> If the point is removing irrelevant variability from the data and I'm not interested in RT per se, why
> should I put it as a random effect and not a fixed effect in the first place? It seems more relevant
> to put Cond as the random effect (adjust it by subjects).

Ideally you want to model both the cross-subject variability in terms of RT effect and the cross-subject variability among those conditions (and their variance-covariance structure). However, 3dLME currently does not have the functionality to specify cross random effects (possible in the future), so you'd have to choose one at the moment.

> I actually have Cond1 (2 levels) and Cond2 (2 levels). So the fact that I have four RTs per subject -
> one per Cond1*Cond2 combination - is unclear to me... Can RT be both a within- and a between-
> predictor? If so then this is new to me, I can't think how this is implemented mathematically. After
> centering, we expect no within-subject effect of RT anyway.

No, centering would not change anything about the RT effect itself. Instead it artificially shift the center value among the subjects, and in the end it won't have any impact on the RT effect estimate. What it changes is the following: 1) maintaining the integrity of the interpretation for other effects such as condition effect in your case, 2) avoiding potential correlation (collinearity) between RT and conditions.

> So why don't we just give one number which is the average for each subject (across Con1 and
> Cond2), thus it is just a between-subject covariate?

Such a combining step would involve information loss and render inferior modeling.

Gang