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.

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

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