I don't know if I qualify as an expert, so take it with a grain of salt. Based on what you're said, the more complex model, which correct/control for (almost) everything would be something like this.
Denoting each dependent variable by $y_{1}, y_{2}$ and $y_{3}$, denoting treatment assigned to subject $i$ by $T_{i,1}, T_{i,2}$ and $T_{i,3}$ and the performance of each tasks by subject $i$ by $t_{i,1}, t_{i,2}, ..., t_{i,6}$, we have:
$[y_{i,1}, y_{i,2}, y_{i,3}]$ ~ $ N([\mu_{i,1}, \mu_{i,2}, \mu_{i,3}], \sigma^{2})$
Now, you have a multivariate normal and you can model the variance as being correlated among each marginal distribution.
Also, you can have a multivariate distribution on the means and model them as correlated as well.
Maybe you will need something like the inverse wishart distribution (I guess). I'd take a look at the book by Gelman and Hill about Multilevel Models, since they have some discussion about using inverse Wishart distributions as priors.
In any case, this seems to me to be a quite complex model. I'd go first with a simple model, not correcting for correlation among the dependent variables (i.e. run 3 separate analysis) neither correcting for the independent variables (the within error you mentioned). Then, I'd asses the fit of the model and, if there is clear room for improvement with a multilevel analysis, I'd move on to the simple multilevel model correcting for correlation withing subject. And then, if necessary, I"d move on to a full multilevel model like the one I outlined above.
I'm not quite sure if that's what you're looking for. Hope it helps a bit.
ps.: Gelman has some papers relating anova and multilevel that may help you.
Don't try to come up with a complicated analysis to deal with this, just do within and between analyses separately on the relevant subsets of your data. That way you will not compromise the interpretability of the outcome and you will be able to make sure that the result is both sensible and robust.
Depending on the nature of your data you could potentially combine the outcomes of the two arms of the experiment. For example, if you can generate likelihood functions for each arm then the combined function is simply their product.
Best Answer
I would go for package
lme4
.If I understand correctly,
mF
below should be your model. It hascondition
as a fixed effect, whilejudge
andsubject
as a random intercepts.Edit: Per discussion below.