.g., Cox proportional hazards). Certainly if your data has no events at all then that's a problem. – purple51 Oct 13 '14 at 0:33
• agreed, in a PhD survival analysis class right now. We finished section on parametric models and just started non-parametric models. Have yet to see an assumption that P(event) = 1. – bdeonovic Oct 13 '14 at 0:48
• I guess the issue is when you have a process that generates structural zeros, and then a time to event process. In other words, how do you decompose the two problems? How do you determine the probability that a patient will ever get sick, vs the time to getting sick for those patients that are not structural zeros. – Ben Rollert Oct 13 '14 at 1:01
• @purple51 that is not quite correct. Most survival models deal with the time till event and assume that everybody eventually experiences the event, this includes the Cox model. What the covariates modify is how long it takes for you to die, not whether or not you die. This does not mean that everybody in your dataset will have to have experienced the event. Censoring is a central theme in these models. In fact, the way many of these models deal with censoring is exactly the assumption that everybody will eventually experience the event. – Maarten Buis Oct 13 '14 at 8:25
• Yes, a critical issue is integrating over survival function; the traditional assumption is that this should yield the expected lifetime. If you assume that all events eventually occur, you can integrate over the survival function. This interpretation doesn't make sense if many will never experience an event. How can something have an expected lifetime, if it is likely to live forever? – Ben Rollert Oct 14 '14 at 17:22