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JUNE 1999
Tell Me Your Secret - Random Response Methodology
Charles R. Mann, Ph.D.
Have you stolen from your employer?
Are you currently using illegal drugs?
Do you typically read at least one article each month from the principal publication in your field?
Introduction
What do you do when you want to estimate the prevalence of an incriminating, embarrassing or otherwise sensitive condition such as those referenced above? When there is little or no benefit to the respondents, how likely are they to accept an assurance that their answers will be kept confidential or that the identifier on their allegedly anonymous response form is for statistical purposes only?
We present here a statistical methodology for obtaining a measure of the prevalence of such a condition under the assumption that respondents will be willing to be of assistance as long as they are convinced that their privacy is protected to the extent that there is no possibility of any specific individual being known to have the negative attribute.
For simplicity of presentation the procedure, known as "random response" methodology, will be described in the context of a large employer concerned with the incidence of illegal drug usage. However, nothing in the presentation will be dependent on this application and the technique described may be directly generalized to any other similar use.
A Typical Application - Drug Usage
Employers faced with decisions as to whether to institute drug testing, treatment or rehabilitation programs may want to establish levels of commitment commensurate with the extent of the drug problem in their workforce. Clearly, it would be unreasonable to expect to determine the magnitude of such a problem by simply questioning employees. Those who are drug users will be unwilling to step forward, or even to fill out an unsigned form, out of fear that it will somehow be traced to them by their employer or coworkers.
We will use this situation to demonstrate the use of the random response methodology. In this case, the goal is to estimate the proportion of drug users among the employees. It is important to note at the outset, however, that it cannot be used to determine which, if any, of the employees is a drug user. This is, in fact, the source of its strength.
Random response analysis was introduced by Warner2 (1965) and first presented in the form discussed here, known as the Simmons Model, by Horvitz, Shah and Simmons3 (1967). A mathematical discussion of the Simmons Model appears in Greenberg, Abdul-Ela, Abdel-Latif, Simmons and Horvitz4. A large scale application of the procedure to examine drug usage among high school students appears in Goodstadt and Gruson5.
Warner's model asks each individual to respond YES or NO to either the question "Is it true that you have made use of an illegal substance within the past thirty days?" or the alternative question "Is it true that you have not made use of an illegal substance within the past thirty days?". (The specific formulation of the question of interest may, of course, be modified to suit the situation. It is sufficient that it is dichotomous, i.e. that it must be answered in one of two known ways which we shall consider to be YES or NO.) The choice of which question the employee is to answer is made by the employee using a chance mechanism so that the employer knows only the probability that the employee is answering each question. The Simmons model is distinguished by its use of a question unrelated to the one of interest, i.e. the employee answers either the question of interest or a question which is innocuous in the sense that public knowledge of an employee's response would be of no concern to that employee.
For the sake of simplicity, the methodology will be described in terms of a completely specified procedure involving the flipping of coins. There are, of course, many possible variations of this process, but the principle remains the same regardless of the specific formulation utilized.
The Procedure
Possibly the totality, but at least a representative sample, of the employees for whom an estimate is to be made are assembled. Each is provided with two coins, say a penny and a nickel. They are asked to flip the coins and remember how each comes up (heads or tails) without letting anyone else see or know their results. Note that it is important that the coins actually be flipped in the air as opposed to being spun on a flat surface or having the employees decide in their minds the outcomes of imaginary flips.6 The participants may, of course, be asked to flip the coins in advance of the meeting or during an intermission in order to assure that the results they obtain are known only to them.
The employees are then asked to consider the outcome of the flip of the penny and, depending on what it was, answer either Question A or Question B. These questions may be presented to them either orally or in writing. If the flip of the penny resulted in heads, the employee is to answer Question A. If it was tails, the employee is to answer Question B. Thus, each question may be answered either YES or NO and in no other way.
- The questions to be answered are:
- Question A: Did the flip of the nickel result in heads?
- Question B: Have you made use of any illegal substance at any time during the past thirty days?7
We are now in a position to count the total number of employees answering YES even though we do not know which question is being answered for any individual. To see how the procedure works, suppose that 1000 employees are assembled. It is expected that 50% of the penny flips will result in heads and 50% will result in tails. Thus, we expect 50%, or 500, of the employees to answer Question B. Further, we expect 50% of those who answer Question B to again obtain heads when they flip their nickels. We, therefore, expect 250 employees to answer YES because they flipped heads with both the penny and the nickel.
Suppose now that a total of 350 employees out of the 1000 said YES. (Remember that no one other than the individual respondent know which question is being answered by any individual.) Since we expect 250 YES responses from the 500 employees who are expected to be answering Question A, we estimate that the remaining 100 YES responses came from the estimated 500 employees who answered Question B. Thus, we may estimate the proportion of employees who have used an illegal substance in the past thirty days to be 100/500 or 20%, even though we are unable to point to any one individual who is known to have used an illegal substance.8
Symbolically, if we denote by p the actual but unknown proportion of employees who are drug users, and denote by f the proportion saying YES in the sample, then we estimate p as (2 x f) - .5. In the example f=.35 and (2 x f) - .5 = (2 x .35) - .5 = .2. Thus, p is estimated to be 20%.
Conclusion
Random Response methodology will, under two major assumptions, provide a useful estimate of the proportion of employees who are drug users. One is that the employees questioned are either the totality of employees for whom an estimate is desired or a representative sample of that group. If sampling is required for the sake of convenience or in order to meet cost constraints then it will be necessary to make use of, and correctly implement, a properly designed sampling procedure. To the extent that sampling is necessary, it is possible to determine the minimum sample size necessary to assure, at least approximately, any specific required accuracy and confidence in the result.
The second assumption is that the procedure is sufficiently well understood by the employees that they cooperate. In the example presented, the employer must make it clear that the purpose of the study is to measure the prevalence of drug usage in order to determine the need for a drug treatment program and that individuals can not be identified as drug users.
References
- Dr. Mann is President, Charles R. Mann Associates, Inc. a Washington, D.C. based statistical consulting and data processing firm which frequently works with Beta Research.
- Warner, S.L.(1965), "Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias," Journal of the American Statistical Association, 60, 63-69.
- Horvitz, D.G., Shah, B.V., and Simmons, W.R. (1967), "The Unrelated Randomized Response Model," in Proceedings of the Social Statistics Section, American Statistical Association, pp. 65-72.
- Greenberg, B.G., Abdul-Ela, Abdel-Latif, A., Simmons, W.R., and Horvitz, D.G.: "The Unrelated Question Randomized Response Model, Theoretical Framework," Journal of the American Statistical Association, 1969, 64, 520-539.
- Goodstadt, M.S. and Gruson, V. (1975), "The Randomized Response Technique: A Test on Drug Use," Journal of the American Statistical Association, 70, 814-818.
- The reason for this is that we make use of the fact that a flipped coin is expected to land face up half the time. This cannot be assumed to be the case for either a spun coin or an imaginary decision.
- The wording of Question B may reflect any particular definition or concept of drug usage. The only restriction is that the subject must be able to answer it either yes or no. The procedure will be used to estimate the proportion of employees for whom the answer is yes.
- It should be noted that this explanation bypasses certain subtleties which could affect the results. Nevertheless, it provides sufficient precision for the purpose of approximating the extent of drug usage.
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