# Randomized response

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Randomized response is a research method used in structured survey interview. It was first proposed by S. L. Warner in 19651 and later modified by B. G. Greenberg in 1969.2 It allows respondents to respond to sensitive issues (such as criminal behavior or sexuality) while maintaining confidentiality. Chance decides, unknown to the interviewer, whether the question is to be answered truthfully, or "yes", regardless of the truth.

For example, social scientists have used it to ask people whether they use drugs, whether they have illegally installed telephones, or whether they have evaded paying taxes. Before abortions were legal, social scientists used the method to ask women whether they had had abortions.

## Example

Ask a man whether he had sex with a prostitute this month. Before he answers ask him to flip a coin. Instruct him to answer "yes" if the coin comes up tails, and truthfully, if it comes up heads. Only he knows whether his answer reflects the toss of the coin or his true experience. It is very important to assume that people who get heads will answer truthfully, otherwise the surveyor are not able to speculate.

Half the people—or half the questionnaire population—get tails and the other half get heads when they flip the coin. Therefore, half of those people will answer "yes" regardless of whether they have done it. The other half will answer truthfully according to their experience. So whatever proportion of the group said "no", the true number who did not have sex with a prostitute is double that, because we assume the two halves are probably the same as it is a large randomized sampling. For example, if 20% of the population surveyed said "no", then the true fraction that did not have sex with a prostitute is 40%.

## Original version

Warner's original version (1965) is slightly different: The sensitive question is worded in two dichotomous alternatives, and chance decides, unknown to the interviewer, which one is to be answered honestly. The interviewer gets a "yes" or "no" without knowing to which of the two questions. For mathematical reasons chance cannot be "fair" (½ and ½). Let p be the probability to answer the sensitive question and EP the true proportion of those interviewed bearing the embarrassing property, then the proportion of "yes"-answers YA is composed as follows:

• $YA = p\times EP + (1 - p)(1 - EP)$

Transformed to yield EP:

• $EP = \frac{YA + p - 1}{2p - 1}$

### Example

• Alternative 1: "I have consumed marijuana."
• Alternative 2: "I have never consumed marijuana."

The interviewed are asked to secretly throw a dice and answer the first question only if they throw a 6, otherwise the second question ($p=\tfrac{1}{6}$). The "yes"-answers are now composed of consumers who have thrown a 6 and non-consumers who have thrown a different number. Let the result be 75 "yes"-answers out of 100 interviewed ($YA=\tfrac{3}{4}$). Inserted into the formula you get

• $EP = (\tfrac{3}{4} + \tfrac{1}{6} - 1) / (2\times \tfrac{1}{6} - 1) = \tfrac{1}{8}$

If all interviewed have answered honestly then their true proportion of consumers is 1/8 (= 12.5%).

## References

• ^1 Warner, S. L. (1965). Randomized response: a survey technique for eliminating evasive answer bias. Journal of the American Statistical Association 60, 63–69.
• ^2 Greenberg, B. G., et al. (1969). The Unrelated Question Randomized Response Model: Theoretical Framework. Journal of the American Statistical Association 64(326), 520–39.
• Arijit Chaudhuri, Rahul Mukerjee: Randomized response: theory and techniques (Google Scholar).
• Lee, Cheon-Sig, Sedory, S.A. and Singh, Sarjinder (2013). Estimating at least seven measures for qualitative variables using randomized response sampling. Statistics and Probability Letters, 83, 399-409.
• M. Ostapczuk, M. Moshagen, Z. Zhao & J. Musch (2009). Assessing sensitive attributes using the randomized-response-technique: Evidence for the importance of response symmetry. Journal of Educational and Behavioral Statistics 34, 267–87.
• M. Ostapczuk, J. Musch & M. Moshagen (2009). A randomized-response investigation of the education effect in attitudes towards foreigners. European Journal of Social Psychology 39, 920–31.
• D. Quercia, Ilias Leontiadis, Liam McNamara, Cecilia Mascolo, Jon Crowcroft (2011). SpotME If You Can: Randomized Responses for Location Obfuscation on Mobile Phones. IEEE ICDCS