Journal of Probability and Statistics
Volume 2013 (2013), Article ID 827048, 11 pages
Research Article

A Survey Design for a Sensitive Binary Variable Correlated with Another Nonsensitive Binary Variable

1School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China
2School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China
3Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong

Received 30 August 2012; Accepted 19 May 2013

Academic Editor: Zhidong Bai

Copyright © 2013 Jun-Wu Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Tian et al. (2007) introduced a so-called hidden sensitivity model for evaluating the association of two sensitive questions with binary outcomes. However, in practice, we sometimes need to assess the association between one sensitive binary variable (e.g., whether or not a drug user, the number of sex partner being 1 or >1, and so on) and one nonsensitive binary variable (e.g., good or poor health status, with or without cervical cancer, and so on). To address this issue, by sufficiently utilizing the information contained in the non-sensitive binary variable, in this paper, we propose a new survey scheme, called combination questionnaire design/model, which consists of a main questionnaire and a supplemental questionnaire. The introduction of the supplemental questionnaire which is indeed a design of direct questioning can effectively reduce the noncompliance behavior since more respondents will not be faced with the sensitive question. Likelihood-based inferences including maximum likelihood estimates via the expectation-maximization algorithm, asymptotic confidence intervals, and bootstrap confidence intervals of parameters of interest are derived. A likelihood ratio test is provided to test the association between the two binary random variables. Bayesian inferences are also discussed. Simulation studies are performed, and a cervical cancer data set in Atlanta is used to illustrate the proposed methods.