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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

    ()

Abstract

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.

Suggested Citation

  • Edgar C. Merkle & Jinyan Fan & Achim Zeileis, 2012. "Testing for Measurement Invariance with Respect to an Ordinal Variable," Working Papers 2012-24, Faculty of Economics and Statistics, University of Innsbruck.
  • Handle: RePEc:inn:wpaper:2012-24
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    References listed on IDEAS

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    1. Zeileis, Achim & Leisch, Friedrich & Hornik, Kurt & Kleiber, Christian, 2002. "strucchange: An R Package for Testing for Structural Change in Linear Regression Models," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 7(i02).
    2. Zeileis, Achim, 2006. "Implementing a class of structural change tests: An econometric computing approach," Computational Statistics & Data Analysis, Elsevier, pages 2987-3008.
    3. Achim Zeileis & Kurt Hornik, 2007. "Generalized M-fluctuation tests for parameter instability," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 61(4), pages 488-508.
    4. Conor Dolan & Han Maas, 1998. "Fitting multivariage normal finite mixtures subject to structural equation modeling," Psychometrika, Springer;The Psychometric Society, pages 227-253.
    5. Zeileis, Achim, 2006. "Object-oriented Computation of Sandwich Estimators," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 16(i09).
    6. Torsten Hothorn & Achim Zeileis, 2008. "Generalized Maximally Selected Statistics," Biometrics, The International Biometric Society, vol. 64(4), pages 1263-1269, December.
    7. Albert Satorra & Peter Bentler, 2001. "A scaled difference chi-square test statistic for moment structure analysis," Psychometrika, Springer;The Psychometric Society, pages 507-514.
    8. Carolin Strobl & Julia Kopf & Achim Zeileis, 2011. "A new method for detecting differential item functioning in the Rasch model," Working Papers 2011-01, Faculty of Economics and Statistics, University of Innsbruck.
    9. Edgar Merkle & Achim Zeileis, 2013. "Tests of Measurement Invariance Without Subgroups: A Generalization of Classical Methods," Psychometrika, Springer;The Psychometric Society, pages 59-82.
    10. Rosseel, Yves, 2012. "lavaan: An R Package for Structural Equation Modeling," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i02).
    11. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer;The Psychometric Society, pages 131-151.
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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 2013-33, Faculty of Economics and Statistics, University of Innsbruck.

    More about this item

    Keywords

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

    JEL classification:

    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General
    • C38 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Classification Methdos; Cluster Analysis; Principal Components; Factor Analysis
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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