# Randomized response

Randomised response is a research method used in structured survey interview. It was first proposed by S. L. Warner in 1965[1] 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 have had abortions[3].

The concept is somewhat similar to plausible deniability. Plausible deniability allows the subject to credibly say he did not make a statement, while the randomized response technique allows the subject to credibly say he had not been truthful when making a statement.

## Example

### With a coin

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 is 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 close to 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%.

### With cards

The same question can be asked with three cards which are unmarked on one side, and bear a question on the other side. The cards are randomly mixed, and laid in front of the subject. The subject takes one card, turns it over, and answers the question on it truthfully with either "yes" or "no".

• One card asks: "Did you have sex with a prostitute this month?"
• Another card asks: "Is there a triangle on this card?" (There is no triangle.)
• The last card asks: "Is there a triangle on this card?" (There is a triangle.)

The researcher does not know which question has been asked.

Under the assumption that the "Yes" and "No" answers to the control questions cancel each other out, the number of subjects who have had sex with a prostitute is triple that of all "Yes" answers in excess of the "No" answers.

## 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 ${\displaystyle p}$ be the probability to answer the sensitive question and ${\displaystyle EP}$ the true proportion of those interviewed bearing the embarrassing property, then the proportion of "yes"-answers ${\displaystyle YA}$ is composed as follows:

• ${\displaystyle YA=p\times EP+(1-p)(1-EP)}$

Transformed to yield EP:

• ${\displaystyle EP={\frac {YA+p-1}{2p-1}}}$

### 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 (${\displaystyle p={\tfrac {1}{6}}}$). 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 (${\displaystyle YA={\tfrac {3}{4}}}$). Inserted into the formula you get

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

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

## References

1. ^ Warner, S. L. (March 1965). "Randomised response: a survey technique for eliminating evasive answer bias". Journal of the American Statistical Association. Taylor & Francis. 60 (309): 63–69. doi:10.1080/01621459.1965.10480775. JSTOR 2283137.
2. ^ Greenberg, B. G.; Abul-Ela, Abdel-Latif A.; Simmons, Walt R.; Horvitz, Daniel G. (June 1969). "The Unrelated Question Randomised Response Model: Theoretical Framework". Journal of the American Statistical Association. Taylor & Francis. 64 (326): 520–39. doi:10.2307/2283636. JSTOR 2283636.
3. ^ Abernathy, James R.; Greenberg, Bernard G.; Horvitz, Daniel G. (February 1970). "Estimates of induced abortion in urban North Carolina". Demography. 7 (1): 19–29. doi:10.2307/2060019. JSTOR 2060019.
• Arijit Chaudhuri, Rahul Mukerjee: Randomized response: theory and techniques (Google Scholar).
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• 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.
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• 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
• S. Aoki & K. Sezaki (2014). Privacy-Preserving Community Sensing for Medical Research with Duplicated Perturbation. IEEE International Conference on Communications, 4252-4257.