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.
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?
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.)