Varying uncertainty in CUB models. (English) Zbl 1414.62327

Summary: This paper presents a generalization of a mixture model used for the analysis of ratings and preferences by introducing a varying uncertainty component. According to the standard mixture model, called CUB model, the response probabilities are defined as a convex combination of shifted Binomial and discrete Uniform random variables. Our proposal introduces uncertainty distributions with different shapes, which could capture response style and indecision of respondents with greater effectiveness. Since we consider several alternative specifications that are nonnested, we suggest the implementation of a Vuong test for choosing among them. In this regard, some simulation experiments and real case studies confirm the usefulness of the approach.


62J12 Generalized linear models (logistic models)
62H30 Classification and discrimination; cluster analysis (statistical aspects)


CUB; catdata
Full Text: DOI


[1] Agresti A (2010) Analysis of ordinal categorical data. Wiley, New York · Zbl 1263.62007
[2] Atkinson, A., A method for discriminating between models, J R Stat Soc Ser B, 32, 323-353, (1970) · Zbl 0225.62020
[3] Baumgartner, H.; Steenback, J-B, Response styles in marketing research: a cross-national investigation, J Market Res, 38, 143-156, (2001)
[4] Benjamini, Y.; Hochberg, Y., Controlling the false discovery rate: a practical and powerful approach to multiple testing, J R Stat Soc Ser B, 57, 289-300, (1995) · Zbl 0809.62014
[5] Buckley J (2009) Cross-national response styles in international educational assessments: evidence from pisa 2006. Technical report, Department of Humanities and Social Sciences in the Professions Steinhardt School of Culture, Education, and Human Development New York University
[6] Cliff N, Keats JA (2003) Ordinal measurement in the behavioral science. Taylor & Francis, UK
[7] Corduas M, Iannario M, Piccolo D (2009) A class of statistical models for evaluating services and performances. In: Bini M et al (eds) Statistical methods for the evaluation of educational services and quality of products. Contributions to statistics. Springer, New York, pp 99-117
[8] Cox D (1961) Tests of separate families of hypotheses. In: Proceeding of the fourth Berkeley symposium on mathematical statistics and probability, vol 1, pp 105-123 · Zbl 0201.52102
[9] Cox, D., Further results on tests of separate families of hypotheses, J R Stat Soc Ser B, 24, 406-424, (1962) · Zbl 0131.35801
[10] Cox, D., A return to an old paper: tests of separate families of hypotheses, J R Stat Soc Ser B (Stat Methodol), 75, 207-215, (2013)
[11] D’Elia, A., The paired comparison mechanism in ranking models: statistical developments and critical issues (in Italian), Quaderni di Statistica, 2, 173-203, (2000)
[12] Eid, M.; Zickar, M.; Davier, M. (ed.); Carstensen, CH (ed.), Detecting response styles and faking in personality and organizational assessments by mixed rasch models, 255-270, (2007), Berlin
[13] Gollwitzer, M.; Eid, M.; Jürgensen, R., Response styles in the assessment of anger expression, Psychol Assessm, 17, 56-69, (2005)
[14] Grilli, L.; Iannario, M.; Piccolo, D.; Rampichini, C., Latent class CUB models, Adv Data Anal Classif, 8, 105-119, (2014)
[15] Iannario, M., A statistical approach for modelling urban audit perception surveys, Quaderni di Statistica, 9, 149-172, (2007)
[16] Iannario, M., Fitting measures for ordinal data models, Quaderni di Statistica, 11, 39-72, (2009)
[17] Iannario, M., Modelling shelter choices in a class of mixture models for ordinal responses, Stat Meth Appl, 21, 1-22, (2012) · Zbl 1333.62181
[18] Iannario, M., Modelling uncertainty and overdispersion in ordinal data, Commun Stat Theory Meth, 43, 771-786, (2014) · Zbl 1287.62001
[19] Iannario, M., Detecting latent components in ordinal data with overdispersion by means of a mixture distribution, Qual Quant, 49, 977-987, (2015)
[20] Iannario, M.; Piccolo, D., A new statistical model for the analysis of customer satisfaction, Qual Technol Quant Manag, 7, 149-168, (2010)
[21] Iannario, M.; Piccolo, D.; Kenett, R. (ed.); Salini, S. (ed.), CUB models: statistical methods and empirical evidence, 231-258, (2012), Chichester
[22] Iannario, M.; Piccolo, D., A framework for modelling ordinal data in rating surveys, Proc Jt Stat Meet Market Res Sect, 7, 1-15, (2012)
[23] Iannario M, Piccolo D (2014) Inference for CUB models: a program in R. Stat & Appl. XII:177-204 · Zbl 1362.60006
[24] Iannario M, Piccolo D (2015a) CUB: a class of mixture models for ordinal data. R package version 0.0. http://CRAN.R-project.org/package=CUB
[25] Iannario M, Piccolo D (2015b) A generalized framework for modelling ordinal data. Stat Meth Appl. doi:10.1007/s10260-015-0316-9 · Zbl 1405.62101
[26] Kokonendji, C.; Zocchi, S., Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions, Stat Probab Lett, 80, 1655-1662, (2010) · Zbl 1195.62007
[27] Kulas, J.; Stachowski, A., Middle category endorsement in odd-numbered Likert response scales: associated item characteristics, cognitive demands, and preferred meanings, J Res Pers, 43, 489-493, (2009)
[28] Lentz, T., Acquiescence as a factor in the measurement of personality, Psychol Bull, 35, 659, (1938)
[29] McCullagh, P., Regression models for ordinal data, J R Stat Soc Ser B, 42, 109-142, (1980) · Zbl 0483.62056
[30] McLachlan G, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, New York · Zbl 1165.62019
[31] McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York · Zbl 0963.62061
[32] Moors, G., Exploring the effect of a middle response category on response style in attitude measurement, Qual Quant, 42, 779-794, (2008)
[33] Pesaran M, Ulloa M (2008) Non-nested hypotheses. In: Durlauf ESN, Blume LE (eds) The new palgrave: a dictionary of economics, , vol 6, 2nd edn, pp 107-114
[34] Piccolo, D., On the moments of a mixture of uniform and shifted binomial random variables, Quaderni di Statistica, 5, 86-104, (2003)
[35] Piccolo, D., Observed information matrix for MUB models, Quaderni di Statistica, 8, 33-78, (2006)
[36] Poulton E (1989) Bias in quantifying judgements. Psychology Press, Hillsdale
[37] Simon H (1957) Models of man; social and rational. Wiley, New York · Zbl 0082.35305
[38] Tutz G (2012) Regression for categorical data. Cambridge University Press, Cambridge · Zbl 1304.62021
[39] Vuong, Q., Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica, 57, 307-333, (1989) · Zbl 0701.62106
[40] White, H., Maximum likelihood estimation of misspecified models, Econometrica, 50, 1-25, (1982) · Zbl 0478.62088
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