Page 321 - Contributed Paper Session (CPS) - Volume 7

CPS2110 Johann Sebastian B. C. et al.
not be able to reach a very high statistical power as the true form of the
variance of the count data is, in fact, a transcendental function, in which a finite
combination of polynomials can only give an approximation to an extent of
error.
As stated in the previous section, the following table presents the
proportion of samples where the estimated coefficient is negative, which
may result in a negative variance of the count data:

Form Constant Linear Quadratic Polynomial
Equidispersed 64% 65% 69% 69%
Weak 53% 53% 58% 58%
Strong 0% 0% 0% 0%
Transcendental 30% 32% 40% 40%
Table 5. Proportion of negative estimates

From the table, for the forms of weak to no overdispersion, the four
assumed variance models estimate a negative coefficient more than half of
the time, which diminishes as the true form of overdispersion becomes
stronger. For the transcendental form, however, a negative estimate happens
only a third of the time. Nonetheless, because the true form of overdispersion
is not made known to the researcher, the current method of modelling and
testing the variance of count data can be said to be prone to producing an
erroneous, negative variance. This has also been observed in the two
pioneering studies of Cameron and Trivedi (1986) (1998).

4. Discussion and Conclusion
The assumption of a certain form of the variance and testing its
significance using the parametric -distribution is the commonly used method
of testing for overdispersion in Poisson regression. However, because of the
dissimilarities of the size and power of the test under the parametric and
nonparametric paradigms, it is indicative that the -distribution may not be
the true distribution of the score test statistic, and thus, is an inappropriate
choice when a regression-based test for overdispersion is to be executed.
Moreover, assuming the form of the variance is very prone to model
misspecification and the poor performance of the model: using less
parameters in the variance model will result in a parsimonious model which
gives a good statistical size when the appropriate distribution is used but fails
to capture the extra-Poisson variation when the null hypothesis is false, thus
resulting in low statistical power of the test. On the other hand, adding more
parameters to the variance model will results in a model which may be able to
capture more of the variation in the data, but fails to be simplified when the

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