I am trying to calculate the sample size (and the repartition between PSUs and SSUs) needed to obtain an estimation of a proportion (that I assume to de around 50%) with a 0.05 standard error in the case of a stratified three degree sample design constructed as follows :
-stratas (16) are all selected
-within each strata, districts are randomly selected with PPS
-within each district, Enumeration areas are selected randomly with PPS
-within each Enumeration Area, households are randomly selected (simple random walk)
So I am looking for a formula that links sample size and cluster repartition to the standard error of the estimator, the expected value of my proportion and clusters' size and intra cluster correlation (These are the only things I know). I have no way of knowing what the proportion I want to estimate is within my clusters.
So far all formulas I found (that calculate estimator variance for instance) either assume total sample size to give repartition between clusters, or assume that the proportion to estimate is known within each cluster.
Any insights or bibliography suggestion would be most helpful.