Modelling Rater Differences in the Analysis of Three-Way Three-Mode Binary Data

  • Michel MeuldersEmail author
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


Using a basic latent class model for the analysis of three-way three- mode data (i.e. raters by objects by attributes) to cluster raters is often problematic because the number of conditional probabilities increases rapidly when extra latent classes are added. To solve this problem, Meulders et al. (J Classification 19:277–302, 2002) proposed a constrained latent class model in which object-attribute associations are explained on the basis of latent features. In addition, qualitative rater differences are introduced by assuming that raters may only take into account a subset of the features. As this model involves a direct link between the number of features F and the number of latent classes (i.e., 2 F ), estimation of the model becomes slow when many latent features are needed to fit the data. In order to solve this problem we propose a new model in which rater differences are modelled by assuming that features can be taken into account with a certain probability which depends on the rater class. An EM algorithm is used to locate the posterior mode of the model and a Gibbs sampling algorithm is developed to compute a sample of the observed posterior of the model. Finally, models with different types of rater differences are applied to marketing data and the performance of the models is compared using posterior predictive checks (see also, Meulders et al. (Psychometrika 68:61–77, 2003)).


Posterior Distribution Latent Class Latent Feature Latent Class Model Rater Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Candel MJJM, Maris E (1997) Perceptual analysis of two-way two-mode frequency data: Probability matrix decomposition and two alternatives. Int J Res Market 14:321–339CrossRefGoogle Scholar
  2. Gelfand AE, Smith AFM (1990) Sampling based approaches to calculating marginal densities. J Am Stat Assoc 85:398–409MathSciNetzbMATHCrossRefGoogle Scholar
  3. Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7:457–472CrossRefGoogle Scholar
  4. Gelman A, Meng XM, Stern H (1996) Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica 4:733–807MathSciNetGoogle Scholar
  5. Meulders M, De Boeck P, Kuppens P, Van Mechelen I (2002) Constrained latent class analysis of three-way three-mode data. J Classification 19:277–302MathSciNetzbMATHCrossRefGoogle Scholar
  6. Meulders M, De Boeck P, Van Mechelen I (2003) A taxonomy of latent structure assumptions for probability matrix decomposition models. Psychometrika 68:61–77MathSciNetCrossRefGoogle Scholar
  7. Vermunt JK (2006) A hierarchical mixture model for clustering three-way data sets. Comput Stat Data Anal 51:5368–5376MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.HUBrusselBrusselBelgium
  2. 2.KULLeuvenBelgium

Personalised recommendations