DESCRIBING SURVIVAL DISTRIBUTIONS

All of the standard approaches to survival analysis are probabilistic or stochastic. That is, the times at which events occur are assumed to be realizations of some random process. It follows that T, the event time for some particular individual, is a random variable having a probability distribution. There are many different models for survival data, and what often distinguishes one model from another is the probability distribution for T. Before looking at these different models, you need to understand three different ways of describing probability distributions.

Cumulative Distribution Function

One way that works for all random variables is the cumulative distribution function, or c.d.f. The c.d.f. of a variable T, denoted by F(t), is a function that tells us the probability that the variable will be less than or equal to any value t that we choose. Thus, F(t) = Pr{T ≤ t}. If we know the value of F for every value of t, then we know all there is to know about the univariate distribution of T. In survival analysis, it is more common to work with a closely related function called the survivor function, defined as S(t) = Pr{T > t} = 1 - F(t). If the event of interest is a death, the survivor function gives the probability of surviving beyond t. Because S is a probability, we know that it is bounded by 0 and 1. And because T cannot be negative, we know that S(0) = 1. Finally, as t gets larger, S never increases (and usually decreases). Within these restrictions, S can have a wide variety of shapes.