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I have a question about interpreting two-way ANOVA including a couple of categorical covariates. As I conduct the analysis, I put $T_1$, $T_2$ and $T_1\times T_2$ to represent the main effects and their interaction in the model. I also put several categorical covariates (baseline differences measured before treatments) into the model to control for them.

  1. How can I interpret the $F$-values of $T_1$, $T_2$ or $T_1\times T_2$ if certain covariates are statistically significant? Is it something like "holding others constant" which is a regression-style interpretation?

  2. Other than the covariates, I think $T_1$ and $T_2$ themselves also serve as covariates to one another. This point makes me even more confused and frustrated. So when I interpret the $F$-value of $T_1$, then should I consider $T_2$ and $T_1\times T_2$ as covariates?

To sum up, "what does it mean to include more categorical variables in the underlying linear model when it comes to interpreting the ANOVA table?" (Obviously, the $F$-values in the ANOVA table differ every time I include or get rid of a variable from the model.)

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  • $\begingroup$ @NatWH my design is unbalanced. Then do you mean that my question has something do with the types of SS?? $\endgroup$ – Kevin Kang May 13 '18 at 1:29
  • $\begingroup$ @NatWH Plus I'm adding a comment from someone in another post. "Hi there, thanks for the comment it really helps. I want to ask you an additional question: if that is the case, then should I apply "holding other factors equal" when interpreting f-values of each factor in ANOVA table???? According to your comment I believe I should, but I can't think of any cases I've seen to date." - me "Yes, KevinKang, that's the correct way to interpret the test." – gung $\endgroup$ – Kevin Kang May 13 '18 at 1:30
  • $\begingroup$ I am not sure about the exact thing gung was commenting on but I believe in this case the interpretation of the test is considering each parameter to be the last entered into the model (which is the type III interpretation I think). I've always thought that it was the parameters, and not tests, which are interpreted ceteris paribus, but I could be wrong. $\endgroup$ – NatWH May 13 '18 at 17:09
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It depends a little on how you conduct the tests (see: How to interpret type I, type II, and type III ANOVA and MANOVA?), but the typical tests are interpreted as "holding others constant", just as you say. That includes covariates, whether they are categorical or not, and it includes the main effects and the interaction effect. The latter is a rather strained interpretation, because what it means to hold the interaction effect, e.g., constant for the test of the main effect is kind of weird (see: What does “all else equal” mean in multiple regression?). Nonetheless, that is what it means. In addition, it does not matter whether the other variables in the model are significant. The test is still of the named variable while "holding others constant".

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