Psychometrika

, Volume 68, Issue 1, pp 61–77 | Cite as

A taxonomy of latent structure assumptions for probability matrix decomposition models

  • Michel Meulders
  • Paul De Boeck
  • Iven Van Mechelen
Articles

Abstract

A taxonomy of latent structure assumptions (LSAs) for probability matrix decomposition (PMD) models is proposed which includes the original PMD model (Maris, De Boeck, & Van Mechelen, 1996) as well as a three-way extension of the multiple classification latent class model (Maris, 1999). It is shown that PMD models involving different LSAs are actually restricted latent class models with latent variables that depend on some external variables. For parameter estimation a combined approach is proposed that uses both a mode-finding algorithm (EM) and a sampling-based approach (Gibbs sampling). A simulation study is conducted to investigate the extent to which information criteria, specific model checks, and checks for global goodness of fit may help to specify the basic assumptions of the different PMD models. Finally, an application is described with models involving different latent structure assumptions for data on hostile behavior in frustrating situations.

Key words

discrete data matrix decomposition Bayesian analysis data augmentation posterior predictive check psychometrics 

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References

  1. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B.N. Petrov & F. Csaki (Eds.),Second international symposium on information theory (pp. 271–281). Budapest: Academiai Kiado.Google Scholar
  2. Akaike, H. (1974). A new look at the statistical model identification.IEEE Transactions on Automatic Control, 19, 716–723.Google Scholar
  3. Bayarri, M. J., & Berger, J. O. (2000). P-values for composite null models.Journal of the American Statistical Association, 95, 1127–1142.Google Scholar
  4. Candel, M. J. J. M., & Maris, E. (1997). Perceptual analysis of two-way two-mode frequency data: Probability matrix decomposition and two alternatives.International Journal of Research in Marketing, 14, 321–339.CrossRefGoogle Scholar
  5. Carlin, B.P., & Louis, T.A. (1996).Bayes and empirical Bayes methods for data analysis. London: Chapman & Hall.Google Scholar
  6. Celeux, G., Hurn, M., & Robert, C. P. (2000). Computational and inferential difficulties with mixture posterior distributions.Journal of the American Statistical Association, 95, 957–970.Google Scholar
  7. Cowles, K., & Carlin, B. P. (1996). Markov Chain Monte Carlo convergence diagnostics: A comparative review.Journal of the American Statistical Association, 91, 883–904.Google Scholar
  8. Cressie, N.A.C., & Read, T.R.C. (1984). Multinomial goodness-of-fit tests.Journal of the Royal Statistical Society, Series B,46, 440–464.Google Scholar
  9. De Boeck, P. (1997). Feature-based classification models with a dominance rule. InProceedings of the Biennial Sessions of the Bulletin of the International Statistical Institute (42nd Session, Book 2, pp. 389–392). Istanbul: International Statistical Institute.Google Scholar
  10. de Bonis, M., De Boeck, P., Pérez-Diaz, F., & Nahas, M. (1999). A two-process theory of facial perception of emotions.Life Sciences, 322, 669–675.PubMedGoogle Scholar
  11. Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion).Journal of the Royal Statistical Society, Series B,39, 1–38.Google Scholar
  12. Efron, B, & Tibshirani, R.J. (1993).An introduction to the bootstrap. New York, NY: Chapmann & Hall.Google Scholar
  13. Endler, N.S., & Hunt, J.M. (1968). S-R inventories of hostility and comparisons of the proportions of variance from persons, behaviors, and situations for hostility and anxiousness.Journal of Personality and Social Psychology, 9, 309–315.PubMedGoogle Scholar
  14. Formann, A.K. (1992). Linear logistic latent class analysis for polytomous data.Journal of the American Statistical Association, 87, 476–486.Google Scholar
  15. Gelfand, A.E., & Smith, A.F.M. (1990). Sampling based approaches to calculating marginal densities.Journal of the American Statistical Association, 85, 398–409.Google Scholar
  16. Gelman, A., & Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences.Statistical Science, 7, 457–472.Google Scholar
  17. Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (1995).Bayesian data analysis. London: Chapman & Hall.Google Scholar
