Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-10T20:28:22.028Z Has data issue: false hasContentIssue false
  • English
  • Français

Hierarchical Multinomial Processing Tree Models: A Latent-Class Approach

Published online by Cambridge University Press:  01 January 2025

Karl Christoph Klauer
Karl Christoph Klauer*
Affiliation:
Albert-Ludwigs-Universität Freiburg, Germany
*
Requests for reprints should be sent to K. C. Klauer, Institut für Psychologie, Albert-Ludwigs-Universität Freiburg, D-79085 Freiburg, Germany. E-mail: christoph.klauer@psychologie.uni-freiburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

hierarchical modelsmultinomial processing-trxee models

Information

Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 1 , March 2006 , pp. 7 - 31
Copyright
Copyright © 2006 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

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.

References

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.CrossRefGoogle Scholar
Batchelder, W.H., & Riefer, D.M. (1999). Theoretical and empirical review of multinomial processing tree modeling. Psychonomic Bulletin & Review, 6, 57–86.10.3758/BF03210812CrossRefGoogle Scholar
Bishop, Y. M.M., Fienberg, S.E., & Holland, P.W. (1975). Discrete multivariate analysis: Theory and practice, Cambridge, MA: MIT Press.Google Scholar
Bollen, K.A., & Stine, R.A. (1993). Bootstrapping goodness-of-fit measures in structural equation models. In Bollen, K.A., & Long, J.S. (Eds.), Testing structural equation models (pp. 111–135). Newbury Park, CA: Sage.Google Scholar
Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Chen, J. (1998). Penalized likelihood-ratio test for finite mixture models with multinomial observations. Canadian Journal of Statistics, 26, 583–599.CrossRefGoogle Scholar
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, 1–38.10.1111/j.2517-6161.1977.tb01600.xCrossRefGoogle Scholar
Efron, B. (1982). The jackknife, the bootstrap and other resampling plans, Philadelphia, PA: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Erdfelder, E. (2000). Multinomiale Modelle in der kognitiven Psychologie [Multinomial models in cognitive psychology]. Unpublished habilitation thesis, Psychologisches Institut der Universität Bonn, Germany.Google Scholar
Hu, X. (1991). Statistical inference program for multinomial binary tree models [Computer software]. University of California at Irvine.Google Scholar
Hu, X. (1998). GPT – HomePage [Computer software and documentation]. Retrieved from http://xhuoffice.psyc.memphis.edu/gpt/.Google Scholar
Hu, X., & Batchelder, W.H. (1994). The statistical analysis of general processing tree models with the EM algorithm. Psychometrika, 59, 21–47.10.1007/BF02294263CrossRefGoogle Scholar
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1997). Discrete multivariate distributions, New York: Wiley.Google Scholar
Johnson, N.L., Kotz, S., & Kemp, A.W. (1993). Univariate discrete distributions, (2nd ed.). New York: Wiley.Google Scholar
Kaplan, D. (2000). Structural equation modeling: Foundations and extensions, Thousand Oaks, CA: Sage.Google Scholar
Linhart, H., & Zuccini, W. (1986). Model selection, New York: Wiley.Google Scholar
Magnus, J.R., & Neudecker, H. (1988). Matrix differential calculus with applications in statistics and econometrics, Chichester, UK: Wiley.Google Scholar
McLachlan, G., & Peel, D. (2000). Finite mixture models, New York: Wiley.10.1002/0471721182CrossRefGoogle Scholar
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, 131–137.10.1080/01621459.1977.10479921CrossRefGoogle Scholar
Muthén, B. (1993). Goodness of fit with categorical and other non-normal variables. In Bollen, K., Long, J.S. (Eds.), Testing structural equation models (pp. 205–234). Newbury Park, CA: Sage.Google Scholar
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, 12–35.10.1080/10618600.1995.10474663CrossRefGoogle Scholar
Powell, M.J.D. (1977). Restart procedures for the conjugate gradient method. Mathematical Programming, 12, 241–254.10.1007/BF01593790CrossRefGoogle Scholar
Rao, C.R. (1973). Linear statistical inference and its applications, New York: Wiley.10.1002/9780470316436CrossRefGoogle Scholar
Raudenbush, S.W., & Bryk, A.S. (2002). Hierarchical linear models. Applications and data analysis methods, Thousand Oaks, CA: Sage.Google Scholar
Riefer, D.M., & Batchelder, W.H. (1991). Statistical inference for multinomial processing tree models. In Doignon, J.-P., & Falmagne, J.-C. (Eds.), Mathematical psychology: Current developments, New York: Springer-Verlag.Google Scholar
Rothkegel, R. (1999). AppleTree: A multinomial processing tree modeling program for Macintosh computers. Behavior Research Methods, Instruments, & Computers, 31, 696–700.CrossRefGoogle ScholarPubMed
Satorra, A. (1992). Asymptotic robust inference in the analysis of mean and covariance structures. In Marsden, P.V. (Eds.), Sociological methodology 1992 (pp. 249–278). Oxford, UK: Blackwell.Google Scholar
Titterington, D.M., Smith, A.F.M., & Makov, U.E. (1985). Statistical analysis of finite mixture distributions, New York: Wiley.Google Scholar