Page 316 - Contributed Paper Session (CPS) - Volume 7

CPS2110 Johann Sebastian B. C. et al.

A simulation study on the score test for poison
overdispersion under different forms of the
variance
Johann Sebastian B. Claveria, Erniel B. Barrios, Joseph Ryan G. Lansangan
University of the Philippines, Diliman, Quezon City, Philippines

Abstract
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
−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.

Keywords
count data; Poisson regression; overdispersion; score test; bootstrap

1. Introduction
The Poisson distribution is the canonical model used in the study of count
data. It arises naturally from a Poisson counting process which only allows
non-negative integers to be observed in a given time , thus making it a more
appropriate model for count data than other real-valued distributions in the
traditional framework of statistical inference. (Karlin & Taylor, 1975) The
Poisson distribution can be characterized by its mass function:

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