What are some examples when the Average Treatment Effect on the Treated/Control (ATT,ATC) is more sought after than the ATE?

I am wondering when and why one may calculate the Average Treatment Effect on the Treated (ATT) or the Average Treatment Effect on the Control (ATC). Is there a specific example or motivation for when one would be interested in these quantities versus the Average Treatment Effect (ATE)?

I'm writing a paper about this very topic, so I'll just summarize here and update with a link to the paper when it's ready. (Edit: Here is the arxiv version.) In short, the ATE, ATT, and ATC can be described as follows:

• The ATE is the average effect of mandating a policy of treatment for everyone vs. mandating a policy of control for everyone
• The ATT is the average effect of withholding treatment from those who would normally receive it
• The ATC is the average effect of expanding treatment to those who would not normally receive it

It's important to also recognize that these are average effects in the study population, which may be narrowly defined (e.g., to eligible patients or patients with equipoise). These effects only differ from each other when treatment assignment is not random and when the treatment effect differs across individuals based on qualities related to treatment assignment.

The ATE might be useful when evaluating a policy that applies to everyone, e.g., whether medical providers should unilaterally prefer surgery A vs. surgery B for eligible patients. The ATE involves comparing the outcomes of a counterfactual world where everyone receives the treatment to a counterfactual world in which no one does. In my opinion, this is almost never a useful comparison unless the population is very narrowly defined to be a group where such policy would make sense. For example, it wouldn't make sense to ask what the ATE of smoking on lung cancer would be for the entire US population (e.g., based on a national survey) because you would never be interested in comparing a policy where nobody smoked to a policy where everyone smoked.

The ATT might be useful when deciding whether to ban a currently implemented practice or continue an experimental program. The ATT is the effect of withholding the treatment from those who receive it, so it is only concerned with those actually being treated. For example, it might make sense to ask what the ATT of smoking is on lung cancer, because you may be interested in a policy of withholding (i.e., preventing) smoking among those currently smoking. You might also be interested in the ATT of a program that would only be eligible to people like the current participants; this would help decide whether you should continue implementing the program (and its selection policy).

The ATC might be useful when deciding whether to extend a currently implemented program to those not currently receiving it. It is only concerned with those not receiving the treatment. The ATC might be a good estimand when challenging current clinical practice, e.g., when treatment is withheld from some patients who might actually benefit from it. It might be useful for understanding the effect of a campaign on an untapped market, e.g., to examine the effect of seeing an ad for a product on purchasing the product among those who don't typically see the ad.

The ATE is an extremely coarse estimand (unless the population is narrowly defined). It considers a policy that makes no reference to how people normally come to receive or not receive the treatment. The ATT and ATC are slightly more specific because they only consider specific relevant subgroups of units and respect the natural process of treatment assignment.

The ATT and ATC also require fewer (unverifiable) assumptions for identification than the ATE. The ATE requires mean conditional exchangeability for both treatment values, i.e., $$E[Y^1|A=a, X] = E[Y^1| X] \\ E[Y^0|A=a, X] = E[Y^0| X]$$ for all $$a$$ in $$\{0,1\}$$, where $$A$$ is the treatment, $$X$$ are the confounders, and $$Y^a$$ is the potential outcome under treatment $$a$$. whereas the ATT only requires $$E[Y^0|A = 1, X] = E[Y^0|A=0, X]$$ and the ATC only requires $$E[Y^1|A = 0, X] = E[Y^1|A=1, X]$$ Similarly, the requirements for positivity are slightly more relaxed for the ATT and ATC. For the ATE, $$0 < P(A=1|X) < 1$$, but the ATT only requires $$P(A = 1|X) < 1$$ and the ATC requires $$P(A = 1|X) > 0$$. Put simply, the ATE requires complete overlap in the covariate space for both groups, whereas the ATT only requires overlap in the covariate space of the treated units and the ATC only requires overlap in the covariate space of the controls.

It's important to formulate your research question in a way that actually answers the substantive question you want to answer. The ATE of smoking is uninteresting. The ATT doesn't help you decide whether you should enroll new types of participants in a program. The ATC doesn't help you decide whether a currently implement treatment is working. The reason this is so important is that different statistical methods target different estimands. Researchers need to decide on the estimand that makes the most sense for them and then use the correct statistical method that corresponds to that estimand.

• Looking forward to that paper! Note you should probably write "identification of the ATE requires mean..." (not: "the ATE" requires it). Apr 30 at 8:56
• Thank you, I clarified this. I will link it here when it is ready!
– Noah
Apr 30 at 19:22