Chi‐Square Tests for Overdispersion with Multiparameter Estimates

When a vector of sample proportions is not obtained through simple random sampling, the covariance matrix for the sample vector can differ substantially from that corresponding to the multinomial model. For example, clustering effects or subject effects in repeated measure experiments can cause the variance of the observed proportions to be much larger than variances under the multinomial model. The phenomenon is generally referred to as overdispersion. This paper presents a method of analysing categorical data in the presence of overdispersion. The actual distribution of the sample proportion need not be known but it is assumed that the covariance matrix is a function of the population proportions and a limited number of scaling parameters. Parameter estimates are obtained through a combination of generalized least squares and moment estimation techniques and large sample chi‐square tests are developed. Tests of fit of the models proposed are also developed and illustrated. Comparisons are made with other methods proposed in the literature, for overdispersed data.