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Marginal likelihood inference for a model for item responses and response times

Authors

  • Cees A. W. Glas,

    Corresponding author
    1. University of Twente, Enschede, The Netherlands
      Correspondence should be addressed to Cees A. W. Glas, Department of Measurement and Data Analysis, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands (e-mail: c.a.w.glas@gw.utwente.nl).
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  • Wim J. van der Linden

    1. CTB/McGraw-Hill, Monterey, California, USA
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Correspondence should be addressed to Cees A. W. Glas, Department of Measurement and Data Analysis, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands (e-mail: c.a.w.glas@gw.utwente.nl).

Abstract

Marginal maximum-likelihood procedures for parameter estimation and testing the fit of a hierarchical model for speed and accuracy on test items are presented. The model is a composition of two first-level models for dichotomous responses and response times along with multivariate normal models for their item and person parameters. It is shown how the item parameters can easily be estimated using Fisher's identity. To test the fit of the model, Lagrange multiplier tests of the assumptions of subpopulation invariance of the item parameters (i.e., no differential item functioning), the shape of the response functions, and three different types of conditional independence were derived. Simulation studies were used to show the feasibility of the estimation and testing procedures and to estimate the power and Type I error rate of the latter. In addition, the procedures were applied to an empirical data set from a computerized adaptive test of language comprehension.

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