Recalculating df in MGCFA testing

Plase cite as follows:
Schroeders, U. & Gnambs, T. (in press). Degrees of freedom in multigroup confirmatory factor analyses: Are models of measurement invariance testing correctly specified? European Journal of Psychological Assessment.
A. Number of indicators
C. Number of cross-loadings
E. Number of resid. covar.
B. Number of factors
D. Number of ortho. factors
F. Number of groups

MI testing constraints df comparison delta(df)
config. (item:factor) 0 - -
metric (loadings) 2 metric-config 2
scalar (loadings+intercepts) 4 scalar-metric 2
residual (loadings+residuals) 5 residual-metric 3
strict (loadings+intercepts+residuals) 7 strict-scalar 3

Additional information

A Indicates the number of indicators or items.
B Indicates the number of latent variables or factors.
C Indicates the number of cross-loadings. For example, in case of a bifactor model the number equals twice the number of indicators (A).
D Indicates the number of orthogonal factors. For example, in case of a nested factor model with six indicators loading on a common factor and three items additionally loading on a nested factors, you have to specify 2 factors (B) and 1 orthogonal factor (D).
E Indicates the number of residual covariances.
F Indicates the number of groups.

Further reading

  • Beaujean, A. A. (2014). Latent variable modeling using R: a step by step guide. New York: Routledge/Taylor & Francis Group.
  • Millsap, R. E. & Olivera-Aguilar, M. (2012). Investigating measurement invariance using confirmatory factor analysis. In R. H. Hoyle (Ed.), Handbook of Structural Equation Modeling (pp. 380-392). New York: Guilford Press.
  • Kline, R. B. (2011). Principles and practice of structural equation modeling. New York: Guilford Press.