Comments on The class of cub models: statistical foundations, inferential issues and empirical evidence by D. Piccolo and R. Simone

  • Gerhard TutzEmail author


I want to congratulate the authors on a comprehensive and stimulating overview on the class of CUB models. The approach was developed about a decade ago, with Domenico Piccolo as the leading researcher in the area. The approach turned out to be very fruitful, various extensions and versions of the basic CUB model have been given since then. I confine myself to some brief remarks with a focus on potential areas of future research.

CUB models as structured mixture models

CUB models are discrete mixture models, they assume that what one can observe is the result of a mixture of unobserved responses. Mixture models are strong tools but challenging with regard to estimation and interpretation. It is not uncommon that data analysts fit a mixture of an unknown number of regression or latent trait models without constraints on the mixture components, that is, regression coefficients and dispersion parameters in the components can vary freely. After fitting various models one selects...



  1. Agresti A (2010) Analysis of ordinal categorical data, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  2. Colombi R, Giordano S, Gottard A, Iannario M (2018) Hierarchical marginal models with latent uncertainty. Scand J Stat. Google Scholar
  3. Efron B (1978) Regression and ANOVA with zero-one data: measures of residual variation. J Am Stat Assoc 73:113–121MathSciNetCrossRefzbMATHGoogle Scholar
  4. Eid M, Rauber M (2000) Detecting measurement invariance in organizational surveys. Eur J Psychol Assess 16(1):20–30CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Ludwig-Maximilians-Universität MünchenMunichGermany

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