I have a somewhat messy experiment that has already been conducted with large test and control groups, and have measured the response variables (sales / individual, etc). A known percentage of the 'test group' (supposed to be exposed to a treatment) did not actually get exposed to the treatment, and I have no way of separating them in the data. I can see whether someone was 'supposed to receive a treatment' and 'definitely did not' (control).
I can directly observe:
- Z = Intended test group (composed of X and Y) - only some received treatment
- C = Control group
- w = proportion of X that makes up Z (but cannot tie to individual records) - e.g. 60% of the intended test group actually recieved the treatment.
I cannot observe:
- X = 'Actual test group' that recieved treatment
- Y = People who were supposed to recieve treatment but did not.
I would like to be able to make statements of significance about whether X differs from C (e.g. with 2-sample t-test)
Intuitively, Y 'should' look like C (they both represent sales/customer for customers who have not received the treatment, though they have different/known sizes), and I want to subtract it out to get an estimate of X I can use in a t-test or similar. Subtracting the means proportionally I think should be fine, but I'm concerned that doing the same for the variance is not correct.
I know 'when random variables subtract, the variances add', but wonder if that will screw up my t-test since I only have the sample variances and not the population parameters.
How should I be trying to make this adjustment?