Interpreting a four-way interaction

An example of using differences in group means to evaluate a simple 2x2x2 3-way interaction separately on each side to explain a four-way interaction in a repeated measures ANOVA. We are interested, in this example, in explaining a four-way interaction by looking at the three-way interaction of rotation (rot/nrot) by spectrum (1 or 2) by position (A(nterior) or POST(erior)) on the left and right side separately.

We can take the two differences below to compare three-way interactions on the left and right sides:

$$[(\mbox{rot-nrot})_text{S1,A} - (\mbox{rot-nrot})_text{S1,POST}] - [\mbox{(rot-nrot)}_text{S2,A} - (\mbox{rot-nrot})_text{S2,POST}]$$ and evaluate on the left side

$$[(\mbox{rot-nrot})_text{S1,A} - (\mbox{rot-nrot})_text{S1,POST}] - [\mbox{(rot-nrot})_text{S2,A} - (\mbox{rot-nrot})_text{S2,POST}]$$ repeating on the right side

The relative sizes of these differences will explain the presence of a four-way interaction.

There are other differences of means that could yield three-way interactions which can be compared to explain a four-way interaction in a similar way to the above. The key in all of these approaches, however, is in splitting the sixteen means (=2x2x2x2) into two comparisons each of four differences as above.

The presence of the three and four-way interactions can be tested, in the usual way, using a repeated measures ANOVA.