Duration Dependence and Dispersion in Count-Data Models

This article explores the relation between nonexponential waiting times between events and the distribution of the number of events in a fixed time interval. It is shown that within this framework the frequently observed phenomenon of overdispersion—that is, a variance that exceeds the mean—is caused by a decreasing hazard function of the waiting times, whereas an increasing hazard function leads to underdispersion. Using the assumption of iid gamma-distributed waiting times, a new count-data model is derived. Its use is illustrated in two applications, the number of births and the number of doctor consultations.