# Randomized Treatment within treated group

I don't know if this situation has a particular name. I'll just give the example and my question.

Suppose we have two naturally concurring groups, $$A$$ and $$B$$. By that, I mean there is a naturalistic mechanism for why the groups are separated and that the groups $$A$$ and $$B$$ are (approximately) independent. I.e., a researcher did not assign them this label. However, suppose that in group $$A$$, a treatment ($$T = 1$$) was randomly given to the members of the group with the intent to measure the effect of the treatment. However, the researchers lost their documentation on who in group $$A$$ received the treatment and who received the control. All they know was that the treatment was assigned completely at random in group $$A$$ to, say, half of the members. They are also sure that no one in group $$B$$ received treatment.

Can the researchers still estimate a treatment effect somehow by taking the labels $$A$$ and $$B$$ as the treated and controlled group? My thinking is perhaps no, but I was curious anyway.

Edit: I forgot to mention that it is a reasonable assumption that groups $$A$$ and $$B$$ come from the same super-population. The only difference is in their labels and that group $$A$$ received a (random) treatment and that group $$B$$ did not.

• Interesting question. Does A/B affect Y through other mechanisms than D? And, out of curiosity: Is there a real-world example for this? – Julian Schuessler Oct 5 '18 at 7:29
• @Marcel Is this different than a regression with binary group variable that has non-classical measurement error? – Dimitriy V. Masterov Oct 8 '18 at 22:29