Hostname: page-component-7dd5485656-g8tfn Total loading time: 0 Render date: 2025-10-30T03:37:18.380Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Estimation of Multinomial Processing Tree Models with Heterogeneity in Participants and Items

Published online by Cambridge University Press:  01 January 2025

Dora Matzke ,
Conor V. Dolan ,
William H. Batchelder  and
Eric-Jan Wagenmakers
Dora Matzke*
Affiliation:
University of Amsterdam
Conor V. Dolan
Affiliation:
University of Amsterdam
William H. Batchelder
Affiliation:
University of California, Irvine
Eric-Jan Wagenmakers
Affiliation:
University of Amsterdam
*
Requests for reprints should be sent to Dora Matzke, Department of Psychology, University of Amsterdam, Weesperplein 4, 1018 XA, Amsterdam, The Netherlands. E-mail: d.matzke@uva.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

Multinomial processing tree (MPT) models are theoretically motivated stochastic models for the analysis of categorical data. Here we focus on a crossed-random effects extension of the Bayesian latent-trait pair-clustering MPT model. Our approach assumes that participant and item effects combine additively on the probit scale and postulates (multivariate) normal distributions for the random effects. We provide a WinBUGS implementation of the crossed-random effects pair-clustering model and an application to novel experimental data. The present approach may be adapted to handle other MPT models.

Keywords

multinomial processing tree modelparameter heterogeneitycrossed-random effects modelhierarchical Bayesian modeling

Information

Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 1 , March 2015 , pp. 205 - 235
Copyright
Copyright © 2013 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

