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In analyzing longitudinal data with counted responses, normal distribution is usually used for distribution of the random efffects. However, in some applications random effects may not be normally distributed. Misspecification of this distribution may cause reduction of efficiency of estimators. In this paper, a generalized log-gamma distribution is used for the random effects which includes the normal one as a special case. As the frquentist analysis faces with complex computation, the Bayesian analysis of this model is investigated and then it is utilized for analyzing two real data sets. Also, some simulation studies are conducted to evaluate the performance of the relevant models.