Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-08T08:01:02.746Z Has data issue: false hasContentIssue false
  • English
  • Français

The Statistical Analysis of General Processing Tree Models with the EM Algorithm

Published online by Cambridge University Press:  01 January 2025

Xiangen Hu  and
William H. Batchelder
Xiangen Hu
Affiliation:
University of California, Irvine
William H. Batchelder*
Affiliation:
University of California, Irvine
*
Requests for reprints and computer programs should be sent to William H. Batchelder, School of Social Sciences, University of California, Irvine, CA 92717.
Get access Rights & Permissions [Opens in a new window]

Abstract

Multinomial processing tree models assume that an observed behavior category can arise from one or more processing sequences represented as branches in a tree. These models form a subclass of parametric, multinomial models, and they provide a substantively motivated alternative to loglinear models. We consider the usual case where branch probabilities are products of nonnegative integer powers in the parameters, 0≤θs≤1, and their complements, 1 - θs. A version of the EM algorithm is constructed that has very strong properties. First, the E-step and the M-step are both analytic and computationally easy; therefore, a fast PC program can be constructed for obtaining MLEs for large numbers of parameters. Second, a closed form expression for the observed Fisher information matrix is obtained for the entire class. Third, it is proved that the algorithm necessarily converges to a local maximum, and this is a stronger result than for the exponential family as a whole. Fourth, we show how the algorithm can handle quite general hypothesis tests concerning restrictions on the model parameters. Fifth, we extend the algorithm to handle the Read and Cressie power divergence family of goodness-of-fit statistics. The paper includes an example to illustrate some of these results.

Keywords

EM algorithmmultinomial modelsprocessing treespower divergence family
Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 1 , March 1994 , pp. 21 - 47
Copyright
Copyright © 1994 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

This research was supported by National Science Foundation Grant BNS-8910552 to William H. Batchelder and David M. Riefer. We are grateful to David Riefer for his useful comments, and to the Institute for Mathematical Behavior Sciences for its support.

