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?
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)?