Two sensitive characteristics and their overlap with two questions per card

In this paper, we developed a new unique unrelated question randomized response model in which each card has two questions, either both questions on the sensitive characteristics or both questions on the two unrelated characteristics. The proposed model is unique in the sense this is the only way of asking two questions printed on each card that leads to protection of the privacy of the respondent. We first develop estimators of the prevalence of the two sensitive characteristics and of their overlap. Then we show that the resultant estimators are unbiased. Next we derive variance expressions for the developed estimators of the proportions. We also compute the relative efficiency and relative privacy protection of the proposed model with respect to its competitors. The variances of the proposed estimators are also verified by comparing them to the Cramer-Rao lower bounds of variance-covariance of the estimators. Estimators of conditional proportion, relative risk, and correlation coefficient are also discussed. Lastly, a real data application of the proposed model is considered, which shows the importance of the use of the proposed model in medical and social science studies.