For a sensitive survey in which the population is comprised of several clusters with a quantitative attribute, we present an additive quantitative randomized response model by cluster sampling that adapts a two-stage cluster sampling instead of a simple random sample based on Himmelfarb-Edgell's additive quantitative attribute model and Gjestvang-Singh's one. We also derive optimum values for the number of 1st stage clusters and the optimum values of observation units in a 2nd stage cluster

more »

... r the condition of minimizing the variance given constant cost. We can see that Himmelfarb-Edgell's model is more efficient than Gjestvang-Singh's model under the condition of cluster sampling.