Estimating a Population Proportion Using Randomized Responses

Abstract

A standard problem in the statistician's repertoire is that of estimating a population proportionp. The usual approach to this problem is to let X represent the number of individuals in a sample of size n who possess the characteristic which defines p, assume that the individual sample responses comprise a binomial experiment (n independent homogeneous trials, each with the same dichotomous responses) so that X is a binomial random variable, and use the estimator XI= n. When the underlying assumptions are satisfied, the estimator p^ is both the minimum variance unbiased estimator and the maximum likelihood estimator of p. The reasonableness of this estimator, as well as its optimality properties, depend crucially on the presumption that individual responses are truthful. If, for example, we are interested in estimating the true proportion of all presently enrolled college students who have seen the movie Star Wars, then there is reason to suppose that a randomly chosen student will respond untruthfully when asked, Have you seen Star Wars? There are, however, many situations in which one or perhaps both responses may embarrass or stigmatize the respondent. Examples include questions regarding drug use, sexual experience, honor code violations, and abortions. Even if a randomly selected student has violated the honor code at his or her college, it is not clear that the response to Have you ever violated the college honor code? will be yes. In such a situation, the estimatorp^ will probably have a bias of unknown, and thus uncorrectable, magnitude (and even direction, if both a or no response can stigmatize.) In recent years a method called the randomized response technique has been introduced to circumvent the problem of untruthful responses. To illustrate this technique, suppose that an investigator has identified a random sample of size n from the population of interest. The investigator has in hand a deck of 100 cards, of which 50 are type I cards and the other 50 are type II cards. The type I cards ask for a (truthful) response to the question of interest (e.g., Have you violated the honor code?), while the other 50 ask for a response to an unrelated or question such as Is the fifth digit of your home telephone number a 0, 1, or 2? A respondent is asked to examine the deck to confirm the stated mix of type I and type II cards, then to shuffle the deck until it is well-mixed, select a card and respond truthfully to the question on the card, and lastly reinsert the card into the deck without having revealed to the investigator which type of card was selected. What prompts a truthful response in this scheme? The key point is that the questioner does not know whether the respondent has answered the question of interest or the irrelevant question, so that the respondent will not be stigmatized by a truthful response. The model presumes that the proportion of type II cards in the deck is high enough to offer enough protection against stigma so that truthful responses are forthcoming. In practice the tendency for an untruthful response may not be completely eliminated by the technique, but should certainly be drastically reduced. Goodstadt and Gruson ([5]) report on the successful use of the technique in a survey on drug use. Presuming that responses using the randomized technique are truthful, an estimator for p can be derived by defining Y as the number of yes responses in the sample using the randomized scheme, and A as the probability of a yes response using the randomized scheme. Then a simple probability argument yields