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On an Improved Bayesian Item Count Technique Using Different Priors
Técnica de conteo de items bayesiana mejorada usando diferentes distribuciones a priori
Keywords:
Bayesian Estimation, Indirect Questioning, Item Count Technique, Population Proportion, Prior Information, Privacy Protection, Randomized Response Technique, Sensitive Attributes (en)atributos sensitivos, estimación Bayesiana, información a priori, preguntas indirectas, proporción poblacional, protección de la privacidad, técnica de conteo de ítems, técnica de respuesta aleatorizada (es)
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Item Count Technique (ICT) serves the purpose of estimating the proportion of the people with stigmatizing attributes using the indirect questioning method. An improved ICT has been recently proposed in the literature (not requiring two subsamples and hence free from finding optimum subsample sizes unlike the usual ICT) in a classical framework that performs better than the usual ICT and the Warner method of Randomized Response (RR) technique. This study extends the scope of this recently proposed ICT in a Bayesian framework using different priors in order to derive posterior distributions, posterior means and posterior variances. The posterior means and variances are compared in order to study which prior is more helpful in updating the item count technique. Moreover, we have compared the Proposed Bayesian estimation with Maximum Likelihood (ML) estimation. We have observed that simple and elicited Beta priors are superior choices (in terms of minimum variance), depending on the sample size, number of items and the sum of responses. Also, the Bayesian estimation provides relatively more precise estimators than the ML Estimation.
La técnica de conteo de ítems (ICT, por sus siglas en inglés) es útil para estimar la proporción de personas que poseen atributos que pueden tener algún grado de estigmatización mediante el uso de un método de preguntas indirectas. Una ICT mejorada ha sido propuesta recientemente en la literatura bajo la inferencia clásica (la cual no requiere dos submuestras y libre de la necesidad de encontrartamaños de muestra óptimos para cada una de ellas como sucede en la ICT usual). Esta ICT mejorada se desempeña mejor que la ICT usual y que el método de Respuesta Aleatorizada (RR, por sus siglas en inglés) de Warner. Este artículo extiende su estudio bajo una visión Bayesiana usando diferentes a priori con el fin de derivar distribuciones, medias y varianzas a posteriori.Las medias y varianzas a posteriori son comparadas con el fin de estudiar cuál a priori es más útil en mejorar la técnica de conteo de ítems. Se observa que a priori simples y Beta elicitadas son las mejores escogencias (en términos dela varianza mínima) dependiendo del tamaño de muestra, el número de ítems y la suma de la respuesta. También, la estimación bayesiana proporciona estimadores relativamente más precisas que la estimación ML.
1Quaid-i-Azam University, Faculty of Natural Sciences, Department of Statistics, Islamabad, Pakistan. King Abdulaziz University, Faculty of Sciences, Department of Statistics, Jeddah, Saudi Arabia. Professor. Email: zhlangah@yahoo.com
2University of Hazara, Faculty of Sciences, Department of Statistics, Mansehra, Pakistan. Professor. Email: alishahejaz@yahoo.com
3Quaid-i-Azam University, Faculty of Natural Sciences, Department of Statistics, Islamabad, Pakistan. Professor. Email: javidshabbir@gmail.com
4Quaid-i-Azam University, Faculty of Natural Sciences, Department of Statistics, Islamabad, Pakistan. King Fahad University of Petroleum and Minerals, Faculty of Sciences, Department of Mathematics and Statistics, Dhahran, Saudi Arabia. Professor. Email: riaz76qau@yahoo.com
Item Count Technique (ICT) serves the purpose of estimating the proportion of the people with stigmatizing attributes using the indirect questioning method. An improved ICT has been recently proposed in the literature (not requiring two subsamples and hence free from finding optimum subsample sizes unlike the usual ICT) in a classical framework that performs better than the usual ICT and the Warner method of Randomized Response (RR) technique. This study extends the scope of this recently proposed ICT in a Bayesian framework using different priors in order to derive posterior distributions, posterior means and posterior variances. The posterior means and variances are compared in order to study which prior is more helpful in updating the item count technique. Moreover, we have compared the Proposed Bayesian estimation with Maximum Likelihood (ML) estimation. We have observed that simple and elicited Beta priors are superior choices (in terms of minimum variance), depending on the sample size, number of items and the sum of responses. Also, the Bayesian estimation provides relatively more precise estimators than the ML Estimation.
Key words: Bayesian Estimation, Indirect Questioning, Item Count Technique, Population Proportion, Prior Information, Privacy Protection, Randomized Response Technique, Sensitive Attributes.
