Redundant Overdispersion Parameters in Multilevel Models for Categorical Responses

In some distributions, such as the binomial distribution, the variance is determined by the mean. However, in practice, overdispersion is often observed where the variance is larger than that predicated by the mean, and underdispersion is sometimes observed where the variance is smaller. It is well known that overdispersion or underdispersion cannot be modeled for dichotomous responses having a Bernoulli distribution. Redundant overdispersion parameters are nevertheless often included when multilevel or hierarchical models for categorical responses are estimated using quasi-likelihood methods and in generalized estimating equations. This may be due to the popularity of an algorithmic model formulation.