# Interpretation of 2-way ANOVA with covariates

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.)

• @NatWH my design is unbalanced. Then do you mean that my question has something do with the types of SS?? – Kevin Kang May 13 '18 at 1:29
• @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 – Kevin Kang May 13 '18 at 1:30
• 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. – NatWH May 13 '18 at 17:09