Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-07T19:25:48.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Hierarchical Multinomial Processing Tree Models: A Latent-Trait Approach

Published online by Cambridge University Press:  01 January 2025

Karl Christoph Klauer
Karl Christoph Klauer*
Affiliation:
Albert-Ludwigs-Universität Freiburg
*
Requests for reprints should be sent to Karl Christoph Klauer, Institut für Psychologie, Universität Freiburg, 79085 Freiburg, Germany. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Multinomial processing tree models are widely used in many areas of psychology. A hierarchical extension of the model class is proposed, using a multivariate normal distribution of person-level parameters with the mean and covariance matrix to be estimated from the data. The hierarchical model allows one to take variability between persons into account and to assess parameter correlations. The model is estimated using Bayesian methods with weakly informative hyperprior distribution and a Gibbs sampler based on two steps of data augmentation. Estimation, model checks, and hypotheses tests are discussed. The new method is illustrated using a real data set, and its performance is evaluated in a simulation study.

Keywords

multinomial processing tree modelshierarchical modelsGibbs sampler
Type
Theory and Methods
Information
Psychometrika , Volume 75 , Issue 1 , March 2010 , pp. 70 - 98
Copyright
Copyright © 2010 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.)

Footnotes

The research reported in this paper was supported by grant Kl 614/31-1 from the Deutsche Forschungsgemeinschaft.

References

Albert, J.H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.CrossRefGoogle Scholar
Ansari, A., Vanhuele, M., & Zemborain, M. (2008). Heterogeneous multinomial processing tree models. Unpublished manuscript. Columbia University, New York.Google Scholar
Atchison, J., & Shen, S.M. (1980). Logistic-normal distributions: Some properties and uses. Biometrika, 67, 261–272.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Batchelder, W.H., & Riefer, D.M. (2007). Using multinomial processing tree models to measure cognitive deficits in clinical populations. In Neufeld, R.W.T. (Eds.), Advances in clinical cognitive science: formal analysis of processes and symptoms (pp. 19–50). Washington: American Psychological Association Books.Google Scholar
Bayarri, M.J., & Berger, J.O. (1999). Quantifying surprise in the data and model verification. Bayesian Statistics, 6, 53–83.CrossRefGoogle Scholar
Boomsma, A., van Duijn, M.A.J., & Snijders, T.A.B. (2000). Essays on item response theory, New York: Springer.Google Scholar
Efron, B. (1982). The jackknife, the bootstrap and other resampling plans, Philadelphia: 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, GermanyGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2004). Bayesian data analyses, (2nd ed.). Boca Raton: Chapman & Hall/CRC.Google Scholar
Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models, New York: Cambridge University Press.Google Scholar
Gill, J. (2008). Bayesian methods: A social and behavioral sciences approach, (2nd ed.). Boca Raton: Chapman & Hall/CRC.Google Scholar
Hu, X., & Batchelder, W.H. (1994). The statistical analysis of general processing tree models with the EM algorithm. Psychometrika, 59, 21–47.CrossRefGoogle Scholar
Karabatsos, G., & Batchelder, W.H. (2003). Markov chain estimation methods for test theory without an answer key. Psychometrika, 68, 373–389.CrossRefGoogle Scholar
Klauer, K.C. (2006). Hierarchical multinomial processing tree models: a latent-class approach. Psychometrika, 71, 1–31.CrossRefGoogle Scholar
O’Hagan, A., & Forster, J. (2004). Kendall’s advanced theory of statistics: Volume 2B: Bayesian inference, London: Arnold.Google Scholar
Rao, C.R. (1973). Linear statistical inference and its applications, New York: Wiley.CrossRefGoogle Scholar
Riefer, D.M., & Batchelder, W.H. (1988). Multinomial modeling and the measurement of cognitive processes. Psychological Review, 95, 318–339.CrossRefGoogle Scholar
Riefer, D.M., & Batchelder, W.H. (1991). Age differences in storage and retrieval: a multinomial modeling analysis. Bulletin of the Psychonomic Society, 29, 415–418.CrossRefGoogle Scholar
Robert, P.C. (1995). Simulation of truncated normal variables. Statistics and Computing, 5, 121–125.CrossRefGoogle Scholar
Raudenbush, S.W., & Bryk, A.S. (2002). Hierarchical linear models: Applications and data analysis methods, (2nd ed.). Thousand Oaks: Sage. PublicationsGoogle Scholar
Rouder, J.N., & Lu, J. (2005). An introduction to Bayesian hierarchical models with an application in the theory of signal detection. Psychonomic Bulletin & Review, 12, 573–604.CrossRefGoogle ScholarPubMed
Rouder, J.N., Lu, J., Morey, R.D., Sun, D., & Speckman, P.L. (2008). A hierarchical process-dissociation model. Journal of Experimental Psychology: General, 137, 370–389.CrossRefGoogle ScholarPubMed
Smith, J.B., & Batchelder, H.W. (2008). Assessing individual differences in categorical data. Psychonomic Bulletin & Review, 15, 713–731.CrossRefGoogle ScholarPubMed
Smith, J., & Batchelder, W. (2009, in press). Beta-MPT: Multinomial processing tree models for addressing individual differences. Journal of Mathematical Psychology.CrossRefGoogle Scholar
Stahl, C., & Klauer, K.C. (2007). HMMTree: A computer program for latent-class hierarchical multinomial processing tree models. Behavior Research Methods, 39, 267–273.CrossRefGoogle ScholarPubMed
Stefanescu, C., Berger, V.W., & Hershberger, S. (2005). Probits. In Everitt, B.S., & Howell, D.C. (Eds.), The encyclopedia of statistics in behavioral science (pp. 1608–1610). Chichester: Wiley.Google Scholar
Tuyl, F., Gerlach, R., & Mengersen, K. (2009). Posterior predictive arguments in favor of the Bayes-Laplace prior as the consensus prior for binomial and multinomial parameters. Bayesian Analysis, 4, 151–158.Google Scholar
Wei, G.C.G., & Tanner, M.A. (1990). A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. Journal of the American Statistical Association, 85, 699–704.CrossRefGoogle Scholar
Zhu, M., & Lu, A.Y. (2004). The counter-intuitive non-informative prior for the Bernoulli family. Journal of Statistics Education, 12. Retrieved June 19, 2009, from http://www.amstat.org/publications/jse/v12n2/zhu.pdf.Google Scholar