12.8 Higher Factorial ANOVA

We’ve seen 1-way ANOVA and 2-way ANOVA but it doesn’t have to stop there. We can have any number of factors, or independent variables. We can have 3-way ANOVA, 4-way ANOVA, etc. In general we can have an m-way ANOVA. An m-way ANOVA will have m IVs (m factors) but still only one DV.

12.8.1 3-way ANOVA

A 3-way ANOVA will have 3 factors (IVs): A, B, and C with a, b and c levels respectively. A 3-way ANOVA will test 7 hypotheses (all of which are one-way ANOVAs) :

  1. Main effect of A (collapse across B and C).
  2. Main effect of B (collapse across A and C).
  3. Main effect of C (collapse across A and B).
  4. 2-way interaction A \times B (collapse across C).
  5. 2-way interaction A \times C (collapse across B).
  6. 2-way interaction B \times C (collapse across A).
  7. 3-way interaction A \times B \times C.

So there will be 7 test statistics to consider:

    \[ F_{A}, \;\; F_{B}, \;\; F_{C}, \;\; F_{A \times B}, \;\; F_{A \times C}, \;\; F_{A \times C}, \;\; F_{B \times C}, \;\; F_{A \times B \times C} \]

The profile plots for a 3-way ANOVA are intrinsically 4-dimensional and so can be difficult to draw. One approach is to make c 2-way style ANOVA plots :

The interpretation of a 3-way interaction can be tough and there will be many post-hoc pairwise comparisons of cells that may be meaningful. For these reasons it is best to be more reductionist in your experiment designs so that you never have to use a 3-way ANOVA. A design that uses preplanned contrasts  is usually better than one that requires a 3 (or higher) way ANOVA.

For an m-way ANOVA, there will be

    \[ \left( \begin{array}{c} m \\ 1 \end{array} \right) \left( \begin{array}{c} m \\ 2 \end{array} \right) \cdots + \left( \begin{array}{c} m \\ m \end{array} \right) = \sum_{i=1}^{m} \left( \begin{array}{c} m \\ i \end{array} \right) \]

hypotheses to test, each with an associated F test statistic. The number of profile plots to consider will be large and will necessarily involve collapsing factors because the data exist in an m+1 dimensional space (number of IVs plus DV). Interpretation will be a nightmare. An m-dimensional ANOVA for m \geq 3 is more of a mathematical curiosity than a useful scientific tool.