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Marginal maximum likelihood estimation for a psychometric model of discontinuous development

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Abstract

Item response theory models posit latent variables to account for regularities in students' performances on test items. Wilson's “Saltus” model extends the ideas of IRT to development that occurs in stages, where expected changes can be discontinuous, show different patterns for different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psychological development and Siegler's rule-based learning. This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm. For individual examinees, empirical Bayes probabilities of learning-stage are given, along with proficiency parameter estimates conditional on stage membership. The MML solution is illustrated with simulated data and an example from the domain of mixed number subtraction.

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Authors and Affiliations

  1. Educational Testing Service, 08541, Princeton, NJ

    Robert J. Mislevy

  2. Graduate School of Education, University of California, Berkeley

    Mark Wilson

Authors
  1. Robert J. Mislevy
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  2. Mark Wilson
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Additional information

The authors' names appear in alphabetical order. We would like to thank Karen Draney for computer programming, Kikumi Tatsuoka for allowing us to use the mixed-number subtraction data, and Eric Bradlow, Chan Dayton, Kikumi Tatsuoka, and four anonymous referees for helpful suggestions. The first author's work was supported by Contract No. N00014-88-K-0304, R&T 4421552, from the Cognitive Sciences Program, Cognitive and Neural Sciences Division, Office of Naval Research, and by the Program Research Planning Council of Educational Testing Service. The second author's work was supported by a National Academy of Education Spencer Fellowship and by a Junior Faculty Research Grant from the Committee on Research, University of California at Berkeley. A copy of the Saltus computer program can be obtained from the second author.

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Mislevy, R.J., Wilson, M. Marginal maximum likelihood estimation for a psychometric model of discontinuous development. Psychometrika 61, 41–71 (1996). https://doi.org/10.1007/BF02296958

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  • DOI: https://doi.org/10.1007/BF02296958

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