Optional versus compulsory randomized response techniques in complex surveys

In estimating the proportion of people bearing a sensitive attribute in a community, to mitigate possible evasive answer biases, Warner (J. Amer. Statist. Assoc. 60 (1965) 63) introduced a technique of randomized response (RR) in human surveys, by way of protecting individual privacy. Chaudhuri and Mukerjee (Calcutta Statist. Assoc. Bull. 34 (1985) 225; Randomized Response: Theory and Techniques, Marcel Dekker, New York) presented a modification allowing a direct response (DR) option to whom the attribute does not appear to be stigmatizing enough. Warner himself and many of his followers restrict the application of their RR devices to surveys with selection exclusively by 'simple random sampling with replacement'. Chaudhuri (J. Statist. Plann. Inference 34 (2001a) 37; Pakistan J. Statist. 17 (3) (2001b) 259; Calcutta Statist. Assoc. Bull. 52 (205-208) (2002) 315) showed the efficacy of some of these devices when sample selection is by general unequal probabilities possibly even without replacement. Here, we present theories for unbiased estimation of the proportion along with unbiased estimation of the variances of the estimators when 'compulsory' or 'optional' RR's are gathered from persons sampled with varying probabilities. Gains in efficiency by allowing DR option rather than RR compulsion are illustrated numerically through simulation from data.