Bayesian estimation of the parameters of the randomized ordinal probit regression model
Keywords:
ordinal probit regression, randomized ordinal response variable, bayesian inference, double exponential, distributionAbstract
A large number of papers estimate the prevalence of a randomized binary variable, but few model the effect of non-sensitive covariates on a randomized binary response variable. A small number of papers study a sensitive ordinal variable, and there are no papers that model the effect of non-sensitive covariates on a randomized ordinal response variable. The objective of this work is to use the Bayesian approach to measure the effect of non-sensitive covariates on a randomized ordinal response variable obtained under the forced response design and evaluate the performance of the proposed estimators. Four prior distributions were used: Double exponential, Normal, t and Cauchy. A simulation was performed to compare the Bayesian estimators studied considering two numbers of categories of the randomized ordinal response variable, two sample sizes, and two numbers of non-sensitive covariates. The comparison criteria were the mean squared error, the length and the coverage of the credible intervals. The proposed Bayesian estimators adequately estimated the true parameters of the randomized ordinal probit regression model, even when the forced response design induced by the Hopkins randomization device was used to produce a randomized ordinal response variable. The Bayesian estimator using the double exponential prior distribution was the best in terms of the criteria used.
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References
Abul-Ela, A.L.A., Greenberg, G.G., Horvitz, D.G. (1967). A Multiproportions Randomized Response Model. Journal of the American Statistical Association, 62, 990-1008. https://doi.org/10.2307/2283687
Ardah, H.I. Oral, E. (2017). Model Selection in Randomized Response Techniques for Binary Responses. Communications in Statistics - Theory and Methods, 47, 3305-3323. https://doi.org/10.1080/03610926.2017.135362 6
Bar-Lev, S.K., Bobovitch, E., Boukai, B. (2004). A Note on Randomized Response Models for Quantitative Data. Metrika, 60, 225-250. https://doi.org/10.1007/s001840300308
Barabesi, L., Franceschi, S., Marcheselli, M. (2012). A Randomized Response Procedure for Multiple Sensitive Questions. Statistical Papers, 53, 703-718.
Blair, G., Imai, K., Zhou, Y.-Y. (2015). Design and Analysis of the Randomized Response Technique. Journal of the American Statistical Association, 110, 1304-1319. https://doi.org/10.1080/01621459.2015.105002 8
Boruch, R.F. (1971). Assuring Confidentiality of Responses in Social Research: a Note on Strategies. The American Sociologist, 6, 308-
311.
Cruyff, M.J.L.F., van den Hout, A., van der Heijden, P.G.M. (2008). The Analysis of Randomized Response Sum Score Variables. Journal of the Royal Statistical Society, Series B, 71, 21-30.
http://dx.doi.org/10.1111/j.1467-9868.2007.00624.x
Drane, W. (1976). N the Theory of Randomized Responses to Two Sensitive Questions. Communications in Statistics - Theory and Methods, 5, 565-574. https://doi.org/10.1080/03610927608827375
Eichhorn, B.H., Hayre, L.S. (1983). Scrambled Randomized Response Methods for Obtaining Sensitive Quantitative Data. Journal of Statistical planning and Inference, 7, 307–316.
Eriksson, S.A. (1973). A New Model for Randomized Response. International Statistical Review, 41, 101-113.
https://doi.org/10.2307/1402791
Ewemooje, O.S., Amahia, G.N. (2015). Improved Randomized Response Technique for Two Sensitive Attributes. Afrika Statistika, 10, 839-
853.
https://doi.org/10.16929/as/2015.639.78 Greenberg, B., Abul-Ela, A., Simmons, W.,
Horvitz, D.G. (1969). The Unrelated Question Randomized Response Model: Theoretical Framework. Journal of the American Statistical Association, 64, 520-539. https://doi.org/10.2307/2283636
Kim, J.-M., Warde, W.D. (2005). Some New Results on the Multinomial Randomized Response Model. Communications in Statistics. Theory and Methods, 847-856. https://doi.org/10.1081/STA-200054378
Kruschke, J.K. (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan. Academic Press.
Kuk, A.Y.C. (1990). Asking Sensitive Question Indirectly. Biometrika, 77, 436-438. https://doi.org/10.1093/biomet/77.2.436
Lee, C.S., Sedory, S.A., Singh, S. (2013). Estimating at Least Seven Measures of Qualitative Variables from a Single Sample using Randomized Response Technique. Statistics and Probability Letters, 83, 399-409. https://doi.org/10.1016/j.spl.2012.10.004
Lensvelt-Mulders, G.J., van der Heijden, P.G., Laudy, O., van Gils, G. (2006). A Validation of a Computer-Assisted Randomized Response
Survey to Estimate the Prevalence of Fraud in Social Security. Journal of the Royal Statistical Society A., 169, 305-318.
A Validation of a Computer-Assisted Randomized Response Survey to Estimate the Prevalence of Fraud in Social Security on JSTOR
Liu, P.T., Chow, L.P. (1976). A new discrete quantitative randomized response model. ACM SIGSIM Simulation Digest, 7, 30-31. https://doi.org/10.1080/01621459.1976.104814 79
Plummer, M., Best, N., Cowles, K., Vines, K. (2006). CODA: Convergence diagnosis and output analysis for MCMC. R News, 6, 7-11.
CODA: Convergence diagnosis and output analysis for MCMC
R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
Scheers, N., Dayton, C. (1988). Covariate Randomized Response Models. Journal of the American Statistical Association, 83, 969-974. https://doi.org/10.1080/01621459.1988.104786 86
Su, Y.-S., Yajima, M. (2015). R2jags: Using R to run JAGS.
Tamhane, A. (1981). Randomized Response Techniques for Multiple Sensitive Attributes. Journal of the American Statistical Association, 76, 916-923.
https://doi.org/10.1080/01621459.1981.104777 41
van den Hout, A., van der Heijden, P.G.M., Gilchrist, R. (2007). The Logistic Regression Model with Response Variables Subject to Randomized Response. Computational Statistics & Data Analysis, 51, 6060-6069. https://doi.org/10.1016/j.csda.2006.12.002
van der Heijden, P., van Gils, G. (1996). Some logistic regression models for randomized response data. In Proceedings of the 11th International Workshop on Statistical Modeling, 341-348.
Van, M.H. (2017). Determinants of the Levels of Development Based on the Human Development Index: Bayesian Ordered Probit Model. International Journal of Economics and Financial Issues, 7, 425.
Warner, S. (1965). Randomized Response: A Survey Technique for Eliminating Evasive
Answer Bias. Journal of the American Statistical Association, 60, 63-69. https://doi.org/10.1080/01621459.1965.104807 75
Xie, Y., Zhang, Y., Liang, F. (2009). Crash Injury Severity Analysis Using Bayesian Ordered Probit Models. Journal of Transportation Engineering, 135, 18-25.
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