The Case of Proportional Cell Frequencies for the Two-Way Cross-Classification with Interaction

The case of proportional cell frequencies for the two-way cross-classification with interaction is considered. Several types of hypotheses for the general unbalanced data that are commonly used in the literature are shown, and they are written out for this particular case. A reparameterized form of the cell means model is defined to establish the reparameterized model, and orthogonal property of the model is shown using the augmented matrix and the numerator sums of squares are computed. Different ways of producing the same analysis of variance tables are shown in both orthogonal and nonorthogonal situations.