Missing cells can destroy factor orthogonality in two-way ANOVA ($C^{\mathsf{T}}(I-P_1)J\ne 0$)

In a fully balanced crossed design, centered indicator columns for the two factors are orthogonal because each pair of levels occurs equally often. Missing cells can change the level-pair counts so that the displayed cross-product becomes nonzero. Then sequential sums of squares depend on term order, although the final additive fit is unchanged. Testing either factor conditional on the other requires its extra sum of squares from the appropriate reduced model.