Two-way analysis of variance ($Y_{ij}=\alpha +r_i+g_j+{\epsilon }_{ij}$)

An additive two-factor Gaussian model separates row and column effects. With one observation in each of $rc$ cells, constraints fixing one row and column effect leave $r+c-1$ mean parameters and $(r-1)(c-1)$ residual degrees of freedom. Balanced least squares gives fitted values equal to row mean plus column mean minus grand mean. With no cell replication, unrestricted interactions are confounded with residual error; the additive interpretation requires an assumption that those interactions are absent or negligible.

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.

For observed cells with means $a_i+b_j$, form a bipartite graph joining row $i$ to column $j$ when that cell is observed. If all levels occur, the design rank is $r+c-k$, where $k$ is the number of connected components. Indeed a coefficient perturbation in the kernel satisfies $a_i=-b_j$ on every observed edge, and hence has exactly one freely chosen constant on each component. A connected design has rank $r+c-1$ and is identifiable after a single reference constraint. Missing cells destroy the balanced-design orthogonality of row and column effects, so sequential sums of squares can depend on order.

A two-way analysis of variance with the full interaction and one observation in each observed factor cell has one independently varying fitted cell mean per observation. A cell-indicator design matrix has full row matrix rank, even when some combinations are missing, so the model is a saturated statistical model. Its residual sum of squares is zero and its residual degrees of freedom are zero. Missing combinations make some coefficients nonidentifiable; they do not restore a pure-error estimate. Standard errors and an ordinary residual-based F-test for interactions therefore cannot be recovered merely by fitting this larger model.