Methods of finding the minimum value and the Lagrange function were applied to deduce the formulae for the optimum sample sizes for polychotomous randomized response technique (RRT) model in stratified two-stage sampling, so as to minimize the cost for specified sampling errors and to minimize the sampling errors under the constraint of a fixed budget. These formulae were successfully applied to sensitive topics survey among men who have sex with men (MSM) in Beijing, China.
"Sample Size Determination for the Polychotomous Randomized Response Model for Sensitive Questions in a Stratified Two-Stage Sampling Survey." J. Appl. Math. 2014 1 - 6, 2014. https://doi.org/10.1155/2014/738097