12.8 Higher Factorial ANOVA

Gordon E. Sarty

12. ANOVA

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) :

Main effect of $A$ (collapse across $B$ and $C$ ).
Main effect of $B$ (collapse across $A$ and $C$ ).
Main effect of $C$ (collapse across $A$ and $B$ ).
2-way interaction $A \times B$ (collapse across $C$ ).
2-way interaction $A \times C$ (collapse across $B$ ).
2-way interaction $B \times C$ (collapse across $A$ ).
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.

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Introduction to Applied Statistics for Psychology Students Copyright © 2022 by Gordon E. Sarty is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

12.8.1 3-way ANOVA

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