Testing for Measurement Invariance with Respect to an Ordinal Variable
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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.
Key wordsmeasurement invariance stochastic process factor analysis ordinal data
This work was supported by National Science Foundation grant SES-1061334.
- Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723. Available from http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1100705. CrossRefGoogle Scholar
- Browne, M.W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K.A. Bollen & J.S. Long (Eds.), Testing structural equation models (pp. 136–162). Newbury Park: Sage. Google Scholar
- Emmons, R.A. & McCullough, M.E. (Eds.) (2004). The psychology of gratitude. New York: Oxford University Press. Google Scholar
- Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., et al. (2012). mvtnorm: multivariate normal and t distributions [Computer software manual]. Available from http://CRAN.R-project.org/package=mvtnorm (R package version 0.9-9992).
- Huber, P.J. (1967). The behavior of maximum likelihood estimation under nonstandard conditions. In L.M. LeCam & J. Neyman (Eds.), Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Berkeley: University of California Press. 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)
- Thomas, M., & Watkins, P. (2003). Measuring the grateful trait: development of the revised GRAT. Poster presented at the Annual Convention of the Western Psychological Association, Vancouver, BC. Google Scholar