**Odd O. Aalen**

*Department *

University of Oslo

Norway

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.