Barnwal and Paul(1988) derived the likelihood ratio statistic and $C(alpha)$ statistic for testing the equality of the means of several groups of count data in the presence of a common dispersion parameter. These tests are generalized to be applicable without the restriction of a common dispersion parameter. And the assumed model of data is also extended from negative binomial to double exponential Poisson model. Monte Carlo simulations show the superiority of $C(alpha)$ statistic based on the double exponential Poisson family which has a very simple form and requires estimates of the parameters only under the null hypothesis.