Do I have to specify all main effects in a factorial ANOVA? I am having an issue specifying main effects and interaction terms in an ANOVA model.
The problem is, lets say that I have 3 factors, A,B and C. I am interested in the main effects of A, B but not C. However, it is very important for me the interaction between A and B. I just want to know if the effects of A and B change within levels of C, but to me is not important if C has an effect or not. 
When I set my model with the following terms:
effect of A
effect of B
effect of AxB
effect of AxC
effect of BxC

I get significant interactions of the terms AxC and BxC. But if I add the main effect C (effect of C) in the model, the intersection terms AxC and BxC are no longer significant. Which I presume is due to the sequential nature of the sum of square procedure. 
My question is: Is it okay to use my first model, where the main effect C is absent, and C only appears interacting with other factors?  
 A: Without the main effect of C the interaction component doesn't really mean anything. Ignoring issues of correlation, even with a fully orthogonal design you need the main effect. 
Consider a simpler model with just A, B and AxB. If you have an interaction term in that model, whether significant or not, it allows the effect of A to vary when B varies and vice versa. However if, for example, the B effect is missing then the interaction term most definitely means something else because the interaction can allow the effect of A to vary over... what?. Further, consider when A = 0. In that case the effect of varying B in the model is also always 0. So, you're changing the meaning of the term, not just it's significance. 
I've seen this done but not liked it. An example might be when one main effect is about a baseline difference. Some assume that any existing baseline difference doesn't really exist and is just sampling error. That said, you then may run into an issue of a misestimated interaction due to baseline differences in the sample.
