To deal with the imperfect compliance of the randomization, you could use the “instrumental variables” approach. In this case, since it is (one-sided) noncompliance in an experiment, this amounts to:
Using all of your data (ie, not subsetting the data to periods in which you complied with randomization)
Dividing the observed treatment effect by the fraction of time in which you complied (if I understand correctly, this is 0.5)
I emphasize that this is a very simple econometric technique and does not rely on unreasonable assumptions (“Wald estimator” is another search term here).
Very cool!
To deal with the imperfect compliance of the randomization, you could use the “instrumental variables” approach. In this case, since it is (one-sided) noncompliance in an experiment, this amounts to:
Using all of your data (ie, not subsetting the data to periods in which you complied with randomization)
Dividing the observed treatment effect by the fraction of time in which you complied (if I understand correctly, this is 0.5)
I emphasize that this is a very simple econometric technique and does not rely on unreasonable assumptions (“Wald estimator” is another search term here).