Exact confidence intervals for randomized response strategies

For surveys with sensitive questions, randomized response sampling strategies are often used to increase the response rate and encourage participants to provide the truth of the question while participants' privacy and confidentiality are protected. The proportion of responding ‘yes’ to the sensitive question is the parameter of interest. Asymptotic confidence intervals for this proportion are calculated from the limiting distribution of the test statistic, and are traditionally used in practice for statistical inference. It is well known that these intervals do not guarantee the coverage probability. For this reason, we apply the exact approach, adjusting the critical value as in 10 J. Frey and A. Pérez, Exact binomial confidence intervals for randomized response, Amer. Statist.66 (2012), pp. 8–15. Available at http://dx.doi.org/10.1080/00031305.2012.663680.Taylor &; Francis Online], Web of Science ®] , Google Scholar]], to construct the exact confidence interval of the proportion based on the likelihood ratio test and three Wilson-type tests. Two randomized response sampling strategies are studied: the Warner model and the unrelated model. The exact interval based on the likelihood ratio test has shorter average length than others when the probability of the sensitive question is low. Exact Wilson intervals have good performance in other cases. A real example from a survey study is utilized to illustrate the application of these exact intervals.