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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.
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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

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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.

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