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Testing for Measurement Invariance with Respect to an Ordinal Variable


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  • Edgar C. Merkle


  • Jinyan Fan


  • Achim Zeileis



Researchers are often interested in testing for measurement invariance with respect to an ordinal auxiliary variable such as age group, income class, or school grade. In a factor-analytic context, these tests are traditionally carried out via a likelihood ratio test statistic comparing a model where parameters differ across groups to a model where parameters are equal across groups. This test neglects the fact that the auxiliary variable is ordinal, and it is also known to be overly sensitive at large sample sizes. In this paper, we propose test statistics that explicitly account for the ordinality of the auxiliary variable, resulting in higher power against "monotonic" violations of measurement invariance and lower power against "non-monotonic" ones. The statistics are derived from a family of tests based on stochastic processes that have recently received attention in the psychometric literature. The statistics are illustrated via an application involving real data, and their performance is studied via simulation.

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Bibliographic Info

Paper provided by Faculty of Economics and Statistics, University of Innsbruck in its series Working Papers with number 2012-24.

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Length: 25
Date of creation: Oct 2012
Date of revision:
Handle: RePEc:inn:wpaper:2012-24

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Keywords: measurement invariance; ordinal variable; parameter stability; factor analysis; structural equation models;

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  1. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer, Springer, vol. 54(1), pages 131-151, March.
  2. Zeileis, Achim & Leisch, Friedrich & Hornik, Kurt & Kleiber, Christian, 2001. "Strucchange: An R package for testing for structural change in linear regression models," Technical Reports 2001,26, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  3. Achim Zeileis & Kurt Hornik, 2007. "Generalized M-fluctuation tests for parameter instability," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, Netherlands Society for Statistics and Operations Research, vol. 61(4), pages 488-508.
  4. Carolin Strobl & Julia Kopf & Achim Zeileis, 2011. "A new method for detecting differential item functioning in the Rasch model," Working Papers, Faculty of Economics and Statistics, University of Innsbruck 2011-01, Faculty of Economics and Statistics, University of Innsbruck.
  5. Zeileis, Achim, 2006. "Implementing a class of structural change tests: An econometric computing approach," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 50(11), pages 2987-3008, July.
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Cited by:
  1. Ting Wang & Edgar C. Merkle & Achim Zeileis, 2013. "Score-Based Tests of Measurement Invariance: Use in Practice," Working Papers, Faculty of Economics and Statistics, University of Innsbruck 2013-33, Faculty of Economics and Statistics, University of Innsbruck.


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