Three-Way ANOVA 22

1 688 Chapter 22 Three-Way ANOVAACONCEPTUALFOUNDATION22 Chapter Three-Way ANOVAYou will need to use the following from previous chapters:Symbolsk:Number of independent groups in a one-way ANOVAc:Number of levels ( , conditions) of an RM factorn:Number of subjects in each cell of a factorial ANOVANT:Total number of observations in an experimentFormulasFormula : SSinter(by subtraction) also Formulas , , : SSbetor one of its componentsConceptsAdvantages and disadvantages of the RM ANOVASS components of the one-way RM ANOVASS components of the two-way ANOVAI nteraction of factors in a two-way ANOVASo far I have covered two types of two-way factorial ANOVAs: two-way inde-pendent (Chapter 14) and the mixed design ANOVA (Chapter 16).

2 There is onlyone more simple two-way ANOVA to describe: the two-way repeated measuresdesign. [There are other two-way designs, such as those including random-effects or nested factors, but they are not commonly used see Hays (1994) fora description of some of these.] Just as the one-way RM ANOVA can bedescribed in terms of a two-way independent-groups ANOVA , the two-way RMANOVA can be described in terms of a Three-Way independent-groups gives me a reason to describe the latter design next.

3 Of course, the Three-Way factorial ANOVA is interesting in its own right, and its frequent use in thepsychological literature makes it an important topic to cover, anyway. I will dealwith the Three-Way independent-groups ANOVA and the two-way RM ANOVAin this section and the two types of Three-Way mixed designs in Section , the Three-Way ANOVA adds nothing new to the proce-dure you learned for the two-way; the same basic formulas are used a greaternumber of times to extract a greater number of SScomponents from SStotal(eight SSs for the Three-Way as compared with four for the two-way).

4 However,anytime you include three factors, you can have a Three-Way interaction, andthat is something that can get quite complicated, as you will see. To give you amanageable view of the complexities that may arise when dealing with threefactors, I ll start with a description of the simplest case: the 2 2 2 Simple Three-Way ExampleAt the end of Section B in Chapter 14, I reported the results of a publishedstudy, which was based on a 2 2 ANOVA . In that study one factor con-trasted subjects who had an alcohol-dependent parent with those who didnot.

5 I ll call this the alcoholfactor and its two levels, at risk(of codepen-dency) and other factor (the experimenterfactor) also had twolevels; in one level subjects were told that the experimenter was an exploitiveperson, and in the other level the experimenter was described as a nurturingperson. All of the subjects were women. If we imagine that the experimentwas replicated using equal-sized groups of men and women, the 8/23/02 11:56 M Page 688two-way design becomes a Three-Way design with gender as the third will assume that all eight cells of the 2 2 2 design contain the samenumber of subjects.

6 As in the case of the two-way ANOVA , unbalanced Three-Way designs can be difficult to deal with both computationally and concep-tually and therefore will not be discussed in this chapter (see Chapter 18,section A). The cell means for a three -factor experiment are often displayedin published articles in the form of a table, such as Table A Conceptual Foundation689 NurturingExploitiveRow MeanControl:Men402834 Women302226 Mean352530At risk:Men364842 Women408864 Mean386853 Column three FactorsThe easiest way to see the effects of this experiment is to graph the cellmeans.

7 However, putting all of the cell means on a single graph would not bean easy way to look at the Three-Way interaction. It is better to use twographs side by side, as shown in Figure With a two-way design one hasto decide which factor is to be placed along the horizontal axis, leaving theother to be represented by different lines on the graph. With a three -waydesign one chooses both the factor to be placed along the horizontal axis andthe factor to be represented by different lines, leaving the third factor to berepresented by different graphs.

8 These decisions result in six different waysthat the cell means of a Three-Way design can be us look again at Figure The graph for the women shows the two-way interaction you would expect from the study on which it is based. Thegraph for the men shows the same kind of interaction, but to a considerablylesser extent (the lines for the men are closer to being parallel). This difference80706050403020 Nurturing0 ExploitiveControlAt riskWomenMen80706050403020 Nurturing0 ExploitiveControlAt riskGraph of Cell Means forData in Table 8/23/02 11:56 M Page 689in amount of two-way interaction for men and women constitutes a three -wayinteraction.

9 If the two graphs had looked exactly the same, the Fratio for thethree-way interaction would have been zero. However, that is not a necessarycondition. A main effect of gender could raise the lines on one graph relativeto the other without contributing to a Three-Way interaction. Moreover, aninteraction of gender with the experimenter factor could rotate the lines onone graph relative to the other, again without contributing to the three -wayinteraction. As long as the difference in slopes ( , the amount of two-wayinteraction) is the same in both graphs, the Three-Way interaction will be Interaction EffectsA Three-Way interaction can be defined in terms of simple effects in a way thatis analogous to the definition of a two-way interaction.

10 A two-way interactionis a difference in the simple main effects of one of the variables as you changelevels of the other variable (if you look at just the graph of the women in Fig-ure , each line is a simple main effect). In Figure each of the twographs can be considered a simple effect of the Three-Way design more specif-ically, a simple interaction effect. Each graph depicts the two-way interactionof alcohol and experimenter at one level of the gender factor. The three -wayinteraction can be defined as the difference between these two simple interac-tion effects.