I'm working on a project investigating the relationship between (let's say) a face's perceived masculinity and its perceived competence. There was a large number of face stimuli (80). Two completely separate groups of subjects rated the faces on each dimension: a given subject rated a random subset of the 80 faces on EITHER masculinity OR competence, never both. In other words, every subject rated multiple faces on a single dimension, and every face was rated by multiple subjects on both dimensions. (This was done to avoid interference effects and demand bias.) The specific face stimulus is considered a nuisance parameter here; face-specific effects are not really of interest.
A very simple modeling approach would be to treat face as the unit of analysis and characterize each face by its mean masculinity and its mean competence. This would result in a dataset with 1 row per face that would be easily amenable to linear regression, etc.
Obviously, this approach loses a lot of information, particularly failing to account for a subject random effect (such that, e.g., some subjects might generally consider faces more competent than other subjects). The problem invites a mixed-model with random intercepts or slopes by subject and face plus a fixed effect of masculinity, such as (in R shorthand):
competence ~ masculinity + (1|id) + (1|face)
However, I am not sure how this would work in light of the experimental design. I am basically having trouble envisioning how the fixed effect of masculinity will be estimable given that no observation contributes both a masculinity AND a competence rating.
Is this problem conducive to mixed-modeling? If so, how?
I tried fitting the mixed model specified as above using
lmer to see what would happen. Not surprisingly, the model throws an error about incorrect grouping structure. This error goes away if I randomly simulate some of the missing-by-design such that there are now observations having both X and Y.
So, I am revising my question:
Is there a modeling approach for this situation that would retain more information than just modeling the face-specific means?