Bayesian Analysis of Warner ’ s Randomized Response Technique

This paper proposed alternative beta estimators of the population proportion of a sensitive attribute when life data were obtained through the administration of survey questionnaires on abortion of some matured women. The results showed that the proposed alternative beta estimators were more efficient in capturing responses from respondents than the simple beta estimator proposed by Winkler and Franklin for relatively small, medium as well as large sample sizes respectively.


Introduction
Asking individuals about sexually-related activity, criminal activity, induced abortion, etc., is an uneasy exercise.Individuals are often reluctant to freely divulge such information in classical survey settings for fear of social disgrace.Yet, for public-health and socio-economic reasons, estimates of the incidence of such activities are essential.The Randomized Response Technique (RRT) can be a useful survey method to find such estimates because individual anonymity is preserved.Initial RRT proposed by Warner [1] and Greenberg et al. [2] presumed that two yes/no questions were provided for each respondent and that a randomization device was used to determine which question would be answered.Since the interviewer would not know the result of the device, participants would be encouraged to give truthful responses to a sensitive question.This technique has generated much interest in the statistical literature since the publication of Warner's RRT.A comprehensive literature review will not be provided here.Instead, the interested reader is referred to the review-oriented references listed herein; namely, Mangat et al. [3,4], Christofides [5], Kim and Warde [6], Adebola and Adepetun [7][8][9].In some situations, prior information about the unknown parameter may be available and can be combined with the sample information for the estimation of that unknown parameter.This is known as the Bayesian approach of estimation.Work done by researchers on Bayesian Analysis of RRTs are not very extensive, however, attempts have been made on the Bayesian Analysis of RRTs.Winkler and Franklin [10], Pitz [11], O'Hagan [12], Oh [13], Migon and Tachibana [14], Unnikrishnan and Kunte [15], Kim et al. [16], Hussain and Shabbir [17][18][19], Adepetun and Adewara [20], Adepetun and Adewara [21,22], are the major references on the Bayesian analysis of the RRTs.This paper presents Bayesian analysis to Warner's RRT using alternative beta priors other than the simple beta prior used by Winkler and Franklin [10] in their published paper.

The Conventional Bayesian Technique of Estimation
Winkler and Franklin [10] in their published paper presented a Bayesian estimation to the RRT put forward by Warner [1] assuming a simple beta prior to estimate the population proportion of respondents possessing sensitive attribute.
Suppose the simple beta prior is defined as follows where (a,b) are the shape parameters of the distribution and π is the population proportion of respondents possessing the sensitive attribute.Let be the total number of the women who have committed abortion for a particular sample of size n selected from the population with simple random sampling.Then the conditional distribution of X given π is Where φ is the probability of "yes response" to the sensitive attribute which was defined as Where P is the preset probability of "yes" response to the sensitive attribute On simplification, we have Recall from binomial series expansion, for x = 0,1,2,…,n Thus, the joint probability density functions (pdf) of X and π was Now the marginal distribution of X can be obtained by integrating the joint distribution of X and π with respect to π.Thus the marginal distribution of X was given by   =  , The posterior distribution of given X was defined as The Bayes estimator of π which is the posterior mean of (8) was given by

The Proposed Bayesian Techniques of Estimation
In this section, we propose an alternative Bayesian estimation to Warner [1] RRT assuming both the Kumaraswamy and the Generalised beta priors in addition to the simple beta prior proposed by Winkler and Franklin [10].

Estimation of π using Kumaraswamy prior
The Kumaraswamy prior distribution of π is given as Using the Kumaraswamy prior in ( 12), the joint probability density function of X and π is derived as where The marginal probability density function (pdf) of X can be obtained as Similarly, the posterior distribution as usual is obtained as follows Under the Square error loss, we proceed to obtain the posterior mean which is the Bayes estimator as follows Therefore, 1 +  + ,  + 1 (18) As a result, the Bias of as well as its Mean Square Error is also given by

Estimation of π using the generalised beta prior
The Generalised Beta prior is defined as where a, b. c are the shape parameters of the prior distribution as given in equation (21).By binomial series expansion, we know that As a result, the joint density function of π and X with the generalized beta prior is The marginal probability density function (pdf) of X can then be obtained from (22) as where Similarly, we obtained the posterior distribution of π given X as In the same way, under the square error loss, the posterior mean which is otherwise known as the Bayes estimator is given by

Results and Discussions
We wrote suitable codes using R-statistical software to evaluate the derived estimators, bias and mean square errors which are given by equations 18-20 and 26-28 at sample sizes 25, 100, and 250 respectively.
From the results presented in Tables and Figs.
1a-6b respectively, when n = 25, P = 0.1 and 0.2, the conventional simple beta estimator is better than the proposed estimators when π lies within the range 0.1 ≤ π < 0.7 while the proposed estimators are better than the conventional simple beta estimator when π lies within the range 0.7 ≤ < 1.However, the proposed Kumaraswamy beta estimator is the best in obtaining more responses from respondents when π lies within the range 0.7 ≤ < 1.
When n = 100, P = 0.1 and 0.2, the conventional simple beta estimator is better than the proposed estimators when π lies within the range 0.1 ≤ π ≤ 0.5 while the proposed estimators are better than the conventional simple beta estimator when π lies within the range 0.6 ≤ π < 1.However, the proposed Kumaraswamy beta estimator is the best in obtaining more responses from respondents when π lies within the range 0.6 ≤ π < 1.
When n = 250, P = 0.1 and 0.2, the conventional simple beta estimator is better than the proposed estimators when π lies within the range 0.1 ≤ π < 0.5 while the proposed estimators are better than the conventional simple beta estimator when π lies within the range 0.6 ≤ π < 1.However, the proposed Kumaraswamy beta estimator is the best in obtaining more responses from respondents when π lies within the range 0.6 ≤ π < 1 respectively.Comment: When n = 25, P= 0.1 and 0.2, the conventional simple beta estimator is better than the proposed estimators when π lies within the range 0.1 ≤ π < 0.7 while the proposed estimators are better than the conventional simple beta estimator when π lies within the range 0.7 ≤ π < 1.However, the proposed Kumaraswamy beta estimator is the best in obtaining more responses from respondents when π lies within the range 0.7 ≤ π < 1 respectively.Comment: When n = 100, P = 0.1 and 0.2, the conventional simple beta estimator is better than the proposed estimators when π lies within the range 0.1 ≤ π ≤ 0.5 while the proposed estimators are better than the conventional simple beta estimator when π lies within the range 0.6 ≤ π < 1.However, the proposed Kumaraswamy beta estimator is the best in obtaining more responses from respondents when π lies within the range 0.6 ≤ π < 1 respectively.Comment: When n = 250, P = 0.1 and 0.2, the conventional simple beta estimator is better than the proposed estimators when π lies within the range 0.1 ≤ π < 0.5 while the proposed estimators are better than the conventional simple beta estimator when π lies within the range 0.6 ≤ π < 1.However, the proposed Kumaraswamy beta estimator is the best in obtaining more responses from respondents when π lies within the range 0.6 ≤ π < 1 respectively.

Conclusion
We have proposed the alternative Bayesian estimators of the population proportion when life data were gathered through the administration of questionnaires on abortion on 300 matured women in addition to the conventional simple beta estimator proposed by Winkler and Franklin [10].We observed clearly from the results that for relatively small, intermediate as well as large sample sizes, the proposed Bayesian estimators are more sensitive in capturing sensitive attribute than the conventional simple beta estimator.In particular, the proposed Kumaraswamy beta estimator is the best in obtaining information from respondents in survey which asks sensitive questions.