Alignment
 Asparouhov, T. & Muthén, B. (2022). Multiple group alignment for exploratory and structural equation models. Accepted for publication in Structural Equation Modeling: A Multidisciplinary Journal.
 Billet, J., Meeusen, C. & Abts, K. (2021). The relationship between (sub)national identity, citizenship conceptions, and perceived ethnic threat in Flanders and Wallonia for the period 19952020: A measurement invariance testing strategy.
Forthcoming in Frontiers in Political Science. DOI: 10.3389/fpos.2021.676551
 DeMars, C.E. (2019). Alignment as an alternative to anchor purification in DIF analyses. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2019.1617151
Special Issue: Measurement Invariance  Sociological Methods & Research
 Cieciuch, J., Davidov, E., Algesheimer, R. & Schmidt, P. (2018). Testing for approximate measurement invariance of human values in the European Social Survey.
Sociological Methods & Research, 47:4 665686. DOI: 10.1177/0049124117701478
 Davidov, E., Dülmer, H., Cieciuch, J., Kuntz, A., Seddig, D. & Schmidt, P. (2018). Explaining measurement nonequivalence using multilevel structure equation modeling: The case of attitudes toward citizenship rights.
Sociological Methods & Research, 47:4 729760. DOI: 10.1177/0049124116672678
 Munck, I., Barber, C., & TorneyPurta, J. (2018). Measurement invariance in comparing attitudes toward immigrants among youth across Europe in 1999 and 2009.
Sociological Methods & Research, 47:4 687728. DOI: 10.1177/0049124117729691
 Muthén, B. & Asparouhov, T. (2018). Recent methods for the study of measurement invariance with many groups: Alignment and random effects.
Sociological Methods & Research, 47:4 637664. DOI: 10.1177/0049124117701488 Mplus scripts.
 Lomazzi, V. (2017). Using alignment optimization to test the measurement invariance of gender role attitudes in 59 countries. methods, data, analyses, [S.l.], p. 27, ISSN 21904936.
 Munck, I., Barber, C., & TorneyPurta, J. (2017). Measurement invariance in comparing attitudes toward immigrants among youth across Europe in 1999 and 2009: The alignment method applied to IEA CIVED and ICCS. Sociological Methods & Research, DOI: 10.1177/0049124117729691
 Flake, J.K. & McCoach, D. B. (2017). An investigation of the alignment method with polytomous indicators under conditions of partial measurement invariance. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2017.1374187
 Marsh, H. W., Guo, J., Nagengast, B., Parker, P. D., Asparouhov, T., Muthén, B., & Dicke, T. (2018). What to do when scalar invariance fails: The extended alignment method for multigroup factor analysis comparison of latent means across many groups. Psychological Methods, 23(3), 524545. DOI: 10.1037/met0000113
 Muthén, B. & Asparouhov, T. (2018). Recent methods for the study of measurement invariance with many groups: Alignment and random effects.
Sociological Methods & Research, 47:4 637664. DOI: 10.1177/0049124117701488 Mplus scripts.
 Asparouhov T. & Muthén, B. (2014). Multiplegroup factor analysis alignment. Structural Equation Modeling: A Multidisciplinary Journal, 21:4, 495508.
Download Mplus files.
 Muthén, B. & Asparouhov T. (2014). IRT studies of many groups: The alignment method. Frontiers in Psychology, Volume 5, DOI: 10.3389/fpsyg.2014.00978
 Muthén, B. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54:4, 557585.
Other Resources
BSEM
 Seddig, D., & Leitgöb, H. (2018a). Approximate measurement invariance and longitudinal confirmatory factor analysis: concept and application with panel data. Survey Research Methods, 12(1), 2941.
 Seddig, D., & Leitgöb, H. (2018b). Exact and Bayesian approximate measurement invariance. In E. Davidov, P. Schmidt, & J. Billiet (Eds.), Cross cultural analysis: Methods and applications (2. Ed.) (pp. 553579). New York: Routledge.
 Shi, D., Song, H., DiStefano, C., MaydeuOlivares, A., McDaniel H.L. & Jiang, Z. (2018). Evaluating factorial invariance: An interval estimation approach using Bayesian structural equation modeling. Multivariate Behavioral Research, DOI: 10.1080/00273171.2018.1514484
 Muthén, B. & Asparouhov, T. (2013). BSEM measurement invariance analysis.
 Muthén, B. & Asparouhov, T. (2012). Bayesian SEM: A more flexible representation of substantive theory. Psychological Methods, 17, 313335.
Download the 2nd version dated April 14, 2011. View the seven web tables referred to in the paper.
View Mplus inputs, data, and outputs used in this version of paper. Download the 1st version dated September 29, 2010 containing a MIMIC section and more tables. Download the corresponding Mplus inputs, data, and outputs.
The seven web tables correspond to tables 8, 10, 17, 18, 19, 20, and 21 of the first version.
Measurement Invariance for Categorical Outcomes
 Svetina, D., Rutkowski, L. & Rutkowski, D. (2020). MultipleGroup Invariance with Categorical Outcomes Using Updated Guidelines: An Illustration Using Mplus and the lavaan/semTools Packages. Structural Equation Modeling: A Multidisciplinary Journal, 27(1), 111130, DOI: 10.1080/10705511.2019.1602776
 Pendergast, L. L., von der Embse, N., Kilgus, S. P., & Eklund, K. R. (2017). Measurement equivalence: A nontechnical primer on categorical multigroup confirmatory factor analysis in school psychology. Journal of School Psychology, 60, 6582. DOI: 10.1016/j.jsp.2016.11.002
 Wu, H. & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika 81(4), 1014–1045 DOI: 10.1007/s1133601695060
Measurement Invariance in EFA
Random Loadings
 Asparouhov, T. & Muthén, B. (2016). General random effect latent variable modeling: Random subjects, items, contexts, and parameters. In Harring, J. R., & Stapleton, L. M., & Beretvas, S. N. (Eds.), Advances in multilevel modeling for educational research: Addressing practical issues found in realworld applications (pp. 163192). Charlotte, NC: Information Age Publishing, Inc.
Mplus scripts.
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