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A generalized approach to randomised response for quantitative variables


  • Antonio Arcos
  • María del Rueda


  • Sarjinder Singh



The methodology of randomised response (RR) has advanced considerably in recent years. Nevertheless, most research in this area has addressed the estimation of qualitative variables, and relatively little attention has been paid to the study of quantitative ones. Furthermore, most studies concern only simple random sampling. In this paper, we present a new model of RR aimed at determining a total that is valid for any sampling design. This general procedure includes several important RR techniques that constitute particular cases. We propose an unbiased estimator, with an analytic expression for its variance. Confidence intervals are also obtained for the parameter, applying analytical formulae such as those based on resampling technologies. A simulation study illustrates the behaviour of the estimator using diverse randomisation devices and for different scrambling distributions. To illustrate the advantages of this method, we obtained a stratified clustered sample of university students, who were questioned to determine the frequency with which they cheated in exams. Their responses to these questions were obtained via an RR technique, and also using a direct questionnaire. We conclude that estimates based on anonymous questionnaires may result in severe underestimation. Copyright Springer Science+Business Media Dordrecht 2015

Suggested Citation

  • Antonio Arcos & María del Rueda & Sarjinder Singh, 2015. "A generalized approach to randomised response for quantitative variables," Quality & Quantity: International Journal of Methodology, Springer, vol. 49(3), pages 1239-1256, May.
  • Handle: RePEc:spr:qualqt:v:49:y:2015:i:3:p:1239-1256
    DOI: 10.1007/s11135-014-0046-3

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    References listed on IDEAS

    1. Kuo-Chung Huang, 2008. "Estimation for sensitive characteristics using optional randomized response technique," Quality & Quantity: International Journal of Methodology, Springer, vol. 42(5), pages 679-686, October.
    2. Ivar Krumpal, 2013. "Determinants of social desirability bias in sensitive surveys: a literature review," Quality & Quantity: International Journal of Methodology, Springer, vol. 47(4), pages 2025-2047, June.
    3. Christopher Gjestvang & Sarjinder Singh, 2007. "Forced quantitative randomized response model: a new device," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(2), pages 243-257, September.
    4. Carlos Bouza, 2009. "Ranked set sampling and randomized response procedures for estimating the mean of a sensitive quantitative character," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 267-277, November.
    5. Elisabeth Coutts & Ben Jann, 2011. "Sensitive Questions in Online Surveys: Experimental Results for the Randomized Response Technique (RRT) and the Unmatched Count Technique (UCT)," Sociological Methods & Research, , vol. 40(1), pages 169-193, February.
    6. Shaul K. Bar-Lev & Elizabeta Bobovitch & Benzion Boukai, 2004. "A note on randomized response models for quantitative data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 60(3), pages 255-260, November.
    7. Oluseun Odumade & Sarjinder Singh, 2010. "An Alternative to the Bar-Lev, Bobovitch, and Boukai Randomized Response Model," Sociological Methods & Research, , vol. 39(2), pages 206-221, November.
    8. Christopher R. Gjestvang & Sarjinder Singh, 2006. "A new randomized response model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 523-530.
    9. van den Hout, Ardo & van der Heijden, Peter G.M. & Gilchrist, Robert, 2007. "The logistic regression model with response variables subject to randomized response," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6060-6069, August.
    10. Giancarlo Diana & Pier Francesco Perri, 2010. "New scrambled response models for estimating the mean of a sensitive quantitative character," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(11), pages 1875-1890.
    11. Amitava Saha, 2007. "A simple randomized response technique in complex surveys," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 59-66.
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