Bounds for the variance, design effect and coefficient of variation of proportions in two-stage cluster sampling with equal sizes

In the estimation of proportions using simple random sampling, the maximum value of the variance can be used to compute the sample size when there is no information of the variable of interest. We extend this result to the estimation of proportions under two-stage cluster sampling with equal sizes, showing the expression for the maximum variance. As a by-product it is immediate to obtain bounds for the design effect and the coefficient of variation of the proportion estimator. Some examples are given related to the computation of the bounds.