Tests of Measurement Invariance Without Subgroups: A Generalization of Classical Methods
- 912 Downloads
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.
Key wordsmeasurement invariance parameter stability factor analysis structural equation models
This work was supported by National Science Foundation grant SES-1061334. The authors thank Jelte Wicherts, who generously shared data for the stereotype threat application, Yves Rosseel, who provided feedback and code for performing the tests with the lavaan package, Kris Preacher, who provided helpful comments on the manuscript, and the participants of the Psychoco 2012 workshop on psychometric computing for helpful discussion.
- Bollen, K.A. (1989). Structural equations with latent variables. New York: Wiley. Google Scholar
- Breiman, L., Friedman, J.H., Olshen, R.A., & Stone, C.J. (1984). Classification and regression trees. Belmont: Wadsworth. Google Scholar
- Brown, R.L., Durbin, J., & Evans, J.M. (1975). Techniques for testing the constancy of regression relationships over time. Journal of the Royal Statistical Society. Series B, 37, 149–163. Google Scholar
- Ferguson, T.S. (1996). A course in large sample theory. London: Chapman & Hall. Google Scholar
- Hansen, B.E. (1997). Approximate asymptotic p values for structural-change tests. Journal of Business & Economic Statistics, 15, 60–67. Google Scholar
- McDonald, R.P. (1999). Test theory: a unified treatment. Mahwah: Erlbaum. Google Scholar
- Millsap, R.E. (2005). Four unresolved problems in studies of factorial invariance. In A. Maydeu-Olivares & J. J. McArdle (Eds.), Contemporary psychometrics (pp. 153–171). Mahwah: Erlbaum. Google Scholar
- Millsap, R.E. (2011). Statistical approaches to measurement invariance. New York: Routledge. Google Scholar
- R Development Core Team (2012). R: a language and environment for statistical computing [Computer software manual]. URL http://www.R-project.org/. Vienna, Austria (ISBN 3-900051-07-0).
- Sánchez, G. (2009). PATHMOX approach: segmentation trees in partial least squares path modeling. Unpublished doctoral dissertation. Universitat Politécnica de Catalunya. Google Scholar
- Shorack, G.R., & Wellner, J.A. (1986). Empirical processes with applications to statistics. New York: Wiley. Google Scholar
- Strobl, C., Kopf, J., & Zeileis, A. (2010). A new method for detecting differential item functioning in the Rasch model (Technical Report No. 92). Department of Statistics, Ludwig-Maximilians-Universität München. URL http://epub.ub.uni-muenchen.de/11915/.
- Wothke, W. (2000). Longitudinal and multi-group modeling with missing data. In T.D. Little, K.U. Schnabel, & J. Baumert (Eds.), Modeling longitudinal and multilevel data: practical issues, applied approaches, and specific examples. Mahwah: Erlbaum. Google Scholar