A mixed model for categorical data from unbalanced designs which is directly analogous to a two-way ANOVA model for quantitative data is proposed. An extension of the fitting constants method is developed to estimate model variance components based on appropriate reductions in sums of squares. The resulting variance component estimators are incorporated into the covariance structure of a general linear models Wald statistic to test for treatment differences. These procedures are illustrated with data obtained from a multicenter clinical trial in which the treatments are regarded as fixed effects and the clinics are regarded as random effects.