A study into mechanisms of attitudinal scale conversion: A randomized stochastic ordering approach

  • Zvi GilulaEmail author
  • Robert E. McCulloch
  • Yaacov Ritov
  • Oleg Urminsky


This paper considers the methodological challenge of how to convert categorical attitudinal scores (like satisfaction) measured on one scale to a categorical attitudinal score measured on another scale with a different range. This is becoming a growing issue in marketing consulting and the common available solutions seem too few and too superficial. A new methodology for scale conversion is proposed, and tested in a comprehensive study. This methodology is shown to be both relevant and optimal in fundamental aspects. The new methodology is based on a novel algorithm named minimum conditional entropy, that uses the marginal distributions of the responses on each of the two scales to produce a unique joint bivariate distribution. In this joint distribution, the conditional distributions follow a stochastic order that is monotone in the categories and has the relevant optimal property of maximizing the correlation between the two underlying marginal scales. We show how such a joint distribution can be used to build a mechanism for scale conversion. We use both a frequentist and a Bayesian approach to derive mixture models for conversion mechanisms, and discuss some inferential aspects associated with the underlying models. These models can incorporate background variables of the respondents. A unique observational experiment is conducted that empirically validates the proposed modeling approach. Strong evidence of validation is obtained.


Categorical conversion Conditional entropy Mixture models Ordinal attitudinal scales Stochastic ordering 



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Zvi Gilula
    • 1
    Email author
  • Robert E. McCulloch
    • 2
  • Yaacov Ritov
    • 3
  • Oleg Urminsky
    • 4
  1. 1.Department of StatisticsHebrew UniversityJerusalemIsrael
  2. 2.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  3. 3.Department of StatisticsUniversity of MichiganAnn ArborUSA
  4. 4.University of Chicago Booth School of BusinessChicagoUSA

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