A note on randomized response models for quantitative data

Abstract

Standard randomized response (RR) models deal primarily with surveys which usually require a ‘yes’ or a ‘no’ response to a sensitive question, or a choice for responses from a set of nominal categories. As opposed to that, Eichhorn and Hayre (1983) have considered survey models involving a quantitative response variable and proposed an RR technique for it. Such models are very useful in studies involving a measured response variable which is highly ‘sensitive’ in its nature. Eichhorn and Hayre obtained an unbiased estimate for the expectation of the quantitative response variable of interest. In this note we propose a procedure which uses a design parameter (controlled by the experimenter) that generalizes Eichhorn and Hayre’s results. Such a procedure yields an estimate for the desired expectation which has a uniformly smaller variance. Copyright Springer-Verlag 2004

A pseudo-empirical log-likelihood estimator using scrambled responses

“…Note that in the case of the additive randomized response model due to Pollock and Bek (1976) and Himmelfarb and Edgell (1980), there is a possibility of having negative scrambled responses which may still result in a negative estimate of mean or total in the presence of positive weights of pseudo-empirical loglikelihoods. This is a reason why only the multiplicative randomized response models accredited to Bar-Lev et al (2004) and Eichhorn and Hayre (1983) have been considered here and seem more practical than the other randomized response models. No doubt the forced quantitative randomized response model due to Gjestvang and Singh (2007), and other additive models in Chaudhuri and Roy (1997) and Chaudhuri and Mukerjee (1988) are another possibilities, but those models may also lead to negative estimates of total.…”

“…A rich growth of literature on such randomized response (RR) methods can be found in Tracy and Mangat (1996). Bar-Lev et al (2004) suggested an improved version of Eichhorn and Hayre's (1983) multiplicative model to collect information on sensitive quantitative variables like income, tax evasion, amount of drug used etc. We first discuss the randomized response model introduced by Bar-Lev, Bobovitch and Boukai (2004), which we call the BBB model hereafter.…”

A new estimator based on auxiliary information through quantitative randomized response techniques

“…The multiplicative model was later investigated in depth by Eichhorn and Hayre (1983), who referred to it as the scrambled responses method. Similarly, Bar-Lev, Bobovitch, and Boukai (2004) proposed a method which uses a partial model that generalizes Eichhorn and Hayre's results and yields an estimate which, under mild conditions, has a uniformly smaller variance. Further developments focused on the use of auxiliary variables to improve the precision.…”

New scrambled response models for estimating the mean of a sensitive quantitative character

“…Gupta et al [12] introduced an optional RR technique which is more efficient than Eichhorn and Hayre's scrambled device, but performs poorly if compared with the two-stage response model proposed by Ryu et al [23]. Bar-Lev et al [2] generalized Eichhorn and Hayre's SRR device by introducing a design parameter which is used for randomizing the responses. Further developments on SRR models can be found in [8,[13][14][15][16][17]19,20,24,26].…”