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# Factorial Analysis of Variance

One-way ANOVA has been introduced as a procedure that is used with one independent variable and two or more levels identified by this variable. It is common, however, to have more than one independent variable in a study. In fact, it is often desirable to have several independent variables because the analysis will provide more information. For example, if a group of researchers investigates the relative effectiveness of three reading curricula, they would probably use a 1 × 3 ANOVA to test the null hypothesis that no difference in achievement exists between any of the three groups. If the researchers were also interested in whether males or females achieved differently, gender would become a second independent variable. Now there are six groups, because for each reading group males and females are analyzed separately.

Example

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In this hypothetical situation, two independent variables are analyzed simultaneously and one dependent variable (achievement). The statistical procedure that would be used to analyze the results would be a two-way ANOVA (two-way because of two independent variables). Because factor is another word for independent variable, factorial means more than one independent variable. Factorial ANOVA, then, is a generic term that means that two or more independent variables are analyzed together. The more specific term, such as two-way or three-way ANOVA, tells the exact number of independent variables. Researchers can be even more precise in indicating what the analysis is by including the levels of each independent variable. As pointed out earlier, levels refers to the subgroups or categories of each independent variable. In the example mentioned above, reading curriculum has three levels and gender two levels. The levels can be shown by numbers that precede the ANOVA. In our reading example, it is a 2×3 ANOVA.

An interaction is the effect of the independent variables together; that is, the impact of one factor on the dependent measure varies with the level of a second factor. Stated differently, it is the joint effect of the independent variables on the dependent variable. An interaction is evident if the differences between levels of one independent variable are inconsistent from one level to another of the other independent variable. In other words, an interaction exists if the effect of one variable differs across different levels of the second variable. Now it is clear how a factorial ANOVA can provide more information than one-way ANOVAs.

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