ABSTRACT
This chapter describes a particular problem, involving an experiment with binary responses. The problem described involves several dispersion components associated with different subgroups or populations, and is not to be confused with simple over-dispersion in which the observations are independent and there is only one dispersion component. Linear models in which there is more than one variance component have been used for many years going back at least to the work of R. A. Fisher and F. Yates in agricultural experiments. There is a parallel but extensive literature on educational testing, which uses ordinary maximum-likelihood estimation for unbalanced Normal-theory linear models. For Normal-theory linear models the previous two methods reduce to restricted maximum likelihood. D. A. Anderson and M. Aitkin develop an unmodified maximum-likelihood estimation procedure for nested variance components in linear logistic and probit models.
