, Volume 81, Issue 4, pp 1014–1045 | Cite as

Identification of Confirmatory Factor Analysis Models of Different Levels of Invariance for Ordered Categorical Outcomes

  • Hao WuEmail author
  • Ryne Estabrook


This article considers the identification conditions of confirmatory factor analysis (CFA) models for ordered categorical outcomes with invariance of different types of parameters across groups. The current practice of invariance testing is to first identify a model with only configural invariance and then test the invariance of parameters based on this identified baseline model. This approach is not optimal because different identification conditions on this baseline model identify the scales of latent continuous responses in different ways. Once an invariance condition is imposed on a parameter, these identification conditions may become restrictions and define statistically non-equivalent models, leading to different conclusions. By analyzing the transformation that leaves the model-implied probabilities of response patterns unchanged, we give identification conditions for models with invariance of different types of parameters without referring to a specific parametrization of the baseline model. Tests based on this approach have the advantage that they do not depend on the specific identification condition chosen for the baseline model.


ordered categorical data invariance testing model identification 



Work on this research by the second author was partially supported by the National Institute of Drug Abuse research education program R25DA026-119 (Director: Michael C. Neale) and by grant R01 AG18436 (20112016, Director: Daniel K. Mroczek) from National Institute on Aging, National Institute on Mental Health.


