Xu Tonghui, Sedory Stephen A, Singh Sarjinder
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX, USA.
Biom J. 2021 Dec;63(8):1688-1705. doi: 10.1002/bimj.202000395. Epub 2021 Jul 23.
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.
在本文中，我们开发了一种全新的独特无关问题随机应答模型，其中每张卡片有两个问题，要么两个问题都关于敏感特征，要么两个问题都关于两个无关特征。所提出的模型具有独特性，因为这是在每张卡片上询问两个问题从而保护应答者隐私的唯一方式。我们首先开发了两种敏感特征的流行率及其重叠部分的估计量。然后我们证明所得估计量是无偏的。接下来，我们推导所开发的比例估计量的方差表达式。我们还计算了所提出模型相对于其竞争模型的相对效率和相对隐私保护。通过将所提出估计量的方差与估计量方差 - 协方差的克拉默 - 拉奥下界进行比较，也验证了所提出估计量的方差。还讨论了条件比例、相对风险和相关系数的估计量。最后，考虑了所提出模型的一个实际数据应用，这表明了在所提出模型在医学和社会科学研究中使用的重要性。
