= Saturation of an unreplicated two-factor regression
{title2=$\operatorname{rank}(X)=\text{number of observed cells}$}
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.
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