First passage time models: Understanding the shape of the hazard rate
In Chapter 6 we saw how the individual variability in hazard can be described in terms of a random frailty variable. For instance, frailty may enter the hazard in a multiplicative fashion, with the individual hazard h described as h(t) = Zα(t). Here, α is a “basic” rate and Z is a nonnegative random variable taking distinct values for each individual. This may be an appropriate way to account for missing (time-independent) covariates such as those genetic and other effects that may remain constant over time. A limitation of the standard frailty approach, however, is that Z is fixed at time zero. Once the level of frailty has been set for an individual, it is retained for the rest of the lifespan. Individuals may be endowed with a set of time-dependent (external) covariates encompassing changes in their environment, but conditional on the covariates the deviation of the individual hazard from the population baseline is completely determined. Thus, knowing the hazard ratio at one instant will completely determine the hazard ratio for the future. However, it is reasonable to believe that for each individual there will be a developing random process influencing the individual hazard and leading up to an event. This fact is usually ignored in the standard models, mostly because the process itself is usually poorly understood and may in fact not be observed at all. However, this does not imply that it should be ignored. A consideration of the underlying process, even in a speculative way, may improve our understanding of the hazard rate.
KeywordsHazard Rate Wiener Process Inverse Gaussian Distribution Phase Type Distribution Positive Real Line
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