First thing to do is operationalise 'importance of predictors'. I shall assume that it means something like 'sensitivity of mean outcome to changes in predictor values'. Since your predictors are grouped then sensitivity of the mean outcome to groups of predictors is more interesting than a variable by variable analysis. I leave it open whether sensitivity is understood causally. That issue is picked up later.
Three version of importance
Lots of variance explained: I'm guessing that psychologists' first port of call is probably a variance decomposition leading to a measure of how much outcome variance is explained by the variance-covarance structure in each group of predictors. Not being an experimentalist I can't suggest much here, except to note that the whole 'variance explained' concept is a bit ungrounded for my taste, even without the 'which sum of which squares' issue. Others are welcome to disagree and develop it further.
Large standardized coefficients: SPSS offers the (misnamed) beta to measure impact in a way that is comparable across variable. There are several reasons not to use this, discussed in Fox's regression textbook, here, and elsewhere. All apply here. It also ignores group structure.
On the other hand, I imagine that one could standardise predictors in groups and use covariance information to judge the effect of a one standard deviation movement in all of them. Personally the motto: "if a things not worth doing, it's not worth doing well" damps my interest in doing so.
Large marginal effects: The other approach is to stay on the scale of the measurements and calculate marginal effects between carefully chosen sample points.
Because you are interested in groups it is useful to choose points to vary groups of variables rather than single ones, e.g. manipulating both cognitive variables at once. (Lots of opportunity for cool plots here). Basic paper here. The
effects package in R will do this nicely.
There are two caveats here:
If you do that you will want to watch out that you are not choosing two cognitive variables that while individually plausible, e.g. medians, are jointly far from any subject observation.
Some variables are not even theoretically manipulable, so the interpretation of marginal effects as causal is more delicate, though still useful.
Different numbers of predictors
Issues arise due to the grouped variables covariance structure, which we normally try not to worry about but for this task should.
In particular when calculating marginal effects (or standardized coefficients for that matter) on groups rather than single variables the curse of dimensionality will for larger groups make it easier for comparisons to stray into regions where there are no cases. More predictors in a group lead to a more sparsely populated space, so any importance measure will depend more on model assumptions and less on observations (but will not tell you that...) But these are the same issues as in the model fitting phase really. Certainly the same ones as would arise in a model-based causal impact assessment.