References

Ashby, F.G., Maddox, W.T., & Lee, W.W. (1994). On the dangers of averaging across subjects when using multidimensional scaling or the similarity-choice model. Psychological Science, 144–151.CrossRefGoogle Scholar
Baayen, R.H. (2008). Analyzing linguistic data: a practical introduction to statistics using R. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Batchelder, W.H. (1975). Individual differences and the all-or-none vs incremental learning controversy. Journal of Mathematical Psychology, 12, 53–74.CrossRefGoogle Scholar
Batchelder, W.H. (1998). Multinomial processing tree models and psychological assessment. Psychological Assessment, 10, 331–344.CrossRefGoogle Scholar
Batchelder, W.H. (2009). Cognitive psychometrics: using multinomial processing tree models as measurement tools. In Embretson, S.E. (Ed.), Measuring psychological constructs: Advances in model based measurement (pp. 71–93). Washington: American Psychological Association.Google Scholar
Batchelder, W.H., & Crowther, C.S. (1997). Multinomial processing tree models of factorial categorization. Journal of Mathematical Psychology, 41, 45–55.CrossRefGoogle Scholar
Batchelder, W.H., & Riefer, D.M. (1980). Separation of storage and retrieval factors in free recall of clusterable pairs. Psychological Review, 87, 375–397.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 & Statistical Psychology, 39, 129–149.CrossRefGoogle Scholar
Batchelder, W.H., & Riefer, D.M. (1990). Multinomial processing models of source monitoring. Psychological Review, 97, 548–564.CrossRefGoogle Scholar
Batchelder, W.H., & Riefer, D.M. (1999). Theoretical and empirical review of multinomial process tree modeling. Psychonomic Bulletin & Review, 6, 57–86.CrossRefGoogle ScholarPubMed
Batchelder, W.H., & Riefer, D.M. (2007). Using multinomial processing tree models to measure cognitive deficits in clinical populations. In Neufeld, R. (Ed.), Advances in clinical cognitive science: formal modeling of processes and symptoms (pp. 19–50). Washington: American Psychological Association.CrossRefGoogle Scholar
Bröder, A., Herwig, A., Teipel, S., & Fast, K. (2008). Different storage and retrieval deficits in normal aging and mild cognitive impairment: a multinomial modeling analysis. Psychology and Aging, 23, 353–365.CrossRefGoogle ScholarPubMed
Brooks, S.B., & Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7, 434–455.CrossRefGoogle Scholar
Clark, H.H. (1973). The language-as-fixed-effect fallacy: a critique of language statistics in psychological research. Journal of Verbal Learning and Verbal Behavior, 12, 335–359.CrossRefGoogle Scholar
Curran, T., & Hintzman, D.L. (1995). Violations of the independence assumption in process dissociation. Journal of Experimental Psychology. Learning, Memory, and Cognition, 21, 531–547.CrossRefGoogle ScholarPubMed
De Boeck, P. (2008). Random item IRT models. Psychometrika, 73, 533–559.CrossRefGoogle Scholar
De Boeck, P., & Partchev, I. (2012). IRTTrees: tree-based item response models of the GLMM family. Journal of Statistical Software, 48, 1–28.CrossRefGoogle Scholar
DeCarlo, L.T. (2002). Signal detection theory with finite mixture distributions: theoretical developments with applications to recognition memory. Psychological Review, 109, 710–721.CrossRefGoogle ScholarPubMed
Deese, J. (1960). Frequency of usage and number of words in free recall: the role of association. Psychological Reports, 337–344.CrossRefGoogle Scholar
DeLosh, E.L., & McDaniel, M.A. (1996). The role of order information in free recall: application to the word–frequency effect. Journal of Experimental Psychology. Learning, Memory, and Cognition, 22, 1136–1146.CrossRefGoogle Scholar
Duncan, C.P. (1974). Retrieval of low-frequency words from mixed lists. Bulletin of the Psychonomic Society, 4, 137–138.CrossRefGoogle Scholar
Erdfelder, E., Auer, T.S., Hilbig, B.E., Aßfalg, A., Moshagen, M., & Nadarevic, L. (2009). Multinomial processing tree models. Zeitschrift für Psychologie, 217, 108–124.CrossRefGoogle Scholar
Estes, W.K. (1956). The problem of inference from curves based on group data. Psychological Bulletin, 53, 134–140.CrossRefGoogle ScholarPubMed
Farrell, S., & Ludwig, C.J.H. (2008). Bayesian and maximum likelihood estimation of hierarchical response time models. Psychonomic Bulletin & Review, 15, 1209–1217.CrossRefGoogle ScholarPubMed
Fischer, G.H., & Molenaar, I.W. (1995). Rasch models: foundations, recent developments, and applications. New York: Springer.CrossRefGoogle Scholar