La técnica de conteo de ítems (ICT, por sus siglas en inglés) es útil para estimar la proporción de personas que poseen atributos que pueden tener algún grado de estigmatización mediante el uso de un método de preguntas indirectas. Una ICT mejorada ha sido propuesta recientemente en la literatura bajo la inferencia clásica (la cual no requiere dos submuestras y libre de la necesidad de encontrartamaños de muestra óptimos para cada una de ellas como sucede en la ICT usual). Esta ICT mejorada se desempeña mejor que la ICT usual y que el método de Respuesta Aleatorizada (RR, por sus siglas en inglés) de Warner. Este artículo extiende su estudio bajo una visión Bayesiana usando diferentes a priori con el fin de derivar distribuciones, medias y varianzas a posteriori.Las medias y varianzas a posteriori son comparadas con el fin de estudiar cuál a priori es más útil en mejorar la técnica de conteo de ítems. Se observa que a priori simples y Beta elicitadas son las mejores escogencias (en términos dela varianza mínima) dependiendo del tamaño de muestra, el número de ítems y la suma de la respuesta. También, la estimación bayesiana proporciona estimadores relativamente más precisas que la estimación ML.
Palabras clave: atributos sensitivos, estimación Bayesiana, información a priori, preguntas indirectas, proporción poblacional, protección de la privacidad, técnica de conteo de ítems, técnica de respuesta aleatorizada.
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References
1. Arnab, R. & Dorffner, G. (2006), 'Randomized response technique for complex survey design', Statistical Papers(48), 131-141.
2. Aslam, M. (2003), 'An application of prior predictive distribution to elicit the prior density', Journal of Statistical Theory and Application(2), 70-83.
3. Bar-Lev, S. K., Bobovitch, E. & Boukai, B. (2004), 'A note on randomized response models for quantitative data', Metrika(60), 255-260.
4. Barabesi, L. & Marcheselli, M. (2010), 'Bayesian estimation of proportion and sensitivity level in randomized response procedures', Metrika(72), 75-88.
5. Chaudhuri, A. (2011), Randomized Response and Indirect Questioning Techniques in Surveys, Chapman & Hall, Florida, United States.
6. Chaudhuri, A. & Christofides, T. C. (2007), 'Item count technique in estimating proportion of people with sensitive feature', Journal of Statistical Planning and Inference(137), 589-593.
7. Dalton, D. R. & Metzger, M. B. (1992), 'Integrity testing for personal selection: an unsparing perspective', Journal of Business Ethics(12), 147-156.
8. Droitcour, J. A., Casper, R. A., Hubbard, M. L., Parsley, T., Visscher, W. & Ezzati, T. M. (1991), The item count technique as a method of indirect questioning: a review of its development and a case study application, 'Measurement Errors in Surveys', Willey, New york.
9. Droitcour, J. A., Larson, E. M. & Scheuren, F. J. (2001), The three card method: estimating sensitive survey items with permanent anonymity of response, 'Proceedings of the Social Statistics Section', American Statistical Association, , , Alexandria, Virginia.
10. Guerts, M. D. (1980), 'Using a randomized response design to eliminate non-response and response biases in business research', Journal of the Academy of Marketing Science(8), 83-91.
11. Gupta, S., Gupta, B. & Singh, S. (2002), 'Estimation of sensitivity level of personal interview survey questions', Journal of Statistical Planning and Inference 100, 239-247.
12. Huang, K. C. (2010), 'Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling', Metrika(71), 341-352.
13. Hubbard, M. L., Casper, R. A. & Lesser, J. T. (1989), Respondent's reactions to item count list and randomized response, 'Proceeding of the Survey Research Section of the American Statistical Association', Washington, D. C., p. 544-448.
14. Hussain, Z. & Shabbir, J. (2010), 'Three stage quantitative randomized response model', Journal of Probability and Statistical Sciences(8), 223-235.
15. Hussain, Z., Shah, E. A. & Shabbir, J. (2012), 'An alternative item count technique in sensitive surveys', Revista Colombiana de Estadistica(35), 39-54.
16. Larkins, E. R., Hume, E. C. & Garcha, B. (1997), 'The validity of randomized response method in tax ethics research', Journal of the Applied Business Research 13(3), 25-32.
17. Liu, P. T. & P., c. L. (1976), 'A new discrete quantitative randomized response model', Journal of the American Statistical Association(71), 72-73.
18. Reinmuth, J. E. & Guerts, M. D. (1975), 'The collection of sensitive information using a two stage randomized response model', Journal of Marketing Research(12), 402-407.
19. Ryu, J. B., Kim, J. M., Heo, T. Y. & Park, C. G. (2005-2006), 'On stratified randomized response sampling', Model Assisted Statistics and Applications(1), 31-36.
20. Tracy, D. & Mangat, N. (1996), 'Some development in randomized response sampling during the last decade-A follow up of review by Chaudhuri and Mukerjee', Journal of Applied Statistical Science(4), 533-544.
