F tz

1 tq(t)/ ¡z Q(y)dy if t < z and as z ^ to, the usual results for stationary processes hold.

5 Discussion

Backward and forward recurrence time distributions play an important role in modeling human disease. Two examples are presented which make use of recurrence times; i.e. modeling both the progressive chronic disease and the early disease detection processes. In these examples, the process is influenced by disease incidence which is usually age related. As a result, the recurrence time distributions are generalized to be functions of disease incidence. A characteristic of the progressive chronic disease model is that the process may have been going on for a long time relative to when the process is beginning to be studied at time t0. As a result, it may be convenient to consider t0 to be far removed from the origin.

A closely related topic is the role of disease incidence in length biased sampling. We have generalized the length biased process to take account of incidence. In our general development, we have also derived the distribution of the "complement" of length biased sampling which may be called "anti-length biased sampling." Our setting is that at time to the process is being observed and individuals who are alive with disease are characterized by being length biased and tend to have longer time in the state in which they are observed at this time. However, individuals who entered and exited the state before time t0 will tend to have a shorter stay in that particular state. In the progressive disease modeling example, those who died of disease before time to will tend to have shorter survival times than those in the disease state at t0.

One can envision a study in which observations are taken within the chronological time period (to, ti). We have considered the situation where to is the beginning of a study. However, if ti represents the time point at which the study stops, observations may be right censored at that time. The conventional approach is to have a model which incorporates a censoring distribution and the observation is the minimum of the censoring distribution and the distribution in that state. However, another formulation is that the time in the state up to ti is a backward recurrence time. Consequently, the likelihood function will be different than the usual formulation for right censored observations.

More generally, the results presented here are applicable when a stochastic process has discrete (or countable) states and is initially being observed. The sojourn times for those observations in the initially observed states are forward recurrence times. Consequently, any modeling of the process using the initial observations must incorporate these forward recurrence times.

References

[CM65] Cox, D.R. and Miller, H.D. : The theory of stochastic processes.

Chapman, London (1965) [FEL71] Feller, W. : An introduction to probability theory and its application. Volume II, 3rd edition. Wiley, New York (1971) [ZF69] Zelen, M. and Feinleib, M. : On the theory of screening for chronic diseases. Biometrika , 56, 601-14 (1969)

This article has appeared in the December 2004 issue of Lifetime Data Analysis

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