Skip to main content
SpringerLink
Log in

Hierarchical Multinomial Processing Tree Models: A Latent-Trait Approach

  • Theory and Methods
  • Published:
Psychometrika Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Albert, J.H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.

    Article  Google Scholar 

  • Ansari, A., Vanhuele, M., & Zemborain, M. (2008). Heterogeneous multinomial processing tree models. Unpublished manuscript. Columbia University, New York.

  • Atchison, J., & Shen, S.M. (1980). Logistic-normal distributions: Some properties and uses. Biometrika, 67, 261–272.

    Article  Google 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.

    Google Scholar 

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

    Google Scholar 

  • Batchelder, W.H., & Riefer, D.M. (2007). Using multinomial processing tree models to measure cognitive deficits in clinical populations. In R.W.T. Neufeld (Ed.), Advances in clinical cognitive science: formal analysis of processes and symptoms (pp. 19–50). Washington: American Psychological Association Books.

    Chapter  Google Scholar 

  • Bayarri, M.J., & Berger, J.O. (1999). Quantifying surprise in the data and model verification. Bayesian Statistics, 6, 53–83.

    Google Scholar 

  • Boomsma, A., van Duijn, M.A.J., & Snijders, T.A.B. (Eds.) (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.

    Google Scholar 

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 Unpublished habilitation thesis, Psychologisches Institut der Universität Bonn, Germany

  • 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.

    Article  Google Scholar 

  • Karabatsos, G., & Batchelder, W.H. (2003). Markov chain estimation methods for test theory without an answer key. Psychometrika, 68, 373–389.

    Article  Google Scholar 

  • Klauer, K.C. (2006). Hierarchical multinomial processing tree models: a latent-class approach. Psychometrika, 71, 1–31.

    Article  Google 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.

    Book  Google Scholar 

  • Riefer, D.M., & Batchelder, W.H. (1988). Multinomial modeling and the measurement of cognitive processes. Psychological Review, 95, 318–339.

    Article  Google 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.

    Google Scholar 

  • Robert, P.C. (1995). Simulation of truncated normal variables. Statistics and Computing, 5, 121–125.

    Article  Google Scholar 

  • Raudenbush, S.W., & Bryk, A.S. (2002). In Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks: Sage Publications.

    Google 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Smith, J.B., & Batchelder, H.W. (2008). Assessing individual differences in categorical data. Psychonomic Bulletin & Review, 15, 713–731.

    Article  Google Scholar 

  • Smith, J., & Batchelder, W. (2009, in press). Beta-MPT: Multinomial processing tree models for addressing individual differences. Journal of Mathematical Psychology.

  • Stahl, C., & Klauer, K.C. (2007). HMMTree: A computer program for latent-class hierarchical multinomial processing tree models. Behavior Research Methods, 39, 267–273.

    PubMed  Google Scholar 

  • Stefanescu, C., Berger, V.W., & Hershberger, S. (2005). Probits. In B.S. Everitt & D.C. Howell (Eds.), The encyclopedia of statistics in behavioral science (Vol. 4, 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.

    Article  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.

    Article  Google 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.

Download references

Author information

Authors and Affiliations

  1. Institut für Psychologie, Universität Freiburg, 79085, Freiburg, Germany

    Karl Christoph Klauer

Authors
  1. Karl Christoph Klauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl Christoph Klauer.

Additional information

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

Rights and permissions

Reprints and permissions

About this article

Cite this article

Klauer, K.C. Hierarchical Multinomial Processing Tree Models: A Latent-Trait Approach. Psychometrika 75, 70–98 (2010). https://doi.org/10.1007/s11336-009-9141-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11336-009-9141-0

Keywords

Search

Navigation