Suppose I am running an experiment to see if a treatment changes the mean weight of a group of people. Note that I am specifically interested in the mean weight: if half the people get heavier, and half get lighter, that is considered a null result. There of course may be some random drift, so I employ a control group:
For the treatment, people are organized into k groups with m subjects each, and I have measurements only of the mean: k independent measurements.
For the control, people are also organized into k groups with m subjects each, but I have measurements on every individual. So now I have k*m independent measurements of height. However, this still produces only k independent measurements of my outcome, since I'm interested in the change in the mean, not the change in individuals.
How can I compare the difference in the change in the mean height between the control and treatment, while taking advantage of the fact that I have more independent measurements for the control groups?
We know the variance on the control measurement is lower, but I can't figure out how to actually take advantage of that in a test, bootstrapped or otherwise. The only thing I can do is measure two separate confidence intervals and see if they overlap, but that's not a proper way to do science.
I actually have individual data for the treatment groups, but the people embedded in a social network, so they are not independent. I can't use nested/hierarchical/multi-level models because that still assumes independence at the lowest level.))