Skip to main content
SpringerLink
Log in

CFA Models with a General Factor and Multiple Sets of Secondary Factors

  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

We propose a class of confirmatory factor analysis models that include multiple sets of secondary or specific factors and a general factor. The general factor accounts for the common variance among manifest variables, whereas multiple sets of secondary factors account for the remaining source-specific dependency among subsets of manifest variables. A special case of the model is further proposed which constrains the specific factor loadings to be proportional to the general factor loadings. This proportional model substantially reduces the number of model parameters while preserving the essential structure of the general model. Furthermore, the proportional model allows for the interpretation of latent variables as the expected values of the observed manifest variables, decomposition of the variances, and the inclusion of interactions, similar to generalizability theory. We provide two applications to illustrate the utility of the proposed class of models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the general model estimation, the estimated factor loading for the 12th manifest variable on the third dimension factor (\(S_1\)) was found to be negative and non-significant (\(\alpha _{123}^{S3} = -0.283, \text {SE}=0.265, p=0.142\)), indicating that this manifest variable did not appear to be associated with the specific factor as expected. Accordingly, we set the factor loading for the 12th manifest variable to zero in estimating the proportional model.

  2. The original wording is slightly revised.

References

  • Barker Lunn, J. (1970). Streaming in the primary school. Sussex: King, Thorne, & Stace.

    Google Scholar 

  • Bauer, D. J., Howard, A. L., Baldasaro, R. E., Curran, P. J., Andrea, M. H., Chassin, L., et al. (2013). A trifactor model for integrating ratings across multiple informants. Psychological Methods, 18, 475–493.

    Article  Google Scholar 

  • Becker, T. E., & Cote, J. A. (1994). Additive and multiplicative method effects in applied psychological research: An empirical assessment of three methods. Journal of Management, 20, 625–641.

    Article  Google Scholar 

  • Bentler, P. M. (1976). Multistructure statistical model applied to factor analysis. Multivariate Bahavioral Research, 11, 3–25.

    Article  Google Scholar 

  • Bollen, K. A., & Bauldry, S. (2010). Model identification and computer algebra. Sociological Methods and Research, 39, 127–156.

    Article  Google Scholar 

  • Bowler, M. C., & Woehr, D. J. (2006). A meta-analytic evaluation of the impact of dimension and exercise factors on assessment center ratings. Journal of Applied Psychology, 91, 1114–1124.

    Article  Google Scholar 

  • Bradlow, E. T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.

    Article  Google Scholar 

  • Brennan, R. L. (2001). Generalizability theory. New York: Springer.

    Book  Google Scholar 

  • Cai, L. (2010). A two-tier full-information item factor analysis model with applications. Psychometrika, 75, 581–612.

    Article  Google Scholar 

  • Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56, 81–105.

    Article  Google Scholar 

  • Cole, D., Ciesla, J. A., & Steiger, J. H. (2007). The insidious effects of failing to include design-driven correlated residuals in latent-variable covariance structure analysis. Psychological Methods, 4, 381–398.

    Article  Google Scholar 

  • Cronbach, L., Gleser, G., Nanda, H., & Rajaratnam, N. (1972). The dependability of behavioral measurements: Theory of generalizability for scores and profiles. New York: Wiley.

    Google Scholar 

  • Davis, W. (1993). The fc1 rule of identification for confirmatory factor analysis. Sociological Methods and Research, 21, 403–407.

    Article  Google Scholar 

  • Eid, M. (2000). A multitrait-multimethod model with minimal assumptions. Psychometrika, 65, 241–261.

    Article  Google Scholar 

  • Eid, M., Geiser, C., Koch, T., & Heene, M. (2017). Anomalous results in g-factor models: Explanations and alternatives. Psychological Methods, 22, 541–562.

    Article  Google Scholar 

  • Eid, M., Nussbeck, F. W., Geiser, C., Cole, D. A., Gollwitzer, M., & Lischetzke, T. (2008). Structural equation modeling of multitrait-multimethod data: Different models for different types of methods. Psychological Methods, 13, 230–253.