References

Batchelder, W. H. (1991). Getting wise about minimum distance measures [Review of Goodness-of-fit statistics for discrete multivariate data by T. R. C. Read & N. A. C. Cressie]. Journal of Mathematical Psychology, 35, 267–273.CrossRefGoogle Scholar
Batchelder, W. H., Hu, X., & Riefer, D. M. (in press) Analysis of a model for source monitoring. In Fischer, G. H. & Laming, D. (Eds.), Mathematical psychology: New developments. Berlin: Springer-Verlag. (Available as Technical Report No. 92-07, Institute for Mathematical Behavior Sciences, School of Social Sciences, UC, Irvine)Google Scholar
Batchelder, W. H., Riefer, D. M. (1986). 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. (1990). Multinomial processing models of source monitoring. Psychological Review, 97, 548–564.CrossRefGoogle Scholar
Bäuml, K.-H. (1991). Experimental analysis of storage and retrieval processes involved in retroactive inhibition: The effect of presentation mode. Acta Psychologica, 77(2), 103–119.CrossRefGoogle Scholar
Bernstein, F. (1925). Zusammenfassende Betrachtungen über die erblichen Blutenstructuren des Menschen [Summarizing considerations on the inheritable blood structures of mankind]. Z. Abstamm. Vererbgsl., 37, 237–270.Google Scholar
Boyles, R. A. (1983). On the convergence of the EM algorithm. Journal of Royal Statistical Society, Series B, 45, 47–50.CrossRefGoogle Scholar
Ceppellini, R., Siniscalco, M., Smith, C. A. B. (1955). The estimation of gene frequencies in random mating populations. Annals of Human Genetics, 20, 97–115.CrossRefGoogle Scholar
Chechile, R., Meyer, D. L. (1976). A Bayesian procedure for separately estimating storage and retrieval components of forgetting. Journal of Mathematical Psychology, 13, 269–295.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M., Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
Efron, B., Hinkley, D. V. (1978). The observed versus expected information. Biometrika, 65, 457–487.CrossRefGoogle Scholar
Elandt-Johnson, R. C. (1971). Probability models and statistical methods in genetics, New York: Wiley & Sons.Google Scholar
Erdfelder, E., Bayen, U. J. (1991). Episodisches Gedächtnis im Alter: Methodologische und empirische Arguments für einen Zugang über mathematische Modelle [Episodic memory in old age: Methodological and empirical arguments for an access through mathematical models]. In Frey, D. (Eds.), Bericht über den 37. Kongreß der Deutschen Gesellschaft für Psychologie in Kiel 1990, Band 2 (pp. 172–180). Göttingen: Hogrefe.Google Scholar
Hartley, H. O. (1958). Maximum likelihood estimation from incomplete data. Biometrics, 14, 174–194.CrossRefGoogle Scholar
Harvey, P. D. (1985). Reality monitoring in mania and schizophrenia. The Journal of Nervous and Mental Disease, 173, 67–72.CrossRefGoogle ScholarPubMed
Hu, X. (1990). Source monitoring program (Version 1.0), Irvine: University of California (available upon request)Google Scholar
Hu, X. (1991). General program for processing tree models (Version 1.0), Irvine: University of California (available upon request)Google Scholar
Humphreys, M. S., Bain, J. D. (1983). Recognition memory: A cue and information analysis. Memory and Cognition, 11, 583–600.CrossRefGoogle ScholarPubMed
Johnson, M. K., Raye, C. L. (1980). Reality monitoring. Psychological Review, 88, 67–85.CrossRefGoogle Scholar
Landsteiner, K. (1901). Über Agglutinationserscheinungen normalen menschlichen Blutes [On agglutination appearances of normal human blood]. Wien. Klin. Wschr., 14, 1132–1134.Google Scholar
Lazarsfeld, P. F., Henry, N. W. (1968). Latent structure analysis, New York: Houghton Mifflin.Google Scholar
Little, R. J. A., Rubin, D. B. (1987). Statistical analysis with missing data, New York: Wiley.Google Scholar
Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of Royal Statistical Society, Series B, 44, 226–233.CrossRefGoogle Scholar
Meng, X. L., Rubin, D. B. (1991). Using EM to obtain asymptotic variance-covariance matrices: The SEM algorithm. Journal of the American Statistical Association, 86, 899–909.CrossRefGoogle Scholar
Read, T. R. C., Cressie, N. A. C. (1988). Goodness-of-fit statistics for discrete multivariate data, New York: Springer-Verlag.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. (Eds.), Mathematical psychology: Current developments (pp. 313–335). New York: Springer-Verlag.CrossRefGoogle Scholar
Riefer, D. M., Rouder, J. M. (1992). A multinomial modeling analysis of the mnemonic benefits of bizarre imagery. Memory and Cognition, 20, 601–611.CrossRefGoogle ScholarPubMed
Rosenbloom, P. S., Laird, J. E., Newell, A., McCarl, R. (1991). A preliminary analysis of the SOAR architecture as a basis for general intelligence. Artificial Intelligence, 47, 289–325.CrossRefGoogle Scholar
Ross, B. H., Bower, G. H. (1981). Comparisons of models of associative recall. Memory & Cognition, 9, 1–16.CrossRefGoogle ScholarPubMed
Rubin, D. B. (1991). EM and beyond. Psychometrika, 56, 241–254.CrossRefGoogle Scholar
Rumelhart, D. E., McClelland, J. L. (1986). Parallel distributed processing (Vol. 1), Cambridge: MIT Press.CrossRefGoogle Scholar
Ruud, P. A. (1991). Extensions of estimation methods using the EM algorithm. Journal of Econometrics, 49, 305–341.CrossRefGoogle Scholar
Smith, C. A. B. (1957). Counting methods in genetical statistics. Annals of Human Genetics, 21, 97–115.CrossRefGoogle ScholarPubMed
Wickens, T. D. (1982). Models for behavior: Stochastic processes in psychology, San Francisco: Freeman.Google Scholar
Weir, B. S. (1990). Genetic data analysis, Sunderland, MA: Sinaver Associates.Google Scholar
Wu, J. (1983). On the convergence properties of the EM algorithm. The Annals of Statistics, 11, 95–103.CrossRefGoogle Scholar