I have a question. In an experiment we test a treatment in three conditions (t1,t2,t3). We split a group of participants so that we have one group for each treatment. So far straight forward. However we want to estimate the interactions between two crossing variables of our participants (gender, major) and our treatment condition. Gender as well as major both have two levels.
So we have three factors:
(C)ondition (between groups, 3 levels)
(G)ender (2 levels)
(M)ajor (2 levels)
Can some one point me in the right direction how to define my model?
I do not feel confident in using: Y = C|G|M.
Thanks a lot in advance!
Markus
your model would be:
yhat = b0 + b1C + b2G + b3M + b4C*G + b5C*M + b6G*M + b7C*G*M
Is that what you were looking for?
Also, to show the 3 way interaction, you might show a series of plots:
Interaction of C*G for each major separately
Interaction of C*M for each gender separately
Assuming you only have one dependent variable, Y, the model you specified Y=C|G|M, is correct. This model has all main effects, all two way interactions, and the three-way interaction. I wouldn't call it "mixed", though, because all three independent variables are between-subject variables, i.e., none are "repeated measures" or nested. As such, it is a relatively "simple" design, within the ANOVA types of designs, in that all factors are between-subject.
The only thing I would say that you need to make sure that you specify all three factors as categorical (or "grouping" variables, depending which software you're using), if you run this as a General Linear Model (GLM; this is because the model parameter estimates will be "coded" differently within GLM). But if you run it as a plain 3-way ANOVA, no problem.
Other than that, what is it that you feel uncomfortable about?
There are lots of resources (e.g., textbooks or online). You could look up "factorial ANOVA", or "3-way between-subjects ANOVA".
Hope this helps a bit, or at least gives you a bit more confidence :-)
Hi Carl and Helena,
Thank you both for your reply. You are absolutely right Helena my Model is in fact a simple Three-way ANOVA NOT a mixed model! I confused myself arguing about the independence of my crossing variables (gender, major). And yes your response gives me a lot more confidence :-).
Special thanks to you Carl one of my students is using your Model for his experiment. His ones is truly a mixed model.
Cheers,
Markus
Markus,
If you are interested in the 3-way interaction, you may want to make sure a model that includes the 3-way interaction is better than one that does not include it.
This can be easily tested, whether in a Fisherian or a Bayesian approach, by a model comparison. That is, you would compare
Ms : yhat = b0 + b1C + b2G + b3M + b4C*G + b5C*M + b6G*M + b7C*G*M
with
M1 : yhat = b0 + b1C + b2G + b3M + b4C*G + b5C*M + b6G*M
If Ms is a better model than M1 (Fisherian approach: p< 0.05; Bayesian approach: BIC-of-Ms < BIC-of-M1), plot the 3-way interaction in a way that lets you 'see' what is happening in your data to make up that 3-way interaction.
To go a little further:
Then maybe you may find a simpler model (e.g., one where the effect of 2 treatments do not differ but differ from the third) by collapsing levels on your Treatment variable. You may have the 'intuition' for such a model by observing the graphs of the 3-way interaction. Then, to confirm whether such a model is better than model Ms, you will have to compare them. The tricky part here is that you cannot compare such 2 models by using the Fisherian approach any more, and you'll have to switch to the Bayesian approach.
HTH.
Cheers,
SCM
Hi Serban,
Thank you for your answer. You are bringing up a very interesting point here. I totally agree to your statement. This questions turns out to be more rewarding than I ever anticipated.
Cheers,
Markus
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