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In randomized trials with non-compliance among the treatment group, a common estimator is the Complier Average Causal Effect (also called the Local Average Treatment Effect), which (conditional on a number of assumptions) provides an unbiased estimate of the causal effect of the treatment. The idea is that both the intention-to-treat and per-protocol estimates of the treatment effect may be biased in the presence of non-compliance, so the CACE is, by my understanding, the causal effect among individuals who were randomized to the treatment and complied with it versus those individuals who would have complied with the treatment if they had been randomized to it.

A common approach to calculating the CACE is using instrumental variables methods, where randomization to treatment is the instrument. So, then, it should be possible to calculate the CACE using a two-stage regression approach, where the first stage we regress the instrument on the compliance indicator, and then we use the predicted values from that regression as a predictor in a regression on the outcome of interest. So, to use a very simplistic notation, the two-stage approach to calculating the CACE is then:

$$C_i = \gamma_0 + \gamma_1Trt_i + \gamma^T{X}$$

$$Y_i = \beta_0 + \beta_1\hat{C_i}+\beta^T{X}$$

where $C_i$ is an indicator for whether or not individual $i$ complied with the treatment, and $X$ is some vector of covariates, and $\beta_1$ should give you the CACE (I think? I am having a hard time following the literature on the subject, since it has a tendency to use notation and language I find rather esoteric).

I see this basic approach used and discussed frequently. However, what I don't understand, is that typically compliance is a "one-sided" issue ... that is, you only observe whether or not someone complies with the treatment if they are in the treatment group; in the control group, compliance status is unobserved.

So how can you even get any estimates from the first stage of the regression? $C_i$ is unobserved for members of the control group, so the only way to get any estimates to plug into the second stage of the regression is to assign them values for $C_i$, but I never see this done or discussed. Am I misunderstanding something about this approach?

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This is an old question and I can't pretend to be able to help you with the maths. But, maybe this will help others looking to understand this conceptually...

The way I've seen CACE used runs like this:

  • Define compliers in the treatment group
  • Use wider data (that is available for the treatment and control arms) to find factors that determine compliance in the treatment group
  • Limit the control group using the wider factors that predict compliance
  • Compare between 'potential' compliers in the control group and actual compliers in the treatment group
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