Does an n-way ANOVA have n independent variables?

Question

I'm trying to swot up before my next lecture of ANOVA models but the n-way has me a little confused. Ended up asking myself this question "Does an n-way ANOVA have n independent variables?" but I can't give myself a definitive answer even though it appears a simple question.

Answer 1

"Does an $n$-way ANOVA have $n$ independent variables?"

Yes, this is exactly correct! See, for instance, in Wikipedia: "two-way analysis of variance (ANOVA) is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable." Three-way or four-way is analogous. One-way ANOVA has only one independent categorical variable, of course.

Now to address some misconceptions. It's important to distinguish between the number of levels that a particular categorical variable (sometimes called a "factor") can have, versus the number of variables. I think this is something that commonly causes learners confusion. For instance, if I measure some response (i.e. dependent) variable for subjects under three different conditions (e.g. Drug A, Drug B, Drug C) then I have only one independent variable (the drug used), which happens to have three levels. That's still only a one-way ANOVA and not (as some students think) a three-way.

Note also that if you run a multiple regression model to estimate this ANOVA, we might use Drug A as a baseline and introduce dummy variables for Drug B and Drug C. These are binary variables, coded as one and zero (e.g. the Drug B variable is one only for those people who took Drug B), So the multiple regression has two independent binary variables... but this doesn't mean that the ANOVA had two independent variables, it still has one (categorical) independent variable.