    Article  Google Scholar 

  • Hoffman, B. J., Melchers, K. G., Blair, C. A., Kleinmann, M., & Ladd, R. (2011). Exercises and dimensions are the currency of assessment centers. Personnel Psychology, 64, 351–395.

    Article  Google Scholar 

  • Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1–55.

    Article  Google Scholar 

  • Jöreskog, K. G. (1970). A general method for analysis of covariance structures. Biometrika, 57, 239–251.

    Article  Google Scholar 

  • Kenny, D. A. (1976). An empirical application of confirmatory factor analysis to the multitrait-multimethod matrix. Journal of Experimental Social Psychology, 12, 247–252.

    Article  Google Scholar 

  • Kenny, D. A. (1979). Correlation and causality. New York, NY: Wiley.

    Google Scholar 

  • Lance, C. E., Lambert, T. A., Gerwin, A. G., Lievens, F., & Conway, J. M. (2004). Revised estimates of dimension and exercise variance components in assessment center postexercise dimension ratings. Journal of Applied Psychology, 89, 377–385.

    Article  Google Scholar 

  • Lievens, F., Dilchert, S., & Ones, D. (2009). The importance of exercise and dimension factors in assessment centers: Simultaneous examinations of construct-related and criterion-related validity. Human Performance, 22, 375–390.

    Article  Google Scholar 

  • MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130–149.

    Article  Google Scholar 

  • Marsh, H. W. (1989). Confirmatory factor analyses of multitrait-multimethod data: Many problems and a few solutions. Applied Psychological Measurement, 13, 335–361.

    Article  Google Scholar 

  • McDonald, R. P. (1982). A note on the investigation of local and global identifiability. Psychometrika, 47, 101–103.

    Article  Google Scholar 

  • McDonald, R. P. (1999). Test theory: A unified approach. Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Mellenbergh, G. J., Kelderman, H., Stijlen, J. G., & Zondag, E. (1979). Linear models for the analysis and construction of instruments in a facet design. Psychological Bulletin, 86, 766–776.

    Article  Google Scholar 

  • Mellon, P., & Crano, W. D. (1977). An extension and application of the multitrait-multimethod matrix technique. Journal oi Educational Psychology, 69, 716–723.

    Article  Google Scholar 

  • Muthén, L., & Muthén, B. (2008). Mplus user’s guide. Angeles, CA: Muthen & Muthen.

    Google Scholar 

  • Pohl, S., & Steyer, R. (2010). Modeling common traits and method effects in multitrait.multimethod analysis. Multivariate Behavioral Research, 45, 45–72.

    Article  Google Scholar 

  • Raykov, T. (1997). Scale reliability, Cronbach’s coefficient alpha, and violations of essential tau-equivalence for fixed congeneric components. Multivariate Behavioral Research, 32, 329–354.

    Article  Google Scholar 

  • Raykov, T., & Marcoulides, G. A. (2006). Estimation of generalizability coefficients via a structural equation modeling approach to scale reliability evaluation. International Journal of Testing, 6, 81–95.

    Article  Google Scholar 

  • Reise, S. P., & Moore, T. M. (2013). Applying unidimensional item response theory models to psychological data. In K. Geisinger (Ed.), APA handbook of testing and assessment in psychology. Test theory and testing and assessment in industrial and organizational psychology (Vol. 1, pp. 101–119). Washington, DC: American Psychological Association.

    Google Scholar 

  • Reise, S. P., Moore, T. M., & Haviland, M. G. (2010). Bifactor models and rotations: Exploring the extent to which multidimensional data yield univocal scale scores. Journal of Personality Assessment, 92, 544–559.

    Article  Google Scholar 

  • Reise, S. P., Scheines, R., Widaman, K. F., & Haviland, M. G. (2013b). Multidimensionality and structural coefficient bias in structural equation modeling: A bifactor perspective. Educational and Psychological Measurement, 73, 5–26.

