I've got a dataset with 3 factor variables with only one interaction. Y is the response and A,B,C are the independent variables. Variable B in particular has a factor that occurs with no other factors, so the design matrix has a 0 in it, leading to an error when trying to run a mixed model (fixed factors only).

I'm trying to figure out how to reduce number of variables so the design matrix doesn't have a 0 in it.

In the case where all variables and all possible interactions are included, this is fairly straightforward, as I can create one new variable with all possible factor values for A,B,C. I've tested this in R and get equivalent values with: Y ~ A * B * C (which expands to Y ~ A + B + C + A:B + A:C + B:C + A:B:C) provides the same output as Y ~ D when A,B,C are put together in a new variable D. Basically I'm reducing it to a one way design.

Idea taken from here – Can you convert three-way ANOVA to one-way ANOVA?

However, what I need to do is only explore two interactions, not all of them. The formula I'm trying to use is Y ~ A + B + C + A:B + A:C.

Can anybody recommend how to reduce this to fewer factor variables so I don't end up with a 0 in the design matrix? I've been trying a lot of combinations but am mentally not able to wrap my head around this.