Randomized Response

Randomized Response

Donald Bentley

Pomona College

THE PROBLEM: How can we get accurate answers to a sensitive question which respondents might be reluctant to answer truthfully?

EXAMPLES: ``Have you ever used illegal drugs?"

``Do you favor a constitutional admendment that would outlaw most abortions?"

``Have you had more than one sexual partner in the past 6 months?"

``Have you ever driven a motor vehicle while intoxicated?"

METHOD #1: Warner (1965)

Let be the sensitive question and be its complement. For example

= ``Have you ever used a sick day leave when you weren't really sick?" YES N0

= ``Have you never used a sick day leave when you weren't really sick?" YES N0

With some (known) probability a subject gets , otherwise (with probability 1 - ) he or she will answer

KEY POINT: The respondent determines which question he or she answers using some probability device which is under his or her control.

Example: Have the respondent roll a die. If the result is { 1, 2, 3, or 4 } answer , if the die is {5 or 6} answer question Since only the respondent knows which question he or she is answering, there should be no stigma attached to a YES or N0 response.

BUT, can we still estimate the proportion who would say YES to the sensitive question?

Let = proportion in the population for which the true response to is YES. So is the chance of getting a YES response to Given the Warner randomized response scheme, the proportion of YES responses should be given by

We solve easily for p to give

If the number of YES responses in a sample of size is , we estimate p with

Question: What happens when

METHOD #2: The Innocuous Question

Replace with an innocuous question, , which has a known probability of yielding a YES response.


= ``Flip a coin. Have you ever shoplifted?" YES N0

= ``Flip a coin. Did you get a head?" YES N0

Again, the respondent does with probability and has a chance to answer . If the known probability of a YES to is , we find that overall

Solve for p to give

and the estimator based on responses of YES in a sample of size becomes


Question: Which method is more efficient?

To compare them we need to examine the variances of each sample proportion estimate.

Note: If we could ask the question directly, we know that

For Warner's Method #1:

For the Innocuous Question Method #2:

If we can calculate

Method #1

Method #2

Innoccuous Question method is almost ten times more efficient than Warner's method.

Some References for Randomized Response:

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