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Mixture of shifted binomial distributions for rating data

Author

Listed:
  • Shaoting Li

    (Dongbei University of Finance and Economics)

  • Jiahua Chen

    (Yunnan University
    University of British Columbia)

Abstract

Rating data are a kind of ordinal categorical data routinely collected in survey sampling. The response value in such applications is confined to a finite number of ordered categories. Due to population heterogeneity, the respondents may have several different rating styles. A finite mixture model is thus most suitable to fit datasets of this nature. In this paper, we propose a two-component mixture of shifted binomial distributions for rating data. We show that this model is identifiable and propose a numerically stable penalized likelihood approach for parameter estimation. We adapt an expectation-maximization algorithm for the penalized maximum likelihood estimation. Our simulation results show that the penalized maximum likelihood estimator is consistent and effective. We fit the proposed model and other models in the literature to some real-world datasets and find the proposed model can have much better fits.

Suggested Citation

  • Shaoting Li & Jiahua Chen, 2023. "Mixture of shifted binomial distributions for rating data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(5), pages 833-853, October.
    • Handle: RePEc:spr:aistmt:v:75:y:2023:i:5:d:10.1007_s10463-023-00865-7
    DOI: 10.1007/s10463-023-00865-7
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    References listed on IDEAS

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    1. Maria Iannario, 2010. "On the identifiability of a mixture model for ordinal data," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 87-94.
    2. 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.
    3. P. Li & J. Chen & P. Marriott, 2009. "Non-finite Fisher information and homogeneity: an EM approach," Biometrika, Biometrika Trust, vol. 96(2), pages 411-426.
    4. Hanfeng Chen & Jiahua Chen & John D. Kalbfleisch, 2001. "A modified likelihood ratio test for homogeneity in finite mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(1), pages 19-29.
    5. Rosaria Simone, 2021. "An accelerated EM algorithm for mixture models with uncertainty for rating data," Computational Statistics, Springer, vol. 36(1), pages 691-714, March.
    6. N. Atienza & J. Garcia-Heras & J. Muñoz-Pichardo, 2006. "A new condition for identifiability of finite mixture distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 63(2), pages 215-221, April.
    7. Zhou, Hua & Lange, Kenneth, 2009. "Rating Movies and Rating the Raters Who Rate Them," The American Statistician, American Statistical Association, vol. 63(4), pages 297-307.
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