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Varying uncertainty in CUB models

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Abstract

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.

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References

  • Agresti A (2010) Analysis of ordinal categorical data. Wiley, New York

    Book  MATH  Google Scholar 

  • Atkinson A (1970) A method for discriminating between models. J R Stat Soc Ser B 32:323–353

    MATH  Google Scholar 

  • Baumgartner H, Steenback J-B (2001) Response styles in marketing research: a cross-national investigation. J Market Res 38:143–156

    Article  Google Scholar 

  • Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B 57(1):289–300

    MathSciNet  MATH  Google Scholar 

  • 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

  • Cliff N, Keats JA (2003) Ordinal measurement in the behavioral science. Taylor & Francis, UK

  • 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

  • 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

  • Cox D (1962) Further results on tests of separate families of hypotheses. J R Stat Soc Ser B 24:406–424

    MathSciNet  MATH  Google Scholar 

  • Cox D (2013) A return to an old paper: tests of separate families of hypotheses. J R Stat Soc Ser B (Stat Methodol) 75:207–215

    Article  MathSciNet  Google Scholar 

  • D’Elia A (2000) The paired comparison mechanism in ranking models: statistical developments and critical issues (in Italian). Quaderni di Statistica 2:173–203

    Google Scholar 

  • Eid M, Zickar M (2007) Detecting response styles and faking in personality and organizational assessments by mixed rasch models. In: von Davier M, Carstensen CH (eds) Multivariate and mixture distribution rasch models extensions and applications. Statistics for social and behavioral sciences. Springer, Berlin, pp 255–270

    Google Scholar 

  • Gollwitzer M, Eid M, Jürgensen R (2005) Response styles in the assessment of anger expression. Psychol Assessm 17:56–69

    Article  Google Scholar 

  • Grilli L, Iannario M, Piccolo D, Rampichini C (2014) Latent class CUB models. Adv Data Anal Classif 8:105–119

    Article  MathSciNet  Google Scholar 

  • Iannario M (2007) A statistical approach for modelling urban audit perception surveys. Quaderni di Statistica 9:149–172

    Google Scholar 

  • Iannario M (2009) Fitting measures for ordinal data models. Quaderni di Statistica 11:39–72

    Google Scholar 

  • Iannario M (2012) Modelling shelter choices in a class of mixture models for ordinal responses. Stat Meth Appl 21:1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Iannario M (2014) Modelling uncertainty and overdispersion in ordinal data. Commun Stat Theory Meth 43:771–786

    Article  MathSciNet  MATH  Google Scholar 

  • Iannario M (2015) Detecting latent components in ordinal data with overdispersion by means of a mixture distribution. Qual Quant 49(3):977–987

    Article  MathSciNet  Google Scholar 

  • Iannario M, Piccolo D (2010) A new statistical model for the analysis of customer satisfaction. Qual Technol Quant Manag 7:149–168

    Article  Google Scholar 

  • Iannario M, Piccolo D (2012a) CUB models: statistical methods and empirical evidence. In: Kenett R, Salini S (eds) Modern analysis of customer surveys: with applications using R. Wiley, Chichester, pp 231–258

    Google Scholar 

  • Iannario M, Piccolo D (2012b) A framework for modelling ordinal data in rating surveys. Proc Jt Stat Meet Market Res Sect 7:1–15

    Google Scholar 

  • Iannario M, Piccolo D (2014) Inference for CUB models: a program in R. Stat & Appl. XII:177–204

  • 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

  • Iannario M, Piccolo D (2015b) A generalized framework for modelling ordinal data. Stat Meth Appl. doi:10.1007/s10260-015-0316-9

  • Kokonendji C, Zocchi S (2010) Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions. Stat Probab Lett 80:1655–1662

    Article  MathSciNet  MATH  Google Scholar 

  • Kulas J, Stachowski A (2009) Middle category endorsement in odd-numbered Likert response scales: associated item characteristics, cognitive demands, and preferred meanings. J Res Pers 43(3):489–493

    Article  Google Scholar 

  • Lentz T (1938) Acquiescence as a factor in the measurement of personality. Psychol Bull 35:659

    Google Scholar 

  • McCullagh P (1980) Regression models for ordinal data. J R Stat Soc Ser B 42:109–142