  18. Gelman, A., Meng, X.M., & Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies.Statistica Sinica, 4, 733–807.Google Scholar
  19. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images.IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.Google Scholar
  20. Goodman, L.A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models.Biometrika, 61, 215–231.Google Scholar
  21. Maris, E. (1999). Estimating multiple classification latent class models.Psychometrika, 64, 187–212.CrossRefGoogle Scholar
  22. Maris, E., De Boeck, P., & Van Mechelen, I. (1996). Probability matrix decomposition models.Psychometrika, 61, 7–29.CrossRefGoogle Scholar
  23. McLachlan, G.J., & Basford, K.E. (1988).Mixture models. New York, NY: Marcel Dekker.Google Scholar
  24. Meng, X.L. (1994). Posterior predictive p-values.The Annals of Statistics, 22, 1142–1160.Google Scholar
  25. Meulders, M. (2000).Probabilistic feature models for psychological frequency data: A Bayesian approach. Unpublished doctoral dissertation, University of Leuven, Belgium.Google Scholar
  26. Meulders, M, De Boeck, P., Van Mechelen, I. (2001). Probability matrix decomposition models and main-effects generalized linear models for the analysis of replicated binary associations.Computational Statistics and Data Analysis, 38, 217–233.CrossRefGoogle Scholar
  27. Meulders, M., De Boeck, P., & Van Mechelen, I. (2002). Rater classification on the basis of latent features in responding to situations. In W. Gaul & G. Ritter (Eds.),Classification, automation, and new media. Proceedings of the 24th Annual Conference of the Gesellschaft für Klassifikation, University of Passau (pp. 453–461). Berlin: Springer-Verlag.Google Scholar
  28. Meulders, M., De Boeck, P., Van Mechelen, I., & Gelman, A. (2000)Hierarchical extensions of probability matrix decomposition models. Manuscript submitted for publication.Google Scholar
  29. Meulders, M., De Boeck, P., Van Mechelen, I., Gelman, A., & Maris, E. (2001). Bayesian inference with probability matrix decomposition models.Journal of Educational and Behavioral Statistics, 26, 153–179.Google Scholar
  30. Raftery, A.E. (1986). A note on Bayes factors for log-linear contingency table models with vague prior information.Journal of the Royal Statistical Society, Series B,48, 249–250.Google Scholar
  31. Robins J.M., Van Der Vaart, A., & Ventura, V. (2000). Asymptotic distribution of p-values in composite null models.Journal of the Statistical Association, 95, 1143–1172.Google Scholar
  32. Rubin, D.B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician.Annals of Statistics, 12, 1151–1172.Google Scholar
  33. Schwarz, G. (1978). Estimating the dimensions of a model.Annals of Statistics, 6, 461–464.Google Scholar
  34. Spiegelhalter, D.J., Best, N.G., & Carlin, B.P. (1998).Bayesian deviance, the effective number of parameters, and the comparison of arbitrarily complex models. Manuscript submitted for publication.Google Scholar
  35. Spiegelhalter, D.J., Thomas, A., Best, N., & Gilks, W.R. (1995).BUGS: Bayesian inference using Gibbs sampling, Version 0.50 (Tech. Rep.). Cambridge, U.K.: Cambridge University, Institute of Public Health, Medical Research Council Biostatistics Unit.Google Scholar
  36. Stephens, M. (2000). Dealing with label switching in mixture models.Journal of the Royal Statistical Society, Series B,62, 795–809.Google Scholar
  37. Tanner, M.A., & Wong, W.H. (1987). The calculation of posterior distributions by data augmentation.Journal of the American Statistical Association, 82, 528–540.Google Scholar
  38. Vansteelandt, K. (1999). A formal model for the competency-demand hypothesis.European Journal of Personality, 13, 429–442.CrossRefGoogle Scholar
  39. Vermunt J.K. (1997).Log-linear models for event histories. Thousand Oaks, CA: Sage.Google Scholar
  40. Von Davier, M. (1997). Bootstrapping goodness-of-fit statistics for sparse categorical data: Results of a Monte Carlo study.Methods of Psychological Research Online, 2, 29–48.Google Scholar

Copyright information

© The Psychometric Society 2003

Authors and Affiliations

  • Michel Meulders
    • 1
  • Paul De Boeck
    • 1
  • Iven Van Mechelen
    • 1
  1. 1.Katholieke Universiteit LeuvenBelgium

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