Page 317 - Contributed Paper Session (CPS) - Volume 7

CPS2110 Johann Sebastian B. C. et al.

The parameter is often called the rate or intensity parameter which is
taken to be positive, while is the measure of space or time where the event
of interest is observed. By modelling the rate at how the counts are generated
as a function of a set of covariates , the Poisson
regression model arises, which is immediately derived from the Poisson
distribution by conditioning the counts on the rate parameter as a linear
combination of the covariates (Cameron & Trivedi, 1998), i.e.:

As with the distribution, the Poisson regression model assumes
equidispersion, where the response variance is equal to its mean.
Underdispersion happens when the mean exceeds the variance, while
overdispersion happens when the variance exceeds the mean. True
overdispersion occurs when the excess variation among the counts is
attributed to a non-Poisson data-generating process (DGP), while apparent
overdispersion occurs as a consequence of a misspecified count model.
The violation of the equidispersion property immediately manifests in the
inflated fit statistics, and it can underestimate the standard errors, thereby
invalidating the inference coming from the count model. However, while the
fit statistics can indicate the presence of overdispersion, the magnitude at
which the data is deemed to suffer from overdispersion is relative. (Hilbe,
2011) Formal tests for detecting overdispersion have been developed by
shifting the focus to different count distributions or variance specifications.
When equidispersion is violated, the response variance can be hypothesized
to be some function of the response mean, i.e.:

With this form, the variance can be modelled via regression. In practice,
ℎ() is assumed to have a closed form – at most, an algebraic expression, say,
ℎ()=() – and a regression-based test for overdispersion can be carried
out by testing the null hypothesis 0: =0 i.e. equidispersion, using the test
statistic (Cameron & Trivedi, 1990):

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