How to tell if a mixed-model will be inherently unidentifiable I am analysing data from an online survey measuring medical use of illicit and licit cannabis.
There are three groups of users: those who use prescribed medical cannabis only, those who use both prescribed and illicit medical cannabis, and those who use illicit medical cannabis only.
We have asked the same question "What is the main condition you treated with your medical cannabis?" twice, one for the illicit version and once for the licit version - each with the exact same response options. The 'prescribed only' group and the 'prescribed plus illicit' group could answer the licit version of this question (but not the illicit only group) and the 'illicit only' group and the 'prescribed plus illicit group' could answer the illicit version of the question (but not the 'prescribed only' group).
Here is a table representing this design

As you can see the 'prescribed only' and 'illicit only groups answer only one version of the question, but the 'prescribed plus illicit' group answers both version.
In one sense this is a mixed design because it includes both between- and within-subjects factors. But in another sense it is not a mixed design , because not all levels of the between-subjects factor are exposed to all levels of the within-subjects factor. Hence my confusion.
Now, my boss would like to test whether there are differences in the incidence of different main conditions depending on whether people use prescribed medical cannabis or illicit medical cannabis. He thinks it should be a simple single-level multinomial logistic regression with symptom type as the outcome and licit vs illicit cannabis as the predictor.
I am not so sure. The fact that members of the 'prescribed plus illicit' group answer the same question twice and thus have one observation in each version makes me think we should use some kind of mixed effects multinomial logistic model, since mixed effects models allow for repeated measures and unbalanced designs. The reason I am posting is I would like to know
(1) Is my boss wrong?  Is a completely between-subjects model inappropriate? Can we pretend that the two observations from 'prescribed + illicit' group are independent observations?
(2) Am I right? Is a mixed-effects model the right way to deal with this data structure? Can a mixed-effects model handle a design than is so unbalanced? Would a model based on this design be inherently unidentifiable?
I should emphasise that we only want to test the difference in outcomes reported between the licit and illicit versions (i.e. the within-subjects factor), not any interaction between this factor and the three-level group (between-subjects) factor.
Any advice much appreciated.
 A: I haven't tried it out, but I expect that something like
qresult <- glmer(answer˜illicit+(1|subject),family=binomial)

should work for a simple logistic regression. illicit here is a dummy variable (or factor) for the version of the question. As far as I know, it's not necessary for the subject identifier subject to have more than one observation for all subjects; it should be enough to have a good number of them (those in the "illicit&prescribed" group). I haven't run multinomial logistic regressions with mixed effects myself; I suspect glmer doesn't do that but it seems package mclogit does it (which apparently requires the specification random=1|subject).
What could not be identified is any potential interaction between subject and whether a subject belongs to the "illicit&prescribed" group, i.e., whether the subjects in this group behave systematically different from the others. As this may well be the case in reality, I'm somewhat skeptical about the simple mixed effects model above, but this could be explored by suitable plots and particularly by running separate analyses for the "both" group and the other two, and checking whether conclusions are different.
Note by the way the technically this assumes that there is a random subject effect also for the individuals in the "pure" groups, however I don't think this is a problem - it will not affect the general tendency of the coefficient estimator for illicit but may add some additional uncertainty to p-values and the like, which may well be realistic.
As I haven't tried it out myself, tell me how it goes. Sometimes trouble shows up when trying out something that "in principle should work".
