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Testing for Measurement Invariance with Respect to an Ordinal Variable

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  • Edgar 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. Copyright The Psychometric Society 2014

Suggested Citation

  • Edgar Merkle & Jinyan Fan & Achim Zeileis, 2014. "Testing for Measurement Invariance with Respect to an Ordinal Variable," Psychometrika, Springer;The Psychometric Society, vol. 79(4), pages 569-584, October.
    • Handle: RePEc:spr:psycho:v:79:y:2014:i:4:p:569-584
    DOI: 10.1007/s11336-013-9376-7
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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, vol. 50(11), pages 2987-3008, July.
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    5. Torsten Hothorn & Achim Zeileis, 2008. "Generalized Maximally Selected Statistics," Biometrics, The International Biometric Society, vol. 64(4), pages 1263-1269, December.
    6. Rosseel, Yves, 2012. "lavaan: An R Package for Structural Equation Modeling," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i02).
    7. Zeileis, Achim, 2006. "Object-oriented Computation of Sandwich Estimators," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 16(i09).
    8. Albert Satorra & Peter Bentler, 2001. "A scaled difference chi-square test statistic for moment structure analysis," Psychometrika, Springer;The Psychometric Society, vol. 66(4), pages 507-514, December.
    9. 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, Universität Innsbruck.
    10. Edgar Merkle & Achim Zeileis, 2013. "Tests of Measurement Invariance Without Subgroups: A Generalization of Classical Methods," Psychometrika, Springer;The Psychometric Society, vol. 78(1), pages 59-82, January.
    11. Albert Satorra, 1989. "Alternative test criteria in covariance structure analysis: A unified approach," Psychometrika, Springer;The Psychometric Society, vol. 54(1), pages 131-151, March.
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    Cited by:

    1. Felix Zimmer & Clemens Draxler & Rudolf Debelak, 2023. "Power Analysis for the Wald, LR, Score, and Gradient Tests in a Marginal Maximum Likelihood Framework: Applications in IRT," Psychometrika, Springer;The Psychometric Society, vol. 88(4), pages 1249-1298, December.
    2. 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, Universität Innsbruck.
    3. Gutiérrez-Vargas, Álvaro A. & Meulders, Michel & Vandebroek, Martina, 2023. "Modeling preference heterogeneity using model-based decision trees," Journal of choice modelling, Elsevier, vol. 46(C).
    4. Ting Wang & Benjamin Graves & Yves Rosseel & Edgar C. Merkle, 2022. "Computation and application of generalized linear mixed model derivatives using lme4," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 1173-1193, September.
    5. Jones, Payton J. & Mair, Patrick & Simon, Thorsten & Zeileis, Achim, 2019. "Network Model Trees," OSF Preprints ha4cw, Center for Open Science.
    6. Ting Wang & Carolin Strobl & Achim Zeileis & Edgar C. Merkle, 2018. "Score-Based Tests of Differential Item Functioning via Pairwise Maximum Likelihood Estimation," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 132-155, March.
    7. K. B. S. Huth & L. J. Waldorp & J. Luigjes & A. E. Goudriaan & R. J. Holst & M. Marsman, 2022. "A Note on the Structural Change Test in Highly Parameterized Psychometric Models," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 1064-1080, September.
    8. Payton J. Jones & Patrick Mair & Thorsten Simon & Achim Zeileis, 2020. "Network Trees: A Method for Recursively Partitioning Covariance Structures," Psychometrika, Springer;The Psychometric Society, vol. 85(4), pages 926-945, December.

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    More about this item

    Keywords

    measurement invariance; stochastic process; factor analysis; ordinal data;
    All these keywords.

    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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