Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Hierarchical Multinomial Processing Tree Models: A Latent-Class Approach

  • 331 Accesses

  • 63 Citations


Multinomial processing tree models are widely used in many areas of psychology. Their application relies on the assumption of parameter homogeneity, that is, on the assumption that participants do not differ in their parameter values. Tests for parameter homogeneity are proposed that can be routinely used as part of multinomial model analyses to defend the assumption. If parameter homogeneity is found to be violated, a new family of models, termed latent-class multinomial processing tree models, can be applied that accommodates parameter heterogeneity and correlated parameters, yet preserves most of the advantages of the traditional multinomial method. Estimation, goodness-of-fit tests, and tests of other hypotheses of interest are considered for the new family of models.

This is a preview of subscription content, log in to check access.


  1. Batchelder, W.H., & Riefer, D.M. (1986). The statistical analysis of a model for storage and retrieval processes in human memory. British Journal of Mathematical and Statistical Psychology, 39, 129–149.

  2. Batchelder, W.H., & Riefer, D.M. (1999). Theoretical and empirical review of multinomial processing tree modeling. Psychonomic Bulletin & Review, 6, 57–86.

  3. Bishop, Y. M.M., Fienberg, S.E., & Holland, P.W. (1975). Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press.

  4. Bollen, K.A., & Stine, R.A. (1993). Bootstrapping goodness-of-fit measures in structural equation models. In K.A. Bollen & J.S. Long (Eds.), Testing structural equation models (pp. 111–135). Newbury Park, CA: Sage.

  5. Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.

  6. Chen, J. (1998). Penalized likelihood-ratio test for finite mixture models with multinomial observations. Canadian Journal of Statistics, 26, 583–599.

  7. Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, 39, 1–38.

  8. Efron, B. (1982). The jackknife, the bootstrap and other resampling plans. Philadelphia, PA: Society for Industrial and Applied Mathematics.

  9. Erdfelder, E. (2000). Multinomiale Modelle in der kognitiven Psychologie [Multinomial models in cognitive psychology]. Unpublished habilitation thesis, Psychologisches Institut der Universität Bonn, Germany.

  10. Hu, X. (1991). Statistical inference program for multinomial binary tree models [Computer software]. University of California at Irvine.

  11. Hu, X. (1998). GPT – HomePage [Computer software and documentation]. Retrieved from http://xhuoffice.psyc.memphis.edu/gpt/.

  12. Hu, X., & Batchelder, W.H. (1994). The statistical analysis of general processing tree models with the EM algorithm. Psychometrika, 59, 21–47.

  13. Johnson, N.L., Kotz, S., & Balakrishnan, N. (1997). Discrete multivariate distributions. New York: Wiley.

  14. Johnson, N.L., Kotz, S., & Kemp, A.W. (1993). Univariate discrete distributions (2nd ed.). New York: Wiley.

  15. Kaplan, D. (2000). Structural equation modeling: Foundations and extensions. Thousand Oaks, CA: Sage.

  16. Linhart, H., & Zuccini, W. (1986). Model selection. New York: Wiley.

  17. Magnus, J.R., & Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics. Chichester, UK: Wiley.

  18. McLachlan, G., & Peel, D. (2000). Finite mixture models. New York: Wiley.

  19. Moore, D.S. (1977). Generalized inverses, Wald’s method, and the construction of chi-squared tests of fit. Journal of the American Statistical Association, 72, 131–137.

  20. Muthén, B. (1993). Goodness of fit with categorical and other non-normal variables. In K. Bollen & J.S. Long (Eds.), Testing structural equation models (pp. 205–234). Newbury Park, CA: Sage.

  21. Pinheiro, J.C., & Bates, D.M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4, 12–35.

  22. Powell, M.J.D. (1977). Restart procedures for the conjugate gradient method. Mathematical Programming, 12, 241–254.

  23. Rao, C.R. (1973). Linear statistical inference and its applications. New York: Wiley.

  24. Raudenbush, S.W., & Bryk, A.S. (2002). Hierarchical linear models. Applications and data analysis methods. Thousand Oaks, CA: Sage.

  25. Riefer, D.M., & Batchelder, W.H. (1991). Statistical inference for multinomial processing tree models. In J.-P. Doignon & J.-C. Falmagne (Eds.), Mathematical psychology: Current developments. New York: Springer-Verlag.

  26. Rothkegel, R. (1999). AppleTree: A multinomial processing tree modeling program for Macintosh computers. Behavior Research Methods, Instruments, & Computers, 31, 696–700.

  27. Satorra, A. (1992). Asymptotic robust inference in the analysis of mean and covariance structures. In P.V. Marsden (Ed.), Sociological methodology 1992 (pp. 249–278). Oxford, UK: Blackwell.

  28. Titterington, D.M., Smith, A.F.M., & Makov, U.E. (1985). Statistical analysis of finite mixture distributions. New York: Wiley.

Download references

Author information

Correspondence to Karl Christoph Klauer.

Additional information

The author thanks Bill Batchelder, Edgar Erdfelder, Thorsten Meiser, and Christoph Stahl for helpful comments on a previous version of this paper. The author is also grateful to Edgar Erdfelder for making available the data set analyzed in this paper.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Klauer, K.C. Hierarchical Multinomial Processing Tree Models: A Latent-Class Approach. Psychometrika 71, 7–31 (2006). https://doi.org/10.1007/s11336-004-1188-3

Download citation


  • hierarchical models
  • multinomial processing-trxee models