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Tests of Measurement Invariance Without Subgroups: A Generalization of Classical Methods


  • Edgar Merkle


  • Achim Zeileis


The issue of measurement invariance commonly arises in factor-analytic contexts, with methods for assessment including likelihood ratio tests, Lagrange multiplier tests, and Wald tests. These tests all require advance definition of the number of groups, group membership, and offending model parameters. In this paper, we study tests of measurement invariance based on stochastic processes of casewise derivatives of the likelihood function. These tests can be viewed as generalizations of the Lagrange multiplier test, and they are especially useful for: (i) identifying subgroups of individuals that violate measurement invariance along a continuous auxiliary variable without prespecified thresholds, and (ii) identifying specific parameters impacted by measurement invariance violations. The tests are presented and illustrated in detail, including an application to a study of stereotype threat and simulations examining the tests’ abilities in controlled conditions. Copyright The Psychometric Society 2013

Suggested Citation

  • 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.
  • Handle: RePEc:spr:psycho:v:78:y:2013:i:1:p:59-82
    DOI: 10.1007/s11336-012-9302-4

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    References listed on IDEAS

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    7. Zeileis, Achim & Shah, Ajay & Patnaik, Ila, 2010. "Testing, monitoring, and dating structural changes in exchange rate regimes," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1696-1706, June.
    8. Rosseel, Yves, 2012. "lavaan: An R Package for Structural Equation Modeling," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i02).
    9. Hansen, Bruce E, 1997. "Approximate Asymptotic P Values for Structural-Change Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 60-67, January.
    10. Ploberger, Werner & Kramer, Walter, 1992. "The CUSUM Test with OLS Residuals," Econometrica, Econometric Society, vol. 60(2), pages 271-285, March.
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    12. 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.
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    Cited by:

    1. Gerhard Tutz & Gunther Schauberger, 2015. "A Penalty Approach to Differential Item Functioning in Rasch Models," Psychometrika, Springer;The Psychometric Society, vol. 80(1), pages 21-43, March.
    2. Peter M. Bentler, 2016. "Covariate-free and Covariate-dependent Reliability," Psychometrika, Springer;The Psychometric Society, vol. 81(4), pages 907-920, December.
    3. Jones, Payton J. & Mair, Patrick & Simon, Thorsten & Zeileis, Achim, 2019. "Network Model Trees," OSF Preprints ha4cw, Center for Open Science.
    4. 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.
    5. 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.
    6. Dylan Molenaar, 2015. "Heteroscedastic Latent Trait Models for Dichotomous Data," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 625-644, September.
    7. Chung-Ying Lin & Meng-Che Tsai, 2016. "Effects of Family Context on Adolescents’ Psychological Problems: Moderated by Pubertal Timing, and Mediated by Self-Esteem and Interpersonal Relationships," Applied Research in Quality of Life, Springer;International Society for Quality-of-Life Studies, vol. 11(3), pages 907-923, September.
    8. Carolin Strobl & Julia Kopf & Achim Zeileis, 2015. "Rasch Trees: A New Method for Detecting Differential Item Functioning in the Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 289-316, June.


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