**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.

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