LMM reduction following marginality taking out "item" before "subject:item" grouping factor
Hi Jake, So, regarding this issue, there is no difference between taking out
variance components for main effects before interactions within the same grouping factor, e.g. reducing (1 + A*B | subject) to (1 + A:B | subject), and taking out the whole grouping factor "item" (i.e. all variance components of it) before "subject:item"?
I think that if you have strong evidence that this is the appropriate random effects structure, then it makes sense to modify your model accordingly, yes.
This makes sense to me. Do all variances of the random slopes (for interactions and main effects)
of a single grouping factor contribute to the standard errors of the fixed main effects and interactions in the same way?
No -- in general, with unbalanced datasets and continuous predictors, it's hard to say much for sure other than "no." But it can be informative to think of simpler, approximately balanced ANOVA-like designs where it's much easier to say much more about which variance components enter which standard errors and how. The standard error for a particular fixed effect is proportional to the (square root of the) corresponding mean square divided by the total sample size, that is, by the product of all the factor sample sizes. So examining the mean square for an effect will tell you which variance components enter its standard error and which sample sizes they are divided by in the expression.
Your app is very useful, too. Just to double-check if I get this right: the entries in each cell of the table are the numbers by which the variance components are divided in the equation of the noncentrality parameter. Is this correct? Regards, Maarten