The number of clusters required for the CRT giving a power of .8 is inluenced by 1) the number of students in a cluster (n), the intra class correlation (icc) the effect size (Cohenโs d) and the explained variance by the covariates on cluster level (rho.Xj). Below a plot is shown for the situation where n=30, icc=.2, icc=.4, icc=.6 and rho.Xj=0.1 (age seems to explain some variance), and a plot is shown for the situation where n=20.
delta<-function(power,J,n,icc,rho.Xj){
icc.cov=(1-rho.Xj^2)/((1-rho.Xj^2)+((1-icc)/icc))
ncp=abs(qt(1-power,df=J-2)+-qt(0.975,df=J-2))
delta=sqrt((ncp^2*(4*(icc.cov+(1-icc.cov)/n)))/J)
return(delta)}The results show the relationship between the total number of clusters (control+intervention) and the effect size. The expected effect size for math and literacy are large, about .9. In this case, with n=20 and icc=.6, 26 clusters are required to have a power of .8. When n=30 and icc=.2,only 12 clusters have to be included.
Since the effect sizes in the Sudan paper are very large, it may be a good idea to look at a smaller effect size as well. When the effect-size is .5, 78 clusters are required in the most conservative situation (n=20 and icc=.6) and 30 clusters are required in the least conservative situation (n=30 and icc=.2).
The question was whether 50 to 60 clusters would be sufficient to have .8 power. The results show that even with effect sizes much smaller than expected, this number of clusters would be more than sufficient to find a significant effect.