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Suppose we run an AB test wherein users in the treatment are shown some content and users in control are not shown the content. For all intents and purposes, it could be a button. Clicking on the content is optional. After the experiment is done, we can compare treatment and control groups on some binary outcome.

The difference in means here is the effect of offering the content to be clicked, not on clicking the content itself. This is known as the ITT effect. Obviously, we can't just take the users who clicked the content and compare them against control because of selection effects.

However, we can use treatment as an instrumental variable. Angrist and Pischke write in their book Mostly Harmless Econometrics...

In many randomized trials, participation is voluntary among those randomly assigned to receive treatment. On the other hand, no one in the control group has access to the experimental intervention. Since the group that receives (i.e., complies with) the assigned treatment is a self-selected subset of those offered treatment, a comparison between those actually treated and the control group is misleading. The selection bias in this case is almost always positive: those who take their medicine in a randomized trial tend to be healthier; those who take advantage of randomly assigned economic interventions like training programs tend to earn more anyway.

[Instrumental variables] using the randomly assigned treatment intended as an instrumental variable for treatment received solves this sort of compliance problem. Moreover, LATE is the effect of treatment on the treated in this case.

If my read on this passage is correct, I can use the treatment assignment as an instrument to estimate the effect of treatment on the treated.

Using R I might be able to do something like ivreg::ivreg(y ~ click | treatment) and interpret the estimate of the coefficient of click as the effect of clicking the content on the outcome y.

Have I understood the use of IV in randomized experiments correctly? If not, what is the interpretation of the click coefficient in this case? Does it have a meaningful interpretation otherwise?

EDIT: I believe the interpretation is "The difference in means between not clicking the content and clicking the content in the group of users who would always click given the option"

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I believe the interpretation of the coefficient for click is the effect of clicking the content on the outcome y for those persons who click when given the treatment, but who would not have clicked otherwise.

See point 1 and 4 here: https://egap.org/resource/10-things-to-know-about-the-local-average-treatment-effect/

Interpretation of LATE is also discussed in "Causal Inference: The Mixtape" (Scott Cunningham) in Chapters 6.2.7 and 7.6 and the following reference is given there (havn't yet read the paper myself though):

Imbens, Guideo W., and Joshua D. Angrist. 1994. “Identification and Estimation of Local Average Treatment Effects.” Econometrica 62 (2): 467–75

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The estimand of interest of interest is the Complier Average Causal Effect (Angrist, Imbens, Rubin 1994), also known as the Local Average Treatment Effect or LATE. Its interpretation is the average treatment effect (ATE) among the subset of subjects who comply with treatment assignment. In a situation of one-sided non-compliance this will be the same as the ATT. It is not guaranteed to be the same in a situation of two sided non-compliance.

Compliers are the subjects who would take treatment (in your case click through when exposed to the prompt) when in the treatment group and would the control treatment when assigned to control. They "comply" with their treatment assignment, and if you did a different version of your experiment the complier population might be different. This population may or may not be theoretically meaningful depending on the question of interest.

A note. Assuming that the estimated number of compliers is not zero the CACE is defined as the ITT/# of compliers. Estimation of this quantity will be biased but consistent, and the bias will be negligible in sufficiently large samples. Mechanically, the absolute value of the CACE will always be at least as big as the ITT, and in any situation of partial compliance will always be strictly larger in absolute terms.

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    $\begingroup$ You talk about CACE and then you say CATE in your answer. $\endgroup$
    – usεr11852
    Nov 14, 2022 at 22:11

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