Gamerman, D., & Lopes, H.F. (2006). Markov chain Monte Carlo: stochastic simulation for Bayesian inference. Boca Raton: Chapman & Hal/CRC..CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2003). Bayesian data analysis. Boca Raton: Chapman & Hall.CrossRefGoogle Scholar
Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge: Cambridge University Press.Google Scholar
Gelman, A., Meng, X., & Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica, 6, 733–807.Google Scholar
Gelman, A., & Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–472.CrossRefGoogle Scholar
Gilks, W.R., Richardson, S., & Spiegelhalter, D.J. (1996). Markov chain Monte Carlo in practice. Boca Raton: Chapman & Hall/CRC..Google Scholar
Gill, J. (2002). Bayesian methods: a social and behavioral sciences approach. New York: Chapman & Hall.CrossRefGoogle Scholar
Golz, D., & Erdfelder, E. (2004). Effekte von L-Dopa auf die Speicherung und den Abruf verbaler Informationen bei Schlaganfallpatienten [Effects of L-Dopa on storage and retrieval of verbal information in stroke patients]. Zeitschrift für Neuropsychologie, 15, 275–286.CrossRefGoogle Scholar
Gregg, V.H. (1976). Word frequency, recognition and recall. In Brown, J. (Ed.), Recall and recognition (pp. 183–216). London: Wiley.Google Scholar
Hall, J.F. (1954). Learning as a function of word-frequency. The American Journal of Psychology, 138–140.CrossRefGoogle Scholar
Heathcote, A., Brown, S., & Mewhort, D.J.K. (2000). The power law repealed: the case for an exponential law of practice. Psychonomic Bulletin & Review, 7, 185–207.CrossRefGoogle ScholarPubMed
Hintze, J.L., & Nelson, R.D. (1998). Violin plots: a box plot-density trace synergism. American Statistician, 52, 181–184.CrossRefGoogle Scholar
Hintzman, D.L. (1980). Simpson’s paradox and the analysis of memory retrieval. Psychological Review, 87, 398–410.CrossRefGoogle Scholar
Hintzman, D.L. (1993). On variability, Simpson’s paradox, and the relation between recognition and recall: reply to Tulving and Flexser. Psychological Review, 100, 143–148.CrossRefGoogle ScholarPubMed
Hu, X., & Batchelder, W.H. (1994). The statistical analysis of general processing tree models with the EM algorithm. Psychometrika, 59, 21–47.CrossRefGoogle Scholar
Hu, X., & Phillips, G.A. (1999). GPT.EXE: a powerful tool for the visualization and analysis of general processing tree models. Behavior Research Methods, 31, 220–234.CrossRefGoogle ScholarPubMed
Karabatsos, G., & Batchelder, W.H. (2003). Markov chain estimation for test theory without an answer key. Psychometrika, 68, 373–389.CrossRefGoogle Scholar
Kass, R.E., & Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
Klauer, K.C. (2006). Hierarchical multinomial processing tree models: a latent-class approach. Psychometrika, 71, 7–31.CrossRefGoogle Scholar
Klauer, K.C. (2010). Hierarchical multinomial processing tree models: a latent-trait approach. Psychometrika, 75, 70–98.CrossRefGoogle Scholar
Kruschke, J.K. (2010). Doing Bayesian data analysis: a tutorial introduction with R and BUGS. Burlington: Academic Press.Google Scholar
Lee, M.D. (2008). Three case studies in the Bayesian analysis of cognitive models. Psychonomic Bulletin & Review, 15, 1–15.CrossRefGoogle ScholarPubMed
Lee, M.D. (2011). How cognitive modeling can benefit from hierarchical Bayesian models. Journal of Mathematical Psychology, 55, 1–7.CrossRefGoogle Scholar
Lee, M.D., & Newell, B.R. (2011). Using hierarchical Bayesian methods to examine the tools of decision-making. Judgment and Decision Making, 6, 832–842.CrossRefGoogle Scholar
Lee, M. D., & Wagenmakers, E.J. (in press). Bayesian modeling for cognitive science: a practical course. Cambridge: Cambridge University Press.Google Scholar
Lee, M.D., & Webb, M.R. (2005). Modeling individual differences in cognition. Psychonomic Bulletin & Review, 12, 605–621.CrossRefGoogle ScholarPubMed
Lord, F.M., & Novick, M.R. (1986). Statistical theories of mental test scores. Reading: Addison-Wesley.Google Scholar
Lunn, D., Jackson, C., Best, N., Thomas, A., & Spiegelhalter, D. (2012). The BUGS book: a practical introduction to Bayesian analysis. Boca Raton: CRC Press/Chapman and Hall.CrossRefGoogle Scholar
Lunn, D., Spiegelhalter, D., Thomas, A., & Best, N. (2009). The BUGS project: Evolution, critique and future directions. Statistics in Medicine, 28, 3049–3067.CrossRefGoogle ScholarPubMed
Lunn, D., Thomas, A., Best, N., & Spiegelhalter, D. (2000). WinBUGS—a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.CrossRefGoogle Scholar
Masson, M.E.J. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing. Behavior Research Methods, 43, 679–690.CrossRefGoogle ScholarPubMed
