Familiar model in the analysis of count data is Poisson which imposes the restrictive assumption that the variance equals the mean. This restriction is not usually satisfied in practice and thus model fitting process leads to invalid estimation results of parameters. This paper introduces the concept of over-distribution and its related tests, and presents several count data regression models, such as the negative binomial and the generalized Poisson, as alternatives to the Poisson model. Furthermore, some models for count data are compared under the assumption of having the same two first order moments. Also, we use the maximum likelihood approach to estimate the model parameters. Finally, we consider a real data set taken from the study of the impact of the metabolic syndrome among obese children in Isfahan. We analyze the number of white globules as responses and then select the best fitted model by using of some standard model choice criteria.