7.4 More than two factors

The analysis extends directly. With three factors \(A\), \(B\), \(C\) at \(a\), \(b\), \(c\) levels and \(n\) replicates of each combination, the model is \[ y_{ijkl} = \mu + \tau _i + \beta _j + \gamma _k + (\tau \beta )_{ij} + (\tau \gamma )_{ik} + (\beta \gamma )_{jk} + (\tau \beta \gamma )_{ijk} + \varepsilon _{ijkl} , \] and the analysis of variance has a line for each term. Each expected mean square has the form \(\sigma ^2 + m\,\Phi \), where \(\Phi \) is the sum of the squared effects divided by the degrees of freedom (for example \(\Phi _A = \sum \tau _i^2/(a - 1)\)) and \(m\) is the multiplier shown:

The pattern is general. The degrees of freedom of an effect are the product of \((\text {levels} - 1)\) over the factors in it. The multiplier in its expected mean square is the number of observations at each combination of its levels, which is \(n\) times the product of the numbers of levels of the factors not in it. And each sum of squares can be computed from the table of totals for the factors in it, minus the sums of squares of all the effects contained in it: for example, \[ SS_{AB} = \frac {1}{cn}\sum _i\sum _jy_{ij\cdot \cdot }^2 - \frac {y_{\cdot \cdot \cdot \cdot }^2}{N} - SS_A - SS_B . \] With fixed factors, every effect is tested against \(MSE\).