Both zero inflation and dependency can often be present in hierarchical discrete data, in which observations are either clustered or repeated, measured from individual subjects. Dependency among responses can be explained by hierarchical structures using random effects, Hall[Hall (2000), Yau and Lee (2001), Hur et al.; (2002), and Wang et al. (2002), Xiang et al.; (2006) and Lee et al., (2006)(2006)]. Although the ZIP regression model can handle zero inflation for a Poisson data, and the non-zero part of the discrete data may be over-dispersed under the Poisson distribution, the mean and the variance should be the same. In some applications, however, the variance often exceeds the mean, causing over-dispersed data. Ridout et al., (2001) considered overdispersionover-dispersion in discrete data and proposed a score test for testingto test the ZIP regression model against ZINB alternatives.