Type II combination questionnaire model: A new survey design for a totally sensitive binary variable correlated with another nonsensitive binary variable

Abstract

Recently, Yu, Lu, and Tian (2013) introduced a combination questionnaire model to investigate the association between one sensitive binary variable and another non-sensitive binary variable. However, in practice, we sometimes need to assess the association between one totally sensitive binary variable (e.g., the number of sex partners being ⩽3 or >3, the annual income being ⩽$25,000 or >$25,000, and so on) and one non-sensitive binary variable (e.g., good or poor health status, with or without cervical cancer, and so on). Although we could directly adopt the four-category parallel model (Liu & Tian, 2013), the information contained in the non-sensitive binary variable cannot be utilized in the design. Intuitively, such information can be used to enhance the degree of privacy protection so that more respondents will not face the sensitive question. The objective of this paper is to propose a new survey design (called Type II combination questionnaire model, which consists of a four-category parallel questionnaire and a supplemental direct questionnaire) and to develop corresponding statistical methods for analyzing sensitive data collected by this technique. Likelihood-based methods including maximum likelihood estimates, asymptotic 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 methods are also presented. Simulation studies are performed and a cervical cancer data set in Atlanta is used to illustrate the proposed methods.