A simulation study on the score test for poison overdispersion under different forms of the variance

In the analysis of count data, overdispersion happens when the response variance of the counts exceeds the response mean; and its presence in the count model often leads to underestimated standard errors and erroneous inferences. The existing method for detecting overdispersion is through the regression-based test, where the estimated response variance is modelled with a function of the mean, whose form is established prior to fitting the variance model; a score test statistic is then compared with the quantiles of a 𝑡𝑛−1 distribution. However, establishing the form of the variance a priori may lead to a mis specified variance, more so that it is not directly observed; consequently, the test statistic is also affected. This study explores the properties of the existing test under different scenarios of specifying the variance of an unknown form. Bootstrap simulations are carried out on different a priori models where the variance is modelled as a constant, a linear function, a quadratic function, and a polynomial function of the mean. Simulations show that while the parametric framework of the score test for overdispersion may seem to yield an improper statistical size yet high statistical power, its nonparametric counterpart yields a better statistical size yet low statistical power. This trend is observed as the assumed form of the variance model becomes more complicated. Moreover, simulations have shown that, as the framework does not consider restrictions in the estimation procedure of the variance model, negative estimates are observed, which may then yield to negative variances.