21. Warner, S. L. (1965), 'Randomized response: A survey for eliminating evasive answer bias', Journal of the American Statistical Association(60), 63-69.
22. Zellner, A. (1996), An Introduction to Bayesian Inference in Econometrics, Chichester, John Wiley, New York.
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv36n2a08,
AUTHOR = {Hussain, Zawar and Ali Shah, Ejaz and Shabbir, Javid and Riaz, Muhammad},
TITLE = {{On an Improved Bayesian Item Count Technique Using Different Priors}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2013},
volume = {36},
number = {2},
pages = {303-317}
}
References
Arnab, R. & Dorffner, G. (2006), ‘Randomized response technique for complex survey design’, Statistical Papers (48), 131–141.
Aslam, M. (2003), ‘An application of prior predictive distribution to elicit the prior density’, Journal of Statistical Theory and Application (2), 70–83.
Bar-Lev, S. K., Bobovitch, E. & Boukai, B. (2004), ‘A note on randomized response models for quantitative data’, Metrika (60), 255–260.
Barabesi, L. & Marcheselli, M. (2010), ‘Bayesian estimation of proportion and sensitivity level in randomized response procedures’, Metrika (72), 75–88.
Chaudhri, A. & Mukerjee, R. (1998), Randomized Response: Theory and Methods, Marcel-Decker, New York.
Chaudhuri, A. (2011), Randomized Response and Indirect Questioning Techniques in Surveys, Chapman & Hall, Florida, United States.
Chaudhuri, A. & Christofides, T. C. (2007), ‘Item count technique in estimating proportion of people with sensitive feature’, Journal of Statistical Planning and Inference (137), 589–593.
Dalton, D. R. & Metzger, M. B. (1992), ‘Integrity testing for personal selection: An unsparing perspective’, Journal of Business Ethics (12), 147–156.
Droitcour, J. A., Casper, R. A., Hubbard, M. L., Parsley, T., Visscher, W. & Ezzati, T. M. (1991), The item count technique as a method of indirect questioning: a review of its development and a case study application, in P. P. Biemer, R. M. Groves, L. Lyberg, N. Mathiowetz & S. Sudeman, eds, ‘Measurement Errors in Surveys’, Willey, New york.
Droitcour, J. A., Larson, E. M. & Scheuren, F. J. (2001), The three card method: estimating sensitive survey items with permanent anonymity of response, in ‘Proceedings of the Social Statistics Section’, American Statistical Association, Alexandria, Virginia.
Guerts, M. D. (1980), ‘Using a randomized response design to eliminate nonresponse and response biases in business research’, Journal of the Academy of Marketing Science (8), 83–91.
Gupta, S., Gupta, B. & Singh, S. (2002), ‘Estimation of sensitivity level of personal interview survey questions’, Journal of Statistical Planning and Inference 100, 239–247.
Huang, K. C. (2010), ‘Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling’, Metrika (71), 341–352.
Hubbard, M. L., Casper, R. A. & Lesser, J. T. (1989), Respondent’s reactions to item count list and randomized response, in ‘Proceeding of the Survey Research Section of the American Statistical Association’, Washington, D. C., pp. 544–448.
Hussain, Z. & Shabbir, J. (2010), ‘Three stage quantitative randomized response model’, Journal of Probability and Statistical Sciences (8), 223–235.
Hussain, Z., Shah, E. A. & Shabbir, J. (2012), ‘An alternative item count technique in sensitive surveys’, Revista Colombiana de Estadistica (35), 39–54.
Larkins, E. R., Hume, E. C. & Garcha, B. (1997), ‘The validity of randomized response method in tax ethics research’, Journal of the Applied Business Research 13(3), 25–32.
Liu, P. T. & chow. L. P. (1976), ‘A new discrete quantitative randomized response model’, Journal of the American Statistical Association (71), 72–73.
Reinmuth, J. E. & Guerts, M. D. (1975), ‘The collection of sensitive information using a two stage randomized response model’, Journal of Marketing Research (12), 402–407.
Ryu, J. B., Kim, J. M., Heo, T. Y. & Park, C. G. (2005-2006), ‘On stratified randomized response sampling’, Model Assisted Statistics and Applications (1), 31–36.
Tracy, D. & Mangat, N. (1996), ‘Some development in randomized response sampling during the last decade-A follow up of review by Chaudhuri and Mukerjee’, Journal of Applied Statistical Science (4), 533–544.
Warner, S. L. (1965), ‘Randomized response: A survey for eliminating evasive answer bias’, Journal of the American Statistical Association (60), 63–69.
Zellner, A. (1996), An Introduction to Bayesian Inference in Econometrics, Chichester, John Wiley, New York.
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