The optimal sample size in the crosswise model for sensitive questions

For interval estimation of the fraction of the population with a stigmatizing characteristic, the nonrandomized response model proposed by Tian, Yu, and Geng (2007) is considered. The most common method for constructing a confidence interval (c.i.) is through the application of the Central Limit Theorem. Unfortunately, such c.i.’s do not maintain the prescribed confidence level, in contradiction to Neyman’s (1934) definition of c.i. In the present paper, the exact c.i. for this fraction is constructed, i.e., the c.i. which keeps the given confidence level. The length of the proposed c.i. depends on the given probability of a positive answer to the neutral question, and on the sample size. For such c.i.’s, the probability of a positive answer to the neutral question is established with respect to the given limit on privacy protection of the interviewee, and the optimal sample size for obtaining the c.i. of a given length is derived.