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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Batchelder, W. H. (1991). Getting wise about minimum distance measures [Review ofGoodness-of-fit statistics for discrete multivariate data by T. R. C. Read & N. A. C. Cressie].Journal of Mathematical Psychology, 35, 267–273.
Batchelder, W. H., Hu, X., & Riefer, D. M. (in press) Analysis of a model for source monitoring. In G. H. Fischer & D. Laming (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)
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.
Batchelder, W. H., & Riefer, D. M. (1990). Multinomial processing models of source monitoring.Psychological Review, 97, 548–564.
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.
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.
Boyles, R. A. (1983). On the convergence of the EM algorithm.Journal of Royal Statistical Society, Series B, 45, 47–50.
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.
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.
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.
Efron, B., & Hinkley, D. V. (1978). The observed versus expected information.Biometrika, 65, 457–487.
Elandt-Johnson, R. C. (1971).Probability models and statistical methods in genetics. New York, Wiley & Sons.
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 D. Frey (Eds.),Bericht über den 37. Kongreß der Deutschen Gesellschaft für Psychologie in Kiel 1990, Band 2 (pp. 172–180). Göttingen: Hogrefe.
Hartley, H. O. (1958). Maximum likelihood estimation from incomplete data.Biometrics, 14, 174–194.
Harvey, P. D. (1985). Reality monitoring in mania and schizophrenia.The Journal of Nervous and Mental Disease, 173, 67–72.
Hu, X. (1990).Source monitoring program (Version 1.0). Irvine: University of California. (available upon request)
Hu, X. (1991).General program for processing tree models (Version 1.0). Irvine: University of California. (available upon request)
Humphreys, M. S., & Bain, J. D. (1983). Recognition memory: A cue and information analysis.Memory and Cognition, 11, 583–600.
Johnson, M. K., & Raye, C. L. (1980). Reality monitoring.Psychological Review.88, 67–85.
Landsteiner, K. (1901). Über Agglutinationserscheinungen normalen menschlichen Blutes. [On agglutination appearances of normal human blood]Wien. Klin. Wschr., 14, 1132–1134.
Lazarsfeld, P. F., & Henry, N. W. (1968).Latent structure analysis. New York: Houghton Mifflin.
Little, R. J. A., & Rubin, D. B. (1987).Statistical analysis with missing data. New York: Wiley.
Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm.Journal of Royal Statistical Society, Series B, 44, 226–233.
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.
Read, T. R. C., & Cressie, N. A. C. (1988).Goodness-of-fit statistics for discrete multivariate data. New York: Springer-Verlag.
Riefer, D. M., & Batchelder, W. H. (1988). Multinomial modeling and the measurement of cognitive processes.Psychological Review, 95, 318–339.
Riefer, D. M., & Batchelder, W. H. (1991). Statistical inference for multinomial processing tree models. In J.-P. Doignon & J.-C. Falmagne (Eds.),Mathematical psychology: Current developments (pp. 313–335). New York: Springer-Verlag.
Riefer, D. M., & Rouder, J. M. (1992). A multinomial modeling analysis of the mnemonic benefits of bizarre imagery.Memory and Cognition, 20, 601–611.
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.
Ross, B. H., & Bower, G. H. (1981). Comparisons of models of associative recall.Memory & Cognition, 9, 1–16.
Rubin, D. B. (1991). EM and beyond.Psychometrika, 56, 241–254.
Rumelhart, D. E., & McClelland, J. L. (1986).Parallel distributed processing (Vol. 1). Cambridge: MIT Press.
Ruud, P. A. (1991). Extensions of estimation methods using the EM algorithm.Journal of Econometrics, 49, 305–341.
Smith, C. A. B. (1957). Counting methods in genetical statistics.Annals of Human Genetics, 21, 97–115.
Wickens, T. D. (1982).Models for behavior: Stochastic processes in psychology. San Francisco: Freeman.
Weir, B. S. (1990).Genetic data analysis. Sunderland, MA: Sinaver Associates.
Wu, J. (1983). On the convergence properties of the EM algorithm.The Annals of Statistics, 11, 95–103.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
Hu, X., Batchelder, W.H. The statistical analysis of general processing tree models with the EM algorithm. Psychometrika 59, 21–47 (1994). https://doi.org/10.1007/BF02294263
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02294263