Randomized response

Randomized response is a research method used in structured survey interview. It was first proposed by S. L. Warner in 1965 and later modified by B. G. Greenberg in 1969. 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" even though they have not 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: Transformed to yield EP:
 * $$YA = p\times EP + (1 - p)(1 - EP)$$
 * $$EP = \frac{YA + p - 1}{2p - 1}$$

Example
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 If all interviewed have answered honestly then their true proportion of consumers is 1/8 (= 12.5%).
 * Alternative 1: "I have consumed marijuana."
 * Alternative 2: "I have never consumed marijuana."
 * $$EP = (\tfrac{3}{4} + \tfrac{1}{6} - 1) / (2\times \tfrac{1}{6} - 1) = \tfrac{1}{8}$$