Duration Models

Survival analysis concerns statistical models of the time until the first occurance of some event. In a biological setting, the event in question might be the death of an organism. In mechanics, it could be the failure of a machine. In economics, it could be entering the workforce.

The survivor function is the fundamental part of such models, denoting the probability that the time of death is later than some time t: S(t)=Pr(T>t) where T is a random variable denoting the time of death. Let f(t) denotes the pdf of T and F(t) denote the cdf of T. We can see that S must be non-increasing since S(t)=1F(t)=1 0 tf(s)ds and f is nonnegative.

Another fundamental object is the hazard function of T, λ(t), which gives the rate at which the probability of death is changing at some time t: λ(t)lim h0Pr(tTt+h|T>t)h. Using the definition of conditional probability and the cdf, we can write λ(t)=f(t)1F(t)=f(t)S(t).

The cumulative hazard function of T, Λ(t), is defined as Λ(t) 0 tλ(t)dt. Λ(t) is also called the integrated hazard. It follows that Λ(t)=lnS(t).

Common Distributions

Some commonly used survivor functions (survival time distributions) include:

F(t)=1e γtorλ(t)=γ.

F(t)=1e γt αorλ(t)=γαt α1.

There is positive or negative duration dependence as α1.