SOME SIMPLE HAZARD MODELS

We have seen that the hazard function is a useful way of describing the probability distribution for the time of event occurrence. Every hazard function has a corresponding probability distribution. But hazard functions can be extremely complicated, and the associated probability distributions may be rather esoteric. This section examines some rather simple hazard functions and discusses their associated probability distributions. These hazard functions are the basis for some widely employed regression models that are introduced briefly here.

The simplest function says that the hazard is constant over time: that is, h(t) = λ or, equivalently, log h(t) = μ. Substituting this hazard into equation (2.6) and carrying out the integration implies that the survival function is S(t) = e^λt. Then, from equation (2.1), we get the p.d.f., f (t) = λe^-λt. This is the p.d.f. for the well-known exponential distribution with parameter λ. Thus, a constant hazard implies an exponential distribution for the time until an event occurs (or the time between events).