Testing the homogeneity of the means of several groups of count data in the presence of unequal dispersions

Extra-dispersion (overdispersion or underdispersion) is a common phenomenon in practice when the variance of count data differs from that of a Poisson model. This can arise when the data come from different subpopulations or when the assumption of independence is violated. This paper develops a procedure for testing the equality of the means of several groups of counts, when extra-dispersions among the treatment groups are unequal, based on the adjusted counts using the concept of the design and size effects employed by Rao and Scott, [Rao, J.N.K., Scott, A.J., 1999. A simple method for analyzing overdispersion in clustered Poisson data. Statist. Med. 18, 1373-1385]. We also obtain the score-type test statistics based on quasi-likelihoods using the mean-variance structure of the negative binomial model, and study the properties and performance characteristics of these. The simulation results indicate that the statistic based on the adjusted count data, which has a very simple form and does not require the estimates of the extra-dispersion parameters, performs best among all the statistics considered in this paper. Finally, the proposed test statistic and the score-type statistic based on double-extended quasi-likelihood are illustrated by an analysis of a set of fetal implants in mice arising from a developmental toxicity study.