Let's say that we have a hypothetical RCT with the following information:
--We are interested in seeing whether a particular belief held among some individuals in a developing country can be challenged with a booklet (the treatment).
--In a region with a population of 4.6 million there are approximately 26,000 with this belief. The population is divided among roughly 3000 villages.
--If we are to ONLY target individuals who have this belief (and we assume this belief is evenly distributed across villages), there will be roughly 9 individuals per village who have this belief. If we were to randomize at the village-level it is unlikely that we could detect a treatment effect given an N=9.
What is the proper way to increase the sample size to achieve sufficient power for detecting an effect? Would it make sense to randomly cluster villages together such that each cluster had an average of, say, 45 individuals with the belief?