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Assessing Joint Distributions with Isoprobability Contours


  • Ali E. Abbas

    () (Department of Industrial and Enterprise Systems Engineering, College of Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801)

  • David V. Budescu

    () (Department of Psychology, Fordham University, Bronx, New York 10458; and Department of Psychology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801)

  • Yuhong (Rola) Gu

    () (Department of Psychology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801)


We present a new method for constructing joint probability distributions of continuous random variables using isoprobability contours--sets of points with the same joint cumulative probability. This approach reduces the joint probability assessment into a one-dimensional cumulative probability assessment using a sequence of binary choices between various combinations of the variables of interest. The approach eliminates the need to assess directly the dependence, or association, between the variables. We discuss properties of isoprobability contours and methods for their assessment in practice. We also report results of a study in which subjects assessed the 50th percentile isoprobability contour of the joint distribution of weight and height. We use the data to show how to use the assessed contours to construct the joint distribution and to infer (indirectly) the dependence between the variables.

Suggested Citation

  • Ali E. Abbas & David V. Budescu & Yuhong (Rola) Gu, 2010. "Assessing Joint Distributions with Isoprobability Contours," Management Science, INFORMS, vol. 56(6), pages 997-1011, June.
  • Handle: RePEc:inm:ormnsc:v:56:y:2010:i:6:p:997-1011
    DOI: 10.1287/mnsc.1100.1161

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    References listed on IDEAS

    1. Woojune Yi & Vicki M. Bier, 1998. "An Application of Copulas to Accident Precursor Analysis," Management Science, INFORMS, vol. 44(12-Part-2), pages 257-270, December.
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    3. Thomas S. Wallsten & David V. Budescu, 1983. "State of the Art---Encoding Subjective Probabilities: A Psychological and Psychometric Review," Management Science, INFORMS, vol. 29(2), pages 151-173, February.
    4. Robert T. Clemen & Terence Reilly, 1999. "Correlations and Copulas for Decision and Risk Analysis," Management Science, INFORMS, vol. 45(2), pages 208-224, February.
    5. Robert T. Clemen & Gregory W. Fischer & Robert L. Winkler, 2000. "Assessing Dependence: Some Experimental Results," Management Science, INFORMS, vol. 46(8), pages 1100-1115, August.
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    Cited by:

    1. David Kaplan & Jianshen Chen, 2012. "A Two-Step Bayesian Approach for Propensity Score Analysis: Simulations and Case Study," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 581-609, July.
    2. Werner, Christoph & Bedford, Tim & Cooke, Roger M. & Hanea, Anca M. & Morales-NĂ¡poles, Oswaldo, 2017. "Expert judgement for dependence in probabilistic modelling: A systematic literature review and future research directions," European Journal of Operational Research, Elsevier, vol. 258(3), pages 801-819.
    3. Tim Bedford & Alireza Daneshkhah & Kevin J. Wilson, 2016. "Approximate Uncertainty Modeling in Risk Analysis with Vine Copulas," Risk Analysis, John Wiley & Sons, vol. 36(4), pages 792-815, April.
    4. Xiaochun Meng & James W. Taylor & Souhaib Ben Taieb & Siran Li, 2020. "Scoring Functions for Multivariate Distributions and Level Sets," Papers 2002.09578,, revised Jun 2020.
    5. Lockwood, Matthew, 2016. "The UK's Levy Control Framework for renewable electricity support: Effects and significance," Energy Policy, Elsevier, vol. 97(C), pages 193-201.


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