Randomized response techniques for a multi-level attribute using a single sensitive question

Collecting reliable responses to sensitive survey questions is challenging, since respondents may be more likely to refuse to respond or to provide biased responses. To address these challenges, Warner (J Am Stat Assoc 60:63-69, 1965) pioneered the randomized response (RR) technique to estimate proportions of individuals in a population with either of two possible attributes. The RR technique can overcome non-response and underreporting biases because it doesn't reveal the respondent's attribute, and a generalization of the random component of the response by Christofides (Metrika 57:195-200, 2003) improves estimation properties. In this study, we develop a new RR model to estimate proportions of individuals with each of multiple categories of an attribute using a single sensitive question by means of only one randomization device based on Christofides's (Metrika 57:195-200, 2003) model. Under the proposed model, a respondent reports the absolute difference between an integer associated with his or her attribute and a random integer. In a part of this research, we conduct a simulation study of the relative efficiency of the proposed methods. The technique is illustrated using data from the 2012 Family and Gender Module of the Taiwan Social Change Survey to estimate the proportions of individuals of different sexual orientations, and the results are compared with the results of direct inquiry from the same survey.