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Stochastic Processes and the Guttman Simplex

Published online by Cambridge University Press:  01 January 2025

Guy J. Groen
Guy J. Groen*
Affiliation:
Carnegie-Mellon University
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Abstract

The problem of whether a precise connection exists between stochastic processes of the type considered in mathematical learning theory and the Guttman simplex is investigated. The approach is to consider a class of models that characterize the sequential properties of discrete data, and to derive a set of conditions which a model must satisfy in order to generate inter-trial correlations rij with the ‘perfect simplex’ property: rik = rijrjk for all trials i < j < k. It is shown that the Chapman-Kolomogorov Equations provide a necessary and sufficient condition for this property to hold. It follows that a process which is Markovian in the errors and successes will have this property. It is also shown that, if certain reliability assumptions are introduced, then the all-or-none model has the simplex property if the appropriate correlation coefficients are corrected for attenuation.

Type
Original Paper
Information
Psychometrika , Volume 36 , Issue 3 , September 1971 , pp. 289 - 302
Copyright
Copyright © 1971 The Psychometric Society

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Footnotes

*

This research was supported by a grant from the United States Public Health Service, NIMH 07722.

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