Power Comparison of Tests for Extra-Variation of Counts Data

We formulate the extension of Poisson exponential family to incorporate the extra-variation of counts data. Assuming a generalized linear mixed model in which the natural parameter identifies a linear predictor including random effects, we discuss several tests for the Poisson assumption against the extra-variation model. As competitors of score test we suggest the likelihood ratio test and the Wald test. A fit of GLMM (generalized linear mixed model) with computational complexity is required in the latter two statistics but they have no serious obstacle in practical use because commonly used statistical packages provide procedures to fit GLMM. The tests have been applied to an example, and through a Monte Carlo study we also compare them in the respects of sizes and powers.