# Are these maximal linear mixed models (within-subject within-item factorial design) really maximal?

What is the maximal linear mixed effects model of a 2 x 2 within-subject and within-item factorial design with multiple observations for each subject-item-factor level combination: m1 (as I have seen quite often) but why not m2 or even m3?

m1: Y ~ 1 + A*B + (1 + A*B|subject) + (1 + A*B|item)

m2: Y ~ 1 + A*B + (1 + A*B|subject) + (1 + A*B|item) + (1|subject:item)

m3: Y ~ 1 + A*B + (1 + A*B|subject) + (1 + A*B|item) + (1 + A*B|subject:item)


I know that these models are often over-parameterized but if one wants to start with the maximal model and try to find an 'optimal' model via reduction, then it should be the correct maximal model in the first place.

Models m2 and m3 can only be estimated if you have repetitions on the level of the participant-item combination. For m3 for the full factorial design. Otherwise this random effect is confounded with the residual variance.

To sum this up, for this and all other random effect parameter you need more than one data point for each level of the random-effect grouping factor and fixed-effect factor. Otherwise it cannot be estimated. lme4 will also produce a corresponding error and not estimate such a model.