Randomized Response

## Randomized Response

#### 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.

Example:

= ``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:

• Zellner, A. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, JASA, June 1962 348-368.

• Warner, S., Randomized response: a survey technique for eliminating evasive answer bias. JASA, March 1965, 63-69.

• Abul-Ela, A., Greenberg, B., and Horvitz, A., A multi-proportions randomized response model. JASA, Sept. 1967, 990-1008.

• Greenberg, B., Abdul-Ela, A., Simmons, W., and Horvitz, D., The unrelated question randomized response model: theorectical framework. JASA, June 1969, 520-539.

• Gould,A. Shah, B., and Abernathy, J., Unrelated question randomized response techniques with two trials per respondent. Proceedings of the Social Statistics Section of the ASA, 1971.

• Grrenberg, B., Abernathy, J., and Horvitz, D., Application of the randomized response technique in obtaining quantitative data JASA, June 1971, 243-250.

• Warner, S., The linear randomized response model. JASA, Dec. 1971, 884-888.

• Cambell, C. and Joiner, B., How to get the answer without being sure you've asked the question. The American Statistician, Dec. 1973, 229-231.

• Dowling, T. and Shactman, Ran On the relative efficiency of randomized response models. JASA, March 1975, 84-87.

• Goodstadt, M. and Gruson, V., The randomized response technique: a test on drug use. JASA, Dec. 1975, 814-818.

• Maceli, J., How to ask sensitive questions without getting punched in the nose. Modules in Applied Mathematics, Vol. 2, edited by W. F. Lucas, Springer-Verlag, New York, 1978.