Merritt, P.S., DeLosh, E.L., & McDaniel, M.A. (2006). Effects of word frequency on individual-item and serial order retention: tests of the order-encoding view. Memory & Cognition, 34, 1615–1627.CrossRefGoogle ScholarPubMed
Moshagen, M. (2010). MultiTree: a computer program for the analysis of multinomial processing tree models. Behavior Research Methods, 42, 42–54.CrossRefGoogle ScholarPubMed
Navarro, D.J., Griffiths, T.L., Steyvers, M., & Lee, M.D. (2006). Modeling individual differences using Dirichlet processes. Journal of Mathematical Psychology, 50, 101–122.CrossRefGoogle Scholar
Nilsson, H., Rieskamp, J., & Wagenmakers, E.J. (2011). Hierarchical Bayesian parameter estimation for cumulative prospect theory. Journal of Mathematical Psychology, 55, 84–93.CrossRefGoogle Scholar
Plummer, M. (2003). JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling [Computer software manual]. Retrieved from http://citeseer.ist.psu.edu/plummer03jags.html.Google Scholar
Postman, L. (1970). Effects of word frequency on acquisition and retention under conditions of free-recall learning. The Quarterly Journal of Experimental Psychology, 22, 185–195.CrossRefGoogle Scholar
Pratte, M.S., & Rouder, J.N. (2011). Hierarchical single-and dual-process models of recognition memory. Journal of Mathematical Psychology, 55, 36–46.CrossRefGoogle Scholar
Purdy, B.P., & Batchelder, W.H. (2009). A context-free language for binary multinomial processing tree models. Journal of Mathematical Psychology, 53, 547–561.CrossRefGoogle Scholar
Raftery, A.E. (1995). Bayesian model selection in social research. Sociological Methodology, 25, 111–164.CrossRefGoogle Scholar
Raftery, A.E. (1999). Bayes factors and BIC. Sociological Methods & Research, 27, 411–417.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). Statistical inference for multinomial processing tree models. In Doignon, J.P., & Falmagne, J.C.G. (Eds.), Mathematical psychology: current developments (pp. 313–335). New York: Springer.CrossRefGoogle Scholar
Riefer, D.M., Knapp, B.R., Batchelder, W.H., Bamber, D., & Manifold, V. (2002). Cognitive psychometrics: assessing storage and retrieval deficits in special populations with multinomial processing tree models. Psychological Assessment, 14, 184–200.CrossRefGoogle ScholarPubMed
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
Rouder, J.N., Lu, J., Sun, D., Speckman, P., Morey, R., & Naveh-Benjamin, M. (2007). Signal detection models with random participant and item effects. Psychometrika, 72, 621–642.CrossRefGoogle Scholar
Rouder, J.N., Sun, D., Speckman, P.L., Lu, J., & Zhou, D. (2003). A hierarchical Bayesian statistical framework for response time distributions. Psychometrika, 68, 589–606.CrossRefGoogle Scholar
Schmittmann, V., Dolan, C., Raijmakers, M., & Batchelder, W.H. (2010). Parameter identification in multinomial processing tree models. Behavior Research Methods, 42, 836–846.CrossRefGoogle ScholarPubMed
Sheu, C., & O’Curry, S.L. (1998). Simulation-based Bayesian inference using BUGS. Behavior Research Methods, 30, 232–237.CrossRefGoogle Scholar
Shiffrin, R.M., Lee, M.D., Kim, W., & Wagenmakers, E.J. (2008). A survey of model evaluation approaches with a tutorial on hierarchical Bayesian methods. Cognitive Science, 32, 1248–1284.CrossRefGoogle ScholarPubMed
Smith, J.B., & Batchelder, W.H. (2008). Assessing individual differences in categorical data. Psychonomic Bulletin & Review, 15, 713–731.CrossRefGoogle ScholarPubMed
Smith, J.B., & Batchelder, W.H. (2010). Beta-MPT: multinomial processing tree models for addressing individual differences. Journal of Mathematical Psychology, 54, 167–183.CrossRefGoogle Scholar
Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R., & Lunn, D. (2003). BUGS: Bayesian inference using Gibbs sampling [Computer software manual]. Retrieved from http://www.mrc-bsu.cam.ac.uk/bugs/.Google 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
Stan Development Team (2012). Stan modeling language [Computer software manual]. Retrieved from http://mc-stan.org/.Google Scholar
Sumby, W.H. (1963). Word frequency and serial position effects. Journal of Verbal Learning and Verbal Behavior, 1, 443–450.CrossRefGoogle Scholar
Wagenmakers, E.J. (2007). A practical solution to the pervasive problems of p values. Psychonomic Bulletin & Review, 14, 779–804.CrossRefGoogle Scholar
Wickelmaier, F. (2011). Mpt: multinomial processing tree (MPT) models [Computer software manual]. Retrieved from http://cran.r-project.org/web/packages/mpt/index.html.Google Scholar