  1. Babakus, E., Ferguson, C. E., & Jöreskog, K. G. (1987). The sensitivity of confirmatory maximum likelihood factor analysis to violations of measurement scale and distributional assumptions. Journal of Marketing Research, 24(2), 222–228.CrossRefGoogle Scholar
  2. Baker, F. B. (1992). Item response theory parameter estimation techniques. New York: Marcel Dekker.Google Scholar
  3. Bernstein, I. H., & Teng, G. (1989). Factoring items and factoring scales are different: Spurious evidence for multidimensionality due to item categorization. Psychological Bulletin, 105, 467–477.CrossRefGoogle Scholar
  4. Boker, S., Neale, M., Maes, H., Wilde, M., Spiegel, M., Brick, T., et al. (2011). OpenMx: An open source extended structural equation modeling framework. Psychometrika, 76, 306–317.CrossRefPubMedPubMedCentralGoogle Scholar
  5. Carey, G. (2005). Cholesky problems. Behavioral Genetics, 35, 653–665.CrossRefGoogle Scholar
  6. Cheung, G. W., & Lau, R. S. (2012). A direct comparison approach for testing measurement invariance. Organizational Research Methods, 15(2), 167–198.CrossRefGoogle Scholar
  7. Cheung, G. W., & Rensvold, R. (1998). Cross cultural comparisons using non-invariant measurement items. Applied Behavioral Science Review, 6, 93–110.CrossRefGoogle Scholar
  8. Cheung, G. W., & Rensvold, R. (1999). Testing factorial invariance across groups: A reconceptualization and proposed new method. Journal of Management, 25, 1–27.CrossRefGoogle Scholar
  9. Christoffersson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–32.CrossRefGoogle Scholar
  10. Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64, 247–254.CrossRefGoogle Scholar
  11. Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74, 33–43.Google Scholar
  12. Drton, M. (2009). Likelihood ratio tests and singularities. The Annals of Statistics, 37(2), 979–1012.CrossRefGoogle Scholar
  13. Estabrook, R. (2012). Factorial invariance: Tools and concepts for strengthening research. In G. Tenenbaum, R. Eklund, & A. Kamata (Eds.), Measurement in sport and exercise psychology. Champaign, IL: Human Kinetics.Google Scholar
  14. Jeffries, N. O. (2003). A note on ’Testing the number of components in a normal mixture’. Biometrika, 90(4), 991–994.CrossRefGoogle Scholar
  15. Jöreskog, K. G., & Moustaki, I. (2001). Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behaviorial Research, 36, 347–387.CrossRefGoogle Scholar
  16. Lubke, G. H., & Muthén, B. O. (2004). Applying multiple group confirmatory factor models for continuous outcomes to Likert scale data complicates meaningful group comparisons. Structural Equation Modeling, 11(4), 514–534.CrossRefGoogle Scholar
  17. Mehta, P. D., Neale, M. C., & Flay, B. R. (2004). Squeezing interval change from ordinal panel data: Latent growth curves with ordinal outcomes. Psychological Methods, 9(3), 301–333.CrossRefPubMedGoogle Scholar
  18. Meredith, W. (1964a). Notes on factorial invariance. Psychometrika, 29, 177–185.CrossRefGoogle Scholar
  19. Meredith, W. (1964b). Rotation to achieve factorial invariance. Psychometrika, 29, 186–206.Google Scholar
  20. Meredith, W. (1993). Measurement invariance, factor analysis and factor invariance. Psychometrika, 58, 525–543.CrossRefGoogle Scholar
  21. Millsap, R. E., & Meredith, W. (2007). Factorial invariance: Historical perspectives and new problems. In R. Cudeck & R. C. MacCallum (Eds.), Factor analysis at 100: Historical developments and future directions (pp. 131–152). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  22. Millsap, R. E., & Yun-Tein, J. (2004). Assessing factorial invariance in ordered categorical measures. Multivariate Behavioral Research, 39(3), 479–515.CrossRefGoogle Scholar
  23. Mislevy, R. J. (1986). Recent developments in the factor analysis of categorical variables. Journal of Educational Statistics, 11, 3–31.CrossRefGoogle Scholar
  24. Muthén, B., & Christofferson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46, 407–419.CrossRefGoogle Scholar
  25. Muthén, B. O. (1984). A general structural equation model for dichotomous, ordered categorical and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
  26. Muthén, B. O., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171–189.Google Scholar
  27. Muthén, L. K., & Muthén, B. O. (1998–2012). Mplus Users Guide (7th ed.). Los Angeles, CA: Muthén & Muthén.Google Scholar
  28. Neale, M. C., Hunter, M. D., Pritkin, J., Zahery, M., Brick, T. R., Kirkpatrick, R. M., et al. (2016). OpenMx 2.0: Extended structural equation and statistical modeling. Psychometrika, 81(2), 535–549.CrossRefPubMedGoogle Scholar
  29. Oort, F. J. (1998). Simulation study of item bias detection with restricted factor analysis. Structural Equation Modeling, 5, 107–124.CrossRefGoogle Scholar
  30. R Development Core Team. (2013). R: A language and environment for statistical computing.
  31. Rensvold, R. B., & Cheung, G. W. (2001). Testing for metric invariance using structural equation models, solving the standardization problem. Research in Management, 1, 25–50.Google Scholar
  32. Strom, D. O., & Lee, J. W. (1994). Variance components testing in the longitudinal mixed effects model. Biometrics, 50, 1171–1177 (Corrected in. Biometrics, 51, 1196.)Google Scholar
  33. van der Linden, W. J. and Barrett, M. D. (2015). Linking item response model parameters. Psychometrika. doi: 10.1007/s11336-015-9469-6.
  34. Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4–69.CrossRefGoogle Scholar
  35. Widaman, K. F., & Reise, S. P. (1997). Exploring the measurement invariance of psychological instruments: Applications in the substance use domain. In K. J. Bryant, M. Windle, & S. G. West (Eds.), The science of prevention: Methodological advances from alcohol and substance abuse research (pp. 281–324). Washington, DC: American Psychological Association.CrossRefGoogle Scholar
  36. Wu, H., & Neale, M. C. (2013). On the likelihood ratio tests in bivariate ACDE models. Psychometrika, 78(3), 441–463.CrossRefPubMedGoogle Scholar
  37. Wu, H. (accepted) A note on the identifiability of fixed effect 3PL models. Psychometrika.Google Scholar
  38. Yoon, M., & Millsap, R. E. (2007). Detecting violations of factorial invariance using data-based specification searches: A Monte-Carlo study. Structural Equation Modeling, 14(3), 435–463.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2016

Authors and Affiliations

  1. 1.Boston CollegeChestnut HillUSA
  2. 2.Northwestern UniversityEvanstonUSA

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