[4] Several different types of measurement invariance can be distinguished in the common factor model for continuous outcomes:[5] The same typology can be generalized to the discrete outcomes case: Each of these conditions corresponds to a multiple-group confirmatory factor model with specific constraints.
Meaningful comparisons between groups usually require that all four conditions are met, which is known as strict measurement invariance.
For each model being compared (e.g., Equal form, Equal Intercepts) a χ2 fit statistic is iteratively estimated from the minimization of the difference between the model implied mean and covariance matrices and the observed mean and covariance matrices.
[7] However, there is some evidence the diff χ2 is sensitive to factors unrelated to changes in invariance targeted constraints (e.g., sample size).
[8] Consequently it is recommended that researchers also use the difference between the comparative fit index (ΔCFI) of two models specified to investigate measurement invariance.
[13][14] The well-known political scientist Christian Welzel and his colleagues criticize the excessive reliance on invariance tests as criteria for the validity of cultural and psychological constructs in cross-cultural statistics.
[15][16] Proponents of invariance testing counter-argue that the reliance on nomological linkage ignores that such external validation hinges on the assumption of comparability.