7.3 One observation per cell

If \(n = 1\), there is no replication: \(SSE\) has \(ab(n - 1) = 0\) degrees of freedom, and the interaction and error cannot be separated. The two-factor model with interaction then fits the data exactly. To test the main effects one must assume that the interaction is zero, or small, and use \(SS_{AB}\) as the error sum of squares, with \((a - 1)(b - 1)\) degrees of freedom. The analysis is then that of a randomised complete block design, with one factor in the role of blocks, and Tukey’s one-degree-of-freedom test of Section 5 can be used to check the assumption.

If the assumption is wrong, \(E(MS_{AB}) > \sigma ^2\), and the \(F\) tests for main effects are conservative: they lose power but do not reject too often. The experiment is safe, but may miss real effects. Unreplicated designs are nevertheless widely used when runs are expensive, especially two-level designs with many factors, where the high-order interactions are assumed negligible; this is taken up in the next section.

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