Understanding the shape of the hazard rate: A proce ss point of view

The shape of the hazard rate as a function of time has great variation. Sometimes it is just increasin g, sometimes decreasing, and at other times it is a co mbination of both these features. For instance, the risk of divorce increases after marriage up to a ti me and then decreases. From frailty theory it is kn own 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, describ ed by a Markov process (of diffusion type, or with discrete state space), such that "death" correspond s to reaching a certain limit. The shape of the haz ard rate of the time it takes to reach this limit depen ds on the quasi-stationary distribution on the tran sient state space. It will also be shown that first-passage-time model s (like for instance the inverse gaussian distribution) are useful survival models for analyzing data, also when covariates are present. I n fact, many of the covariates used in survival analyses are indicators of how far some underlying process has advanced. 1. The process point of view Inn survival andd event history analysis one mainly considers only the occurrence of events.. The underlying process leading upp too the event is generally ignored.. The mainn reasonn for this is that the process is usually completely,, or at least partly,, concealed,, a ndd it seems difficult too incorporate something about which one has noo knowledge.. However,, we want to mak e the point that one should consider the underlying process whether it is knownn or not,, and that important insight may be gathered from this.. A reference is Aalenn & Gjessing (2001). It is natural too model the underlying process inn terms of well-e stablishedd stochastic models.. Important examples are the following ones: • Finite state Markov chain. If there is one absorbing state,, withh the remaining states constituting a single transient class,, thenn the time too absorpti onn is denotedd a phase-type distribution.. There is a well-developedd theory for suchh distribution s. • Multistage models of carcinogenesis. There exists a well developedd mathematical theory of carcinogenesis. Essentially this is basedd onn various kinds of time-continuous Markov models