Estimating a single drug effect in a sample of patients: each patient assumes multiple drugs. Is this possible? So here is my scenario:
Patient A takes drugs 1 - 2 - 3
Patient B takes drugs 2 - 3 - 4
Patient C takes drugs 1 - 2- 4
and so on ...
Each patient gets his symptom severity measured two times: first at (T0), then at (T1)
Symptom severity is a number (say from 1 to 10)
Assuming I have got data for many many patients, what methods can tell me what drug yields the best reduction in symptoms severity ?
I cant get my head around this..
is a MME legit ?
 A: There would have to be some assumptions made to start to analyze such a study, but they could in principle be tested from the data with a large enough study.
For concreteness, let's use your example of 4 drugs total with each patient taking exactly 3 of them so that there are 4 groups of patients total. Each patient has severity measured before taking the drugs (T0) and at some defined point after (T1). So if you just wanted to know which combination of 3 drugs worked best that could be done as a simple analysis of T0-T1 severity difference values as a function of treatment-group membership.
You ask, however, which drug (singular) works best. That requires some additional assumptions. For example, if you could assume that there were no interactions among the drugs with respect to outcome, then you could think in reverse: the drug missing from the group that did the worst would presumably be the drug that would be best on its own. But that assumption of no interactions among the drugs is very strong.
With a large enough study you could test the assumption of no interactions among the drugs by performing a regression that includes terms for each drug and interaction terms for each 2-way and 3-way combination of drugs. If the interaction term coefficients were all close to 0 then the regression coefficients for the individual drugs would provide measures of individual drug effectiveness. In practice, though, there would likely be substantial interactions in a study like this, in which case any attempt to identify the "best" drug this way would be fruitless.
