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Say we have data on the total number of units assigned to the control group (1200) and their outcome (20%). We are dealing with one-sided non-compliance, and have data on number of never-takers in the treatment group (500), the outcome of never-takers (25%), and the number of compliers in the treatment group (1500) and their outcome (30%).

We can use excludability to estimate the number and outcome of never-takers in the control group. But how can we estimate these for the compliers in the control group? Also, is it possible to find the treatment effect for compliers (relative effect between compliers in treatment to compliers in control)?

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    $\begingroup$ What is "one sided non-compliance"? And what is "excludability"? $\endgroup$
    – AdamO
    Commented Nov 2, 2022 at 18:31

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I understand your description of the problem to mean you have data from the following table (n (% response)):

\begin{array}{c|ccc} & \text{Never-took} & \text{Complied} & \text{Total} \\ \hline \text{Treatment} & 500 (25) & 1500 (30) & 2000 (31.3) \\ \text{Control} & & &1200 (20.0) \\ \end{array}

And your question is either, a) can I arithmetically fill in the missing control n's and the control %'s response. or b) is it a reasonable assumption to do so? Well, for a, apropos of nothing, you'd have to apportion 2/3s of controls to complied and 1/3 of controls to never took. Since the drug is placebo, you could assume the efficacy is 20% regardless of compliance.

But that brings me to point B: it's totally bogus to make this kind of assumption. It's well known that control subjects are less likely to comply precisely because the drug isn't working.. alternately, regression to the mean might imply subjects are more likely to try treatment if they're more likely to respond. Not having any other data simply means you can't do a per-protocol analysis.

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  • $\begingroup$ Well the efficacy for never-takers in the control group is 25%. So if we know that the mean efficacy for the control group is 20%, then the efficacy for compliers in the control group must be different than 20%, no? The proportion of never-takers in the control group is equal to the proportion of never-takers in the treatment group, which is 25%. So this means that out of the 1200 units in the placebo, 300 (25%) are non-compliers. The trick here is to find the efficacy for the remainding 75% $\endgroup$
    – juanjedi
    Commented Nov 2, 2022 at 21:30
  • $\begingroup$ It wasn't clear from your problem description. Anyway, it's a simple arithmetic problem and you don't know all the variables. When $n_{1,2} + n_{2,2} = n_{., 2}$ you can only solve for the third by knowing two other free parameters. $\endgroup$
    – AdamO
    Commented Nov 3, 2022 at 13:59

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