Understanding the shape of the hazard rate: The use of quasi- stationary distributions in first-passage-time models.



Odd Aalen
Section of Medical Statistics
P.O.Box 1122 Blindern
Norway


Abstract

The shape of the hazard rate as a function of time has great variation. Sometimes it is just increasing, sometimes decreasing, and at other times it is a combination of both these features. For instance, the risk of divorce increases after marriage up to a time and then decreases. From frailty theory it is known that such shapes may have complex explanations, and do not simply reflect a development of risk at the individual level.

To understand these features better it is useful to look at first-passage-time models of survival and "death". One assumes an underlying process, described by a Markov process (of diffusion type, or with discrete state space), such that "death" corresponds to reaching a certain limit. The shape of the hazard rate of the time it takes to reach this limit depends on the quasi-stationary distribution on the transient state space.

It will also be shown that first-passage-time models (like for instance the inverse gaussian distribution) are useful survival models for analyzing data, also when covariates are present. In fact, many of the covariates used in survival analyses are indicators of how far some underlying process has advanced.