1. Introduction

Warner [8] proposed the randomized response technique as a survey technique to reduce potential bias due to non-response and social desirability when asking questions about sensitive behaviors (see Warner [8], for a comprehensive review). The method asks respondents to use a randomization device, such as a coin, deck of cards, spinners whose outcome is not known by the enumerator. The outcome of the randomized device determines which of the two questions the respondent answers. A lot of improvements have been done to Warner’s randomized response model, to mention few, Greenberg et al. [4], Gupta and Shabbir [5], Adebola and Adegoke [1], Adepetun and Adebola [3], Adebola et al. [2].

In this paper, we develop a Modification of Warner’s Randomized Response Techniques by introducing the concept of sub-sample of non-respondent. Randomized Response Techniques helps to reduce response and non-response bias while our model further reduces the non-response bias. Item non-response occurs when the respondent refuses to answer the sensitive part of the question which is the major concern of the interviewer. In sections that follow, we present the Warner’s [8] Randomized Response Model, propose Randomized Response Model and thereafter its relative efficiency over the existing one.

2. Warner’s Randomized Response Model

Warner [7] gave a genius idea by using randomized device to encourage truthful answer from the respondent with respect to a sensitive behavior. The randomizing device, such as a spinning arrow, dice or coins is used to select one of the two questions; such as,

“I am HIV positive” (class A, presented with probability P)

“I am HIV negative” (class B, presented with probability 1-P)

The respondents have the options “Yes” or “No” presented to him or her. The interviewer does not know which question any respondent has answered but knows the probability P and 1-P with which the two statements are presented. Here, with a random sample of n respondents, the interviewer records a binomial estimate of the proportion θ of “Yes” answers, where y is the number of yes answers.

If the questions are answered truthfully, the relation between and in the population is given as:

(2.1)

Where is the proportion of people with the stigmatized or sensitive behavior using Warner’s techniques and is the probability of selecting the sensitive question.

(2.2)

Proof

From equation (2.1), we have

Making the subject of the relation, we have

The unbiased estimator of a sensitive proportion is given by:

(2.3)

The Variance is given by

Recall that where c is a constant.

Then,

We have,

Thus,

(2.4)

Where follows a binomial distribution,

Now to find the unbiased estimator of the variance

The Variance is given by

The unbiased variance estimator of a sensitive proportion is given by:

(2.5)

The second term in is the variance that would have if all n respondents answered truthfully a direct question about class A membership.

Except by chosen near 0.5 and p > 0.85, the first term is greater than the second, often much greater. The method is thus quite imprecise in general. This might be expected since the interviewer does not know whether a “yes” answer implies membership in a class A or the opposite.

However, Warner’s method may give a smaller mean square error (MSE) than a direct sensitive question would, if the latter produced numerous refusals or false answers.

3. Our Model

Several randomized response techniques has been developed, the models developed do not take into consideration of item non-response (refusal to answer the sensitive part of the question). Non-response is an important source of non-sampling error in survey sampling, it occurs when some but not all the required information is collected from the sample unit. The most damaging is unit non-response where a sampling unit refuses to answer the sensitive part of the randomized response techniques designed questionnaire.

In view of this, we proposed an improved Warner’s randomized response model that is based on sub-sample of non-respondent so as to induce a better estimate of the proportion of people with the stigmatized or sensitive behavior. Questionnaires were sent out and the number of useable responses were recorded (useable responses at the first interview given as) while the remaining were referred to non-response given as In order to further reduce the non-response bias then a survey on sub-sample of non-respondent come to place. It is assumed that the whole of the sub-sample responded to the survey and are useable responses which would be achieved by close supervision. Close supervision in this context doesn’t mean the interviewer knows the question answered by the respondent but it means the respondent is properly instructed and monitored on a one to one basis. Let be the number “yes” response from the respondent at the first interview. Let be the number of “yes” answer from the sub-sample of non-respondent. Let be the sample size of the Survey. Let be the sub-sample size. Let be the unit which is used to take the sub-sample. The proportion of “yes” response from our model is given by:

(3.1)

By simplifying, we have (see theorem 1 for proof):

(3.2)

Where be the proportion of “yes” answer, be the proportion of “yes” answer form the respondent at the first interview and be the proportion of “yes” answer from the sub-sample of non-respondent.

From the Warner’s randomized response techniques, the proportion of “yes” response from our model is given by:

(3.3)

Let then we have:

Solving for we have,

Then we have,

Recall that

Hence, the unbiased estimator of is given by:

(3.4)

The variance of the estimator is given by:

(3.5)

By simplifying, we have

By further simplification,

The unbiased variance estimator of a sensitive proportion is given by:

(3.6)

Then,

(3.7)

Where is the variance for a Warner’s model.

Theorem 1: The proportion of “yes’ response, is given by:

Proof: From the Hansen and Hurwitz [6], which introduces the concept of subsample of non-respondent, we have

Where

Then, we have

(3.8)

Recall that

By substituting for m in equation (3.8), we have

The unbiased estimator of is given by

Where

Theorem 2: Given that is an integer value and k >1

Show that

Proof:

Given that, Thus when

That is,

By re-arranging,

(1)

Divide through by

Multiply through by

We have,

Then,

(2)

From equation (1), we have

Divide through by we have

Subtract 1 from both sides, we have

Divide through by we have

Then,

(3)

From equation (2) and (3), using transitivity law

We have,

4. Comparative Study of our Model

Here we performed the comparative study of our model; this can be achieved mathematically and empirically. Mathematically, it follows that the proposed model is more efficient than the Warner’s randomized response model if we have;

By simplifying, we have

Since theorem 2 holds, then the variance of our proposed model is less than the variance of Warner’s RRT.

Empirically, to also validate our conclusion on the proposed model we present the tables below.

Table 1. Table showing the relative efficiency when n= 250,      = 0.1, p=0.7

Table 2. Table showing the relative efficiency when n= 500,       = 0.1, p=0.7

We can deduce from the empirical comparison that the choice of k plays a major role in the comparative study. It can be derived from the table that k = 2 gave the minimum variance in the proposed model; conclusively, the smaller the choice of k, the more efficient the proposed model is over the conventional Warner’s model.

5. Conclusions

This paper presented an improved Warner’s randomized response model; the proposed strategy further reduces the non-response bias by introducing the concept of sub-samples of non-respondent. The proposed model is likely to induce better estimate with a reduced variance. Moreover, the proposed model is more efficient than the Warner’s model. Lastly, we are able to conclude that the smaller the choice of k (the unit which is used to divide the non-respondent so as to have the sub-sample size), the higher the gain in efficiency of the proposed model over the conventional Warner’s model.