Factorial ANOVA with confound

Since you haven't gotten a reply here yet, I'd encourage you to be a bit more explicit about your design. Do you perhaps have an example you can walk us through or link to?

Also, though there are many applied statisticians and folks familiar with estimating causal effect from a given experimental design here, a question like this may have more success at Cross Validated (and be sure to link to any cross posts FAQ: Is it OK if I cross-post?).