I am working on a project where N doctors were asked to prescribe the same medication to the same set of n hypothetical patients.

Each doctor could choose a different dose of medication. For the purposes of this question, let’s assume they could choose between “small dose” and “large dose” and that n = 3 (i.e., there are 3 patients, seen by all doctors). Each of the 3 patients presents with a different indication.

As an example:

```
**Patient # 1 Patient # 2 Patient # 3**
(Headache) (Muscle Ache) (Other Aches)
**Doctor # 1** "Small dose" "Small dose" "Large dose"
**Doctor # 2** "Large dose" "Small dose" "Small dose"
etc.
```

There are two types of doctors in the data and interest lies in estimating the effect of doctor type on the probability of prescribing a “Large dose” rather than a “Small dose”, controlling for the amount of the total daily dose.

The first thing that I thought about was that I could treat doctors and patients as crossed random effects in a mixed effects binary logistic

regression model which would relate dose type to doctor type, controlling for total daily dose. However, not only is the number of patients small, but I only have one dose type per doctor by patient combination. So this approach is out of the question.

I also don’t know if it makes sense to drop patient as a random effect from the above model? The fact that all doctors consulted the same patients doesn’t seem to bode well for a model without a random patient effect.

Then I thought that I could analyze the data for each doctor separately (since we are not really interested in making comparisons across indications) via binary logistic regression, but what niggles at me is that the doctors all consult the same (hypothetical) patient for each indication. This likely invalidates the assumption of independence of the dose types prescribed by different doctors to the same patient. Maybe if I used the quasibinomial distribution in the binary logistic regression I would feel slightly better about this simplified approach (presuming the quasibinomial might be able to correct to some extent for the lack of independence), but I can’t because of separation issues for one of the indications.

Not sure what other types of models would make sense here - any ideas would be much appreciated.