    MathSciNet  MATH  Google Scholar 

  • McLachlan G, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, New York

    Book  MATH  Google Scholar 

  • McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York

    Book  MATH  Google Scholar 

  • Moors G (2008) Exploring the effect of a middle response category on response style in attitude measurement. Qual Quant 42:779–794

    Article  Google Scholar 

  • 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

  • Piccolo D (2003) On the moments of a mixture of uniform and shifted binomial random variables. Quaderni di Statistica 5:86–104

    Google Scholar 

  • Piccolo D (2006) Observed information matrix for MUB models. Quaderni di Statistica 8:33–78

    Google Scholar 

  • Poulton E (1989) Bias in quantifying judgements. Psychology Press, Hillsdale

    Google Scholar 

  • Simon H (1957) Models of man; social and rational. Wiley, New York

    MATH  Google Scholar 

  • Tutz G (2012) Regression for categorical data. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Vuong Q (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57:307–333

    Article  MathSciNet  MATH  Google Scholar 

  • White H (1982) Maximum likelihood estimation of misspecified models. Econometrica 50:1–25

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Authors thank Associate Editor and referees for comments and suggestions that lead to a much improved paper, and Alan Agresti for very constructive comments. Authors gratefully acknowledge the support from research projects FIRB 2012 at University of Perugia (code RBFR12SHVV) and SHAPE in the frame of STAR Programme (CUP E68C13000020003) at University of Naples Federico II, financially supported by UniNA and Compagnia di San Paolo. This research has been partly supported by FARO2011 project of University of Naples Federico II. The second Author benefits from a Fulbright scholarship for visiting Department of Statistics and Actuarial Science, University of Iowa, USA.

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Correspondence to Domenico Piccolo.

Appendix

Appendix

EM algorithm

The EM algorithm may be effectively implemented for a finite mixture (McLachlan and Peel 2000) by the following steps, where \(p_r^V\) has to be specified on a priori ground. Hereafter, for a given m, we will denote \(\varvec{\theta }=(\pi ,\xi )'\) and set a small tolerance \(\epsilon \) (\({=}10^{-6}\), for instance).

  1. 0.

    \(k=0\); \(\varvec{\theta }^{(0)}=(\pi ^{(0)},\xi ^{(0)})'\);   \(\ell ^{(0)}=\ell (\varvec{\theta }^{(0)})\).

  2. 1.

    \(bb^{(k)}=b_r(\xi ^{(k)});\,\tau ^{(k)}=\left[ 1+p_r^V\, \frac{1-\pi ^{\left( k \right) }}{\pi ^{\left( k\right) }\,b_r\left( r;\xi ^{\left( k\right) }\right) } \right] ^{-1},\quad r=1,2,\ldots ,m\).;

  3. 2.

    \(\overline{R}_n(\varvec{\theta }^{(k)})= \frac{\sum \nolimits _{r=1}^{m}\,r\,n_r\,\tau (r; \varvec{\theta }^{(k)})}{\sum \nolimits _{r=1}^{m} n_r\,\tau (r; \varvec{\theta }^{(k)})}\).

  4. 3.

    \(\pi ^{(k+1)}=(1/n)\sum _{r=1}^{m}\,n_r\,\tau (r; \varvec{\theta }^{(k)})\);   \(\xi ^{(k+1)}=\frac{m-\overline{R}_n(\varvec{\theta }^{(k)})}{m-1}\).

  5. 4.

       \(\varvec{\theta }^{(k+1)}=(\pi ^{(k+1)},\xi ^{(k+1)})';\,\,\quad \ell ^{(k+1)}=\ell (\varvec{\theta }^{(k+1)})\).

  6. 5.
    $$\begin{aligned} \left\{ \begin{array}{ll} \text {if }\mid \ell ^{(k+1)}-\ell ^{(k)}\mid \ge \epsilon ,\,k \rightarrow k+1; \hbox {go to} 1;\\ \text {if }\mid \ell ^{(k+1)}-\ell ^{(k)}\mid < \epsilon , \hat{\varvec{\theta }}=\varvec{\theta }^{(k+1)}; \hbox {stop}.\\ \end{array} \right. \end{aligned}$$

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Gottard, A., Iannario, M. & Piccolo, D. Varying uncertainty in CUB models. Adv Data Anal Classif 10, 225–244 (2016). https://doi.org/10.1007/s11634-016-0235-0

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