Quality & Quantity

, Volume 49, Issue 3, pp 977–987 | Cite as

Detecting latent components in ordinal data with overdispersion by means of a mixture distribution

  • Maria IannarioEmail author


The paper describes a mixture distribution generated by Beta Binomial and Uniform random variables to allow for a possible overdispersion in surveys when the response of interest is an ordinal variable. This approach considers the joint presence of feeling, uncertainty and a possible dispersion sometimes present in the evaluation contexts. After a discussion of the main properties of this class of models, asymptotic likelihood methods have been applied for efficient statistical inference. The implementation on the survey on household income and wealth (SHIW) will confirm the versatility of this distribution and the usefulness to distinguish the determinants of uncertainty and overdispersion in real data.


Latent components Overdispersion cub model cube model 



The author thanks the Editor and the referees for suggestions which improved the paper. This research has been supported by Programme STAR (CUP E68C13000020003) at University of Naples Federico II and FIRB 2012 project (code RBFR12SHVV) at University of Perugia.


  1. Baumgartner, H., Steenback, J.B.: Response styles in marketing research: a cross-national investigation. J. Mark. Res. 38, 143–156 (2001)Google Scholar
  2. Chatfield, C., Goodhart, G.J.: The beta-binomial model for consumer purchasing behavior. Appl. Statist. 19, 240–250 (1970)CrossRefGoogle Scholar
  3. Cochran, W.G.: Sampling techniques, 3rd edn. John Wiley & Sons, New York (1977)Google Scholar
  4. Cox, D.R.: Some remarks on overdispersion. Biometrika 70, 269–274 (1983)CrossRefGoogle Scholar
  5. De Finetti, B., Paciello, U.: Calcolo della differenza media. METRON 8, 1–6 (1930)Google Scholar
  6. Farewell, V.T.: A note on regression analysis of ordinal data with variability of classification. Biometrika 69, 533–538 (1982)CrossRefGoogle Scholar
  7. Finney, D.J.: Probit Analysis. Cambridge University Press, Cambridge (1971)Google Scholar
  8. Fitzmaurice, G.M., Heath, A.F., Cox, D.R.: Detecting overdispersion in large scale surveys: application to a study of education and social class in Britain. J. Roy. Statist. Soc. Ser. C 46, 415–432 (1997)CrossRefGoogle Scholar
  9. Gerstenkorn, T., Gerstenkorn, J.: Gini’s mean difference in the theory and application to inflated distributions. Statistica LXIII, 469–488 (2003)Google Scholar
  10. Greenleaf, E.: Improving rating scale measures by detecting and correcting bis components in some response styles. J. Marketing Res. 29, 176–188 (1992)CrossRefGoogle Scholar
  11. Hinde, J., Demétrio, C.G.B.: Overdispersion: Models and Estimation. ABE, Sao Paulo (1998)Google Scholar
  12. Iannario, M.: Preliminary estimators for a mixture model of ordinal data. Adv. Data Anal. Classif. 6, 163–184 (2012)CrossRefGoogle Scholar
  13. Iannario, M.: Modelling uncertainty and overdispersion in ordinal data. Comm. Stat. Theory Methods 43, 771–786 (2014a)CrossRefGoogle Scholar
  14. Iannario, M.: Testing overdispersion in CUBE models. Comm. Stat. Simul. Comput. 43, 771–786 (2014b)Google Scholar
  15. Iannario, M., Piccolo, D.: cub models: statistical methods and empirical evidence. In: Kenett, R.S., Salini, S. (eds.) Modern Analysis of customer surveys: with applications using R, pp. 231–258. Wiley, Chichester (2012)Google Scholar
  16. McCullagh, P., Nelder, J.A.: Generalized Linear Models, \(2^{nd}\) edition. Chapman & Hall, London (1989)CrossRefGoogle Scholar
  17. McLachlan, G., Krishnan, T.: The EM algorithm and extensions. Wiley, New York (1997)Google Scholar
  18. Piccolo, D.: On the moments of a mixture of uniform and shifted binomial random variables. Quad. Stat. 5, 85–104 (2003)Google Scholar
  19. Piccolo, D.: Observed information matrix for MUB models. Quad. Stat. 8, 33–78 (2006)Google Scholar
  20. Piccolo, D.: Inferential issues on \(CUBE\) models with covariates. Comm. Stat. Theory Methods, 44, forthcoming (2014)Google Scholar
  21. Rossi, P.E., Gilula, Z., Allenby, G.M.: Overcoming scale usage heterogeneity: a Bayesian hierachical approach. J. Amer. Statist. Assoc. 96, 20–31 (2001)CrossRefGoogle Scholar
  22. Sartori, R., Ceschi, A.: Uncertainty and its perception: experimental study of the numeric expression of uncertainty in two decisional contexts. Qual. Quant. 45, 187–198 (2011)CrossRefGoogle Scholar
  23. Tripathi, R.C., Gupta, R.C., Gurland, J.: Estimation of parameters in the Beta Binomial model. Ann. Inst. Statist. Math. 46, 317–331 (1994)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Political SciencesUniversity of Naples Federico IINaplesItaly

Personalised recommendations