    Article  Google Scholar 

  • Rijmen, F. (2010). Formal relations and an empirical comparison between the bi-factor, the testlet, and a second-order multidimensional IRT model. Journal of Educational Measurement, 47, 361–372.

    Article  Google Scholar 

  • Rijmen, F., Jeon, M., von Davier, M., & Rabe-Hesketh, S. (2014). A third-order item response theory model for modeling the effects of domains and subdomains in large-scale educational assessment surveys. Journal of Educational and Behavioral Statistics, 39, 235–256.

    Article  Google Scholar 

  • Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016). Evaluating bifactor models: Calculating and interpreting statistical indices. Psychological Methods, 21, 137–150.

    Article  Google Scholar 

  • Schmid, J., & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53–61.

    Article  Google Scholar 

  • Schneider, J. R., & Schmitt, N. (1992). An exercise design approach to understanding assessment center dimension and exercise constructs. Journal of Applied Psychology, 77, 32–41.

    Article  Google Scholar 

  • Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. Boca Raton, FL: Chapman & Hall/CRC.

    Book  Google Scholar 

  • Steyer, R., Ferring, D., & Schmitt, M. J. (1992). States and traits in psychological assessment. European Journal of Psychological Assessment, 8, 79–98.

    Google Scholar 

  • Stucky, B. D., Thissen, D., & Edelen, M. O. (2013). Using logistic approximations of marginal trace lines to develop short assessments. Applied Psychological Measurement, 37, 41–57.

    Article  Google Scholar 

  • Wainer, H. (1995). Precision and differential item functioning on a testlet-based test: The 1991 Law School Admissions Test as an example. Applied Psychological Measurement, 8, 157–186.

    Google Scholar 

  • Wainer, H., Bradlow, E., & Wang, X. (2007). Testlet response theory and its applications. New York, NY: Cambridge University Press.

    Book  Google Scholar 

  • Wainer, H., & Thissen, D. (1996). How is reliability related to the quality of test scores? What is the effect of local dependence on reliability? Educational Measurement: Issues and Practice, 15, 22–29.

    Article  Google Scholar 

  • Wald, A. (1950). A note on the identification of econometric relations. In T. C. Koopmans (Ed.), Statistical inference in dynamic economic models (pp. 238–244). New York: Wiley.

    Google Scholar 

  • Widaman, F. (1985). Hierarchically nested covariance structure models for multitrait-multimethod data. Applied Psychological Measurement, 9, 1–26.

    Article  Google Scholar 

  • Woehr, D. J., Putka, D. J., & Bowler, M. C. (2012). An examination of G-theory methods for modeling multitrait multimethod data: Clarifying links to construct validity and confirmatory factor analysis. Organizational Research Methods, 15, 134–161.

    Article  Google Scholar 

  • Wolfram Research, Inc (2010). Mathematica Edition: Version 8.0. Champaign, IL: Wolfram Research, Inc.

  • Yung, Y.-F., Thissen, D., & McLeod, L. (1999). On the relationship between the higher-order factor model and the hierarchical factor model. Psychometrika, 64, 113–128.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Education, University of California, Los Angeles, 457 Portola Plaza, Los Angeles, CA, 90095, USA

    Minjeong Jeon

  2. American Institutes for Research, Washington, USA

    Frank Rijmen

  3. University of California, Berkeley, Berkeley, USA

    Sophia Rabe-Hesketh

Authors
  1. Minjeong Jeon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Rijmen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sophia Rabe-Hesketh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minjeong Jeon.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 111 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jeon, M., Rijmen, F. & Rabe-Hesketh, S. CFA Models with a General Factor and Multiple Sets of Secondary Factors. Psychometrika 83, 785–808 (2018). https://doi.org/10.1007/s11336-018-9633-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11336-018-9633-x

Keywords

Search

Navigation