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Factorial ANOVA tests have multiple factors (ie, independent variables).

For example, can we say the names two-way ANOVA and factorial ANOVA are equivalent and interchangeable? Yes, the former name provides more information (namely, specifying we have two factors), but do both describe the same test?

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My understanding is that "two-way" implies that there are two factors and that "factorial" means that you are including a full factorial interaction structure (i.e., all 2, 3, ..., n - way interactions). You can include factors in a ANOVA that don't interact with the others fully, in which case I might not that a factorial ANOVA. This might be the case if some effects are known not to interact with others or if the design doesn't allow for a full factorial design because of missing cells.

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