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A mixture model for ordinal variables measured on semantic differential scales

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  • Manisera, Marica
  • Zuccolotto, Paola

Abstract

Subjective perceptions and attitudes are usually measured by administering questionnaires with ordered response scales. Among them, a particular case are semantic differential scales, where the respondent has to declare his/her position between two bipolar adjectives. To model ordinal variables measured on semantic differential scales, a novel model is introduced as an extension in the framework of the CUB (Combination of discrete Uniform and shifted Binomial random variables) class of models. The proposed model addresses the analysis of ordinal variables measured on semantic differential scales. However, it is definitely well suited to all the rating scales that have a middle option that means indifference between two extremes. This is a circumstance that occurs in the main part of the most commonly used Likert scales. The proposal is based on a mixture of a discrete Uniform and a - linearly transformed - Multinomial random variable, so it is called CUM. Parameter estimation is carried out using the expectation-maximization algorithm, and the parameters can be represented in a triangular space with a ternary plot. A simulation study is carried out and, finally, applications on real data are examined in order to show limits and potentialities of the proposal.

Suggested Citation

  • Manisera, Marica & Zuccolotto, Paola, 2022. "A mixture model for ordinal variables measured on semantic differential scales," Econometrics and Statistics, Elsevier, vol. 22(C), pages 98-123.
    • Handle: RePEc:eee:ecosta:v:22:y:2022:i:c:p:98-123
    DOI: 10.1016/j.ecosta.2021.07.002
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    References listed on IDEAS

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    1. Manisera, Marica & Zuccolotto, Paola, 2014. "Modeling rating data with Nonlinear CUB models," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 100-118.
    2. Simone, Rosaria & Tutz, Gerhard & Iannario, Maria, 2020. "Subjective heterogeneity in response attitude for multivariate ordinal outcomes," Econometrics and Statistics, Elsevier, vol. 14(C), pages 145-158.
    3. Karlis, Dimitris & Xekalaki, Evdokia, 2003. "Choosing initial values for the EM algorithm for finite mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 577-590, January.
    4. Domenico Piccolo & Rosaria Simone, 2019. "The class of cub models: statistical foundations, inferential issues and empirical evidence," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 389-435, September.
    5. Maria Iannario & Marica Manisera & Domenico Piccolo & Paola Zuccolotto, 2020. "Ordinal Data Models for No-Opinion Responses in Attitude Survey," Sociological Methods & Research, , vol. 49(1), pages 250-276, February.
    6. D'Elia, Angela & Piccolo, Domenico, 2005. "A mixture model for preferences data analysis," Computational Statistics & Data Analysis, Elsevier, vol. 49(3), pages 917-934, June.
    7. Marica Manisera & Paola Zuccolotto, 2019. "Discussion of “The class of cub models: statistical foundations, inferential issues and empirical evidence” by Domenico Piccolo and Rosaria Simone," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 465-470, September.
    8. Anna Gottard & Maria Iannario & Domenico Piccolo, 2016. "Varying uncertainty in CUB models," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 10(2), pages 225-244, June.
    9. Maria Iannario, 2012. "Modelling shelter choices in a class of mixture models for ordinal responses," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(1), pages 1-22, March.
    10. Rosaria Simone & Gerhard Tutz, 2018. "Modelling uncertainty and response styles in ordinal data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 72(3), pages 224-245, August.
    11. repec:cup:judgdm:v:12:y:2017:i:1:p:42-59 is not listed on IDEAS
    12. Domenico Piccolo & Rosaria Simone, 2019. "Rejoinder to the discussion of “The class of cub models: statistical foundations, inferential issues and empirical evidence”," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(3), pages 477-493, September.
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