Event Counts

Poisson Regression Model

The most basic event count model is the Poisson regression model, which assumes that the count data are produced by a Poisson process, with a conditional mean captured by the covariates in the model:

where μi = exp(xiβ). Because the Poisson regression model is a loglinear model, the expected count changes by a factor of exp(βk) for a one-unit change in xk. The Poisson regression model can be estimated by maximum likelihood. Although it is natural to conceive of count processes as following a Poisson process, the Poisson regression model actually contains three quite restrictive limitations that limit its applicability. First, the Poisson process is premised on events being independent of each other—the occurrence of an event does not increase the probability of the unit experiencing the same type of event in the future. This assumption will often be violated in practice. Second, related to this, the Poisson regression model also assumes that the conditional variance is equal to the conditional mean, a condition known as equidispersion. In many applications, the conditional variance will exceed the conditional mean, a condition known as overdispersion. Third, the Poisson regression model underpredicts the number of zeros (absences of events) that are observed in many applications. More sophisticated count models than the Poisson regression model account for these divergences from the assumptions of the Poisson model.

Overdispersion

In an event count model, the dependent variable is the number of events, not the occurrence of an event. As a consequence, the researcher needs to make assumptions about the process that aggregates to the event count. When overdispersion is present, estimates from the Poisson regression model generally will still be consistent if the conditional mean is properly specified. (Note, however, that truncated or censored data can result in inconsistency, even if the conditional mean is properly specified). Standard errors from Poisson regressions, however, will be biased downward, leading to Type I errors (rejecting the null hypothesis when it is true) in inference. As a consequence, overdispersion must be incorporated into the model specification when present.

There are two principal sources of overdispersion in count data. On the one hand, unmodeled sources of heterogeneity may produce a conditional variance that is larger than the conditional mean. On the other hand, the assumption of independence of events inherent in the Poisson process may be violated. Typically, researchers conceive of this nonindependence as a form of contagion within units—the experience of an initial event increases the probability of the same unit experiencing subsequent events. For example, a country's [Page 854]experience of an interstate conflict may increase the probability of the country experiencing subsequent interstate conflicts.

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