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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?
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.
