Background: Survival analysis

Abstract

We recall some elementary definitions concerning probability distributions, putting an emphasis toward one minus the usual cumulative distribution function, i.e., the survival function. This is also sometimes called the survivorship function. The closely related hazard function has, traditionally, been the most popular function around which to construct models. For multistate models it can be helpful to work with intensity functions, rather than hazard functions since these allow the possibility of moving in and out of states. This is facilitated by the very important function, Y(t), the “at-risk” indicator. A number of special parametric cases of proportional hazards models are presented. The issue of censoring and the different kinds of censoring is discussed. The “at-risk” indicator Y i (w, t), taking the value one when the subject iis at risk of making a transition of a certain kind, indicated by w, makes it particularly simple to address more complex issues in survival such as repeated events, competing risks, and multistate modelling. We consider some tractable parametric models, the exponential model in particular.