Identifiability of an incomplete two-factor additive design ($\mathrm{rank}X=r+c-k$)

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.