7.1 Main effects and interaction

For two factors, write \(\mu _{ij}\) for the mean response at level \(i\) of \(A\) and level \(j\) of \(B\), and use dots for averages: \(\bar \mu _{i\cdot } = \frac 1b\sum _j\mu _{ij}\), \(\bar \mu _{\cdot j} = \frac 1a\sum _i\mu _{ij}\) and \(\bar \mu _{\cdot \cdot }\) the average of all \(ab\) cell means.

By construction \(\mu _{ij} = \mu + \tau _i + \beta _j + (\tau \beta )_{ij}\) with \(\mu = \bar \mu _{\cdot \cdot }\), and \begin {gather*} \sum _i\tau _i = 0 , \qquad \sum _j\beta _j = 0 , \\ \sum _i(\tau \beta )_{ij} = 0 \text { for each } j , \qquad \sum _j(\tau \beta )_{ij} = 0 \text { for each } i . \end {gather*} The factors do not interact exactly when all \((\tau \beta )_{ij} = 0\), that is, when the difference \(\mu _{ij} - \mu _{i'j}\) between two levels of \(A\) is the same at every level of \(B\). On an interaction plot of \(\mu _{ij}\) against \(j\), one line for each \(i\), the lines are then parallel. For two levels of each factor, \((\tau \beta )_{11}\) is one quarter of the difference of differences, and the interaction \(AB\) of Section 1 is \(2(\tau \beta )_{22}\); the definitions agree up to a constant.

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