Regression with data from different sources: Nested fixed effects possible?

#1
Assume I have collected my data with 3 different methods, each method share the same outcome Y as well as the predictor X. I am interested in the dependency of Y on X via regression analysis.

However, each method has additionally a unique set of fixed or random effects (A,B,...), which are related to the special method and correct for bias (c.f. attached figure). The general structure of the data thus would be :


Y Method X A B C
-----------------------------------------
3 M1 2 6 NA NA
4 M1 4 4 NA NA
4 M2 1 NA 5 NA
3 M3 4 NA NA 4
...

Is it possible to analyze these pooled data simultaneously, introducing the method as a fixed or random coefficient, although we have these “exclusive variables”? I don’t know how, since each unique variable creates NA’s for two of the methods, thus we have NA’s in every row leading to an exclusion of all data from the regression analysis...

Thanks for your suggestions
 
#2
Hi all,

I have possibly got an idea: I could additionally introduce a dummy-variable for each method (which is "1" if we have used this method, otherwise "0") and subsequently I introduce all method-specific covariates as an interaction term with this dummy variable. Thus, these covariates would only be non-zero if the special method have been used. Does this make sense?

Best regards
 
#3
Hey there! I have two notes here. What would be the logic to introduce interactions? Multiplying the variables have to be justified somehow. And another thought, if you have "NA" as a value and multiply it by a dummy, you'd still have "NA", would you not? Correct me if I am missing something here. Trying to help :)

On a side note, possibly you could use some missing values imputation approach (e.g., MLMV) and check the consistency of the estimates.
 
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