In this thesis, we introduce the robust likelihood function proposed by Royall and Tsou (2003). Based on the method we first establish a robust parametric likelihood ratio test about regression parameters for the correlation coefficients modeled in a generalized linear model fashion. Next, we construct a robust parametric score test to compare means of several dependent populations of count. Furthermore, a robust parametric likelihood ratio test for regression parameters of means for correlated count data is proposed. The validity of the proposed likelihoods requires no knowledge of the true underlying distributions, so long as they have finite fourth or second moments. The efficacy of the robust methodology is demonstrated via simulations and real examples.