Q:

What is the proper way to increase power when randomizing at the village level?

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?  

A:

For a cluster randomized trial to increase number of clusters is more efficient to increase the power than increase the number of individuals in each cluster when your total sample size is fixed.
For example, if you want to sample total 1000 people for the CRT trial, you select 50 clusters (villages) each with 20 villagers will achieve more power than select 20 clusters (villages) each with 50 villagers.
You can read the following paper for your reference.
Advanced Statistics: Statistical Methods for Analyzing Cluster and Cluster-randomized Data 

