A new randomized response device for sensitive characteristics: an application of the negative hypergeometic distributio
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A new randomized response device for sensitive characteristics: an application of the negative hypergeometic distribution Sarjinder Singh · Stephen A. Sedory
Received: 10 October 2011 / Accepted: 8 August 2012 / Published online: 28 May 2013 © Sapienza Università di Roma 2013
Abstract In this paper, a new randomized response device is proposed based on the use of negative hypergeometric distribution while estimating the proportion of persons possessing a sensitive characteristic in a population. Situations where the proposed randomization device can perform better than the Warner (J Am Stat Assoc 60, 63–69, 1965), the Kuk (Biomerika 77(2), 436–438, 1990) and the Mangat (J R Statist Soc B 56, 93–95, 1994) estimators are simulated and discussed. Keywords
Sensitive variables · Estimation of proportion · Relative efficiency
1 Introduction Warner [5] proposed an interviewing technique, called Randomized Response, to protect an interviewee’s privacy and to reduce a major source of bias (evasive answers or refusing to respond) in estimating the prevalence of sensitive characteristics in surveys of human populations. Warner [5] designed a randomization device, for example a spinner or a deck of cards that consists of two mutually exclusive outcomes. In the case of cards, each card has one of the following statements: (i) I possess attribute A; (ii) I do not possess attribute A. The maximum likelihood estimator of π, the proportion of respondents in the population possessing the attribute A, is given by: πˆ w =
(n 1 /n) − (1 − P) 2P − 1
(1.1)
where n 1 is the number of individuals responding “yes”, n is the number of respondents selected by a simple random and with replacement sample (SRSWR), and P is the proportion of cards bearing the statement, “I possess an attribute A.” The variance of πˆ w is given by:
S. Singh (B) · S. A. Sedory Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA e-mail: [email protected]
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S. Singh, S. A. Sedory
V (πˆ w ) =
P(1 − P) π(1 − π) + n n(2P − 1)2
(1.2)
Kuk [1] introduced an ingenious randomized response model in which respondents belonging to a sensitive group A are instructed to use a deck of cards having the proportion θ1 of cards with the statement, “I belong to group A” and if respondents belong to non-sensitive group Ac then they are instructed to use a different deck of cards having the proportion θ2 of cards with the statement, “I do not belong to group A”. Again let π be the true proportion of persons belonging to the sensitive group A. Then, the probability of a ‘yes’ answer in the Kuk [1] model is given by: θkuk = θ1 π + (1 − π)θ2
(1.3)
Further assume that a simple random with replacement sample of n respondents is selected from the population, and that n 1 is the number of observed “yes” answers. The number of people n 1 that answer “yes” is binomially distributed with parameters θkuk = θ1 π +(1−π)θ2 and n. For the Kuk [1] model, an unbiased estimator of the population proportion π is given by: πˆ kuk =
n 1 /n − θ2 , θ1
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