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A growing number of social experiments are conducted outside of the laboratory, and by assigning the treatment condition through emails (e.g., often the content of the email is the intervention itself).

Nothing guarantees that the recipient is indeed receiving or even reading the email. This creates a problem of compliance to the assigned treatment condition. Sometimes, however, one may have additional information regarding whether or not the subject has actually seen the email. So the general question would how to best incorporate this additional information in the analysis of the data, and more specifically the estimation of the average treatment effect?

For example, consider the following experimental setting. I have a set of treated units and a set of controls, and I am interested in the effect of the treatment on a particular outcome variable Y. What I call "treated units" are in effect subjects randomly assigned to an experimental condition T=0,1, but I don't know if they have actually received the treatment, or not. However, I observe also a variable X=0,1 that tells me those who, if assigned, are more likely to have received the treatment.

In the case of emails, I might know whether they have opened the email or not. This is not a perfect monitoring, because sometimes I fail to count people who have indeed opened and read the email, and some other times I make the opposite mistake. What is the best way to estimate the average treatment effect in this case?

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Take a look at this paper on the Oregon Health Experiment. One of the problems there, which is analogous to yours (if I understand the question), was that some winners of the randomized lottery did not enroll in the program, so the authors used an instrumental variable strategy to evaluate the causal effect of Medicaid for those actually covered, in addition to calculating the intent-to-treat estimator of comparing the outcomes for the treatment and control groups.

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