I'm trying to think of a way to test certain hypotheses regarding an experiment I've run. It's a 2 x 3 x 4 within-subjects, repeated-measures design. I would like to fit a (cross-classified) multilevel model with, say, intercepts varying over subjects.
Let's label the independent variables as A (2) x B (3) x C (4). The independent variables are dummy coded, the dependent variable is reaction time to a certain stimulus.
The specific question I'm interested in is whether the difference in reaction times between C1 and (C2 + C3 + C4)/3 is larger in condition A1 than in condition A2 (both averaged over levels of B).
Now, I could calculate the difference scores between C1 and the average of the remaining levels, and then run a t-test with A as the independent variable. However, since there are supposed to be benefits to using multilevel models, I want to try to get an answer out of them.
So, I assumed I have to fit the following model (in
reactionTime ~ (1 | subId) + A + B + C + A:C. If I'm not mistaken, the interaction is necessary to allow the C coefficients to be different with regard to levels of A.
If I fit the model, I get the following coefficients:
I understand that each combination of coefficients represents one experimental condition. However, I'm unsure on how to test the contrast I've specified.
Can I use the obtained coefficients to calculate the difference between C1 and (C2 + C3 + C4)/3 for A1 and A2 separately, and then perform a standard paired samples t-test on those values? If so, what would be the degrees of freedom and how would I find the SE of the difference?
Is there a direct way to test something like this in a multilevel model? I am aware that testing such "main effects" might be dubious, but I'd like to know whether it is even possible.