A Multi-Proportions Randomized Response Model
In surveys of human populations respondents are not likely to participate or tell the truth when the reply may tend to stigmatize them in the eyes of the surveyor. Warner developed a technique for estimating the proportion of persons with such a sensitive characteristic in a binomial situation by having the respondent answer only 'Yes' or 'No' to one of two questions selected at random. This technique is hereby extended to the case where the population can be divided into $t$ categories in which at least one and not more than $(t - 1)$ of them are stigmatizing. The case when $t = 3$ is investigated in detail and the estimates are shown to be unbiased. The variances are larger than those for regular trinomial estimators but this loss may be compensated for by greater cooperation. Under the assumption that respondents may tend to lie under either interviewing method, comparisons are made to determine where the randomizing technique still has an advantage. An example of a field trial is cited where the method was shown to work only with highly skilled interviewers and on more intelligent respondents. Improvements in field technique are also suggested
Abul-Ela, AL., Greenberg, BG., & Horvitz, D. (1967). A Multi-Proportions Randomized Response Model. Journal of the American Statistical Association, 62(319), 990-1008.