As in the case of the two-way ANOVA, unbalanced threeway designs can be difficult to deal with both computationally and conceptually and therefore will not be discussed in this chapter (see Chapter 18, section A). We will assume that all eight cells of the design contain the same number of subjects. If we imagine that the experiment was replicated using equal-sized groups of men and women, the originalĢ Cohen_Chapter22.j.qxd 8/23/02 11:56 M Section A Conceptual Foundation 689 two-way design becomes a three-way design with gender as the third factor. The other factor (the experimenter factor) also had two levels in one level subjects were told that the experimenter was an exploitive person, and in the other level the experimenter was described as a nurturing person. I ll call this the alcohol factor and its two levels, at risk (of codependency) and control. In that study one factor contrasted subjects who had an alcohol-dependent parent with those who did not. A Simple Three-Way Example At the end of Section B in Chapter 14, I reported the results of a published study, which was based on a 2 2 ANOVA. To give you a manageable view of the complexities that may arise when dealing with three factors, I ll start with a description of the simplest case: the ANOVA. However, anytime you include three factors, you can have a three-way interaction, and that is something that can get quite complicated, as you will see. Computationally, the three-way ANOVA adds nothing new to the procedure you learned for the two-way the same basic formulas are used a greater number of times to extract a greater number of SS components from SS total (eight SSs for the three-way as compared with four for the two-way). I will deal with the three-way independent-groups ANOVA and the two-way RM ANOVA in this section and the two types of three-way mixed designs in Section B. Of course, the threeway factorial ANOVA is interesting in its own right, and its frequent use in the psychological literature makes it an important topic to cover, anyway. This gives me a reason to describe the latter design next. Just as the one-way RM ANOVA can be described in terms of a two-way independent-groups ANOVA, the two-way RM ANOVA can be described in terms of a three-way independent-groups ANOVA. There is only one more simple two-way ANOVA to describe: the two-way repeated measures design. 1 Cohen_Chapter22.j.qxd 8/23/02 11:56 M Page Chapter 22 Three-Way ANOVA Three-Way ANOVA 22 Chapter A CONCEPTUAL FOUNDATION 688 You will need to use the following from previous chapters: Symbols k: Number of independent groups in a one-way ANOVA c: Number of levels (i.e., conditions) of an RM factor n: Number of subjects in each cell of a factorial ANOVA N T : Total number of observations in an experiment Formulas Formula 16.2: SS inter (by subtraction) also Formulas 16.3, 16.4, 16.5 Formula 14.3: SS bet or one of its components Concepts Advantages and disadvantages of the RM ANOVA SS components of the one-way RM ANOVA SS components of the two-way ANOVA Interaction of factors in a two-way ANOVA So far I have covered two types of two-way factorial ANOVAs: two-way independent (Chapter 14) and the mixed design ANOVA (Chapter 16).
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