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From my mailbox: should I work with only a subsample of my control group if I have big take-up problems?

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Over the past month I’ve received several versions of the same question, so thought it might be useful to post about it.
 
Here’s one version:
I have a question about an experiment in which we had a very big problem getting the individuals in the treatment group to take-up the treatment. Therefore we now have a treatment much smaller than the control. For efficiency reasons does it still make sense to survey all the control group, or should we take a random draw in order to have an equal number of treated and control?
 
And another version

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