Formulae for interaction sums of squares (SS) in balanced designs
Formulae for interaction sums of squares in a factorial (between subject) analysis of variance can be expressed as a sum of squared residuals. The residual formulae may be found to be easier to use for computing purposes than those (for example in Howell, 2002 as they only involve the squaring of only one term.
Boniface (1995) gives the formula for the two-way interaction SS as
$$n_text{ij} \sum_text{combinations i,j} (\mbox{mean}_text{ij} - \mbox{mean}_text{i+} - \mbox{mean}_text{+j} + \mbox{overall mean})^text{2}$$
where $$n_text{ij}$$ observations have combination of values i and j. These are assumed equal for all i and j (balance). The '+' in the subscripts denotes pooling so, for example, $$\mbox{mean}_text{i+}$$ signifies the mean when the first factor takes the value i.
Using formulae in Howell (2002, p.459) for the three-way interaction and the orthogonality of the sums of squares in the anova for balanced designs we can define the SS(three-way interaction) as
$$n_text{ijk} \sum_text{combinations i,j,k} (\mbox{mean}_text{ijk} + \mbox{mean}_text{i++} + \mbox{mean}_text{+j+} - \mbox{mean}_text{++k} - \mbox{mean}_text{ij+} - \mbox{mean}_text{i+k} -\mbox{mean}_text{+jk} - \mbox{overall mean})^text{2}$$
where $$n_text{ijk}$$ observations have combination of values i, j and k. These are assumed equal for all i, j and k (balance). The '+' in the subscripts denotes pooling so that, for example, $$\mbox{mean}_text{i++}$$ signifies the mean when the first factor takes the value i and $$\mbox{mean}_text{ij+}$$ signifies the mean when the first two factors have values i and j respectively.