I am testing measurement invariance of a questionnaire. I am aware of the fact that some methodologists think it unnecessary to test for invariance of residual variances and covariances. However, the measure I am working with has a highly replicated correlated errors issue, and I would like to test its invariance since possible noninvariance could affect the measure's reliability for some groups. What's the correct sequence I should follow? Should correlated errors invariance be tested before or after intercept invariance? Intuitively, I think it would make sense to test it together or immediately before residual variances invariance, since both are usually considered under the label 'strict invariance'. However, the Mplus tutorial in Byrne's book suggests testing correlated residuals invariance after factor loadings invariance and before intercept invariance, and it also makes sense to me. Readable references would be very appreciated.
After intercept invariance testing! If you have intercept invariance your model would be scalar invarant. Even that is often hard to establish. Usually you need to free some invariant intercepts, and you have a partial scalar invariant model, fully acceptable if at least two items have invariant intercepts. See references in enclosed article. Correlation errors invariance is seldom required but how to is described in the book enclosed (see link). Best of luck.
https://www.amazon.co.uk/gp/aw/d/1119978297/ref=mp_s_a_1_4?ie=UTF8&qid=1502096375&sr=8-4&pi=AC_SX236_SY340_FMwebp_QL65&keywords=structural+equation+modeling
Article Assessing understandings of substance use disorders among No...
After intercept invariance testing! If you have intercept invariance your model would be scalar invarant. Even that is often hard to establish. Usually you need to free some invariant intercepts, and you have a partial scalar invariant model, fully acceptable if at least two items have invariant intercepts. See references in enclosed article. Correlation errors invariance is seldom required but how to is described in the book enclosed (see link). Best of luck.
https://www.amazon.co.uk/gp/aw/d/1119978297/ref=mp_s_a_1_4?ie=UTF8&qid=1502096375&sr=8-4&pi=AC_SX236_SY340_FMwebp_QL65&keywords=structural+equation+modeling
Article Assessing understandings of substance use disorders among No...
Dear Pablo,
I do not understand your doubt well, but I think this can be of help.
The following is from Brown's book:
For the aforementioned reasons, the recommended sequence of multiplegroups CFA invariance evaluation is as follows: (1) Test the CFA model separately in each group; (2) conduct the simultaneous test of equal form (identical factor structure); (3) test the equality of factor loadings; (4) test the equality of indicator intercepts; (5) test the equality of indicator residual variances (optional); and, if substantively meaningful; (6) test the equality of factor variances; (7) test the equality of factor covariances (if applicable, i.e., > 1 latent factor); and (8) test the equality of latent meanAdd your answers. Steps 1–5 are tests of measurement invariance; Steps 6–8 are tests of population heterogeneity (Brown, 2006, p.270)
Brown, T. A. (2014). Confirmatory factor analysis for applied research. Guilford Publications.
To compare latent means is possible if you at least have a partial scalar model, see above. See the mentioned article above.
I also calculated effect size for the between groups differences in latent means to be able to indicate whether the significant differences were substantial or not. :-)
Invariance testing is about testing that the model you use holds for all groups/waves of measurement.
If you have found that it is necessary to correlate some error terms, it would be my suggestion that you keep these error correlations in every step of the invariance testing, since this is the model you intent to use.
However, correlation of error terms is often considered a problem. Sometimes it is possible to allow for 'just a little' correlation by using the Bayesian estimator and set a prior correlation between some error terms with mean=0 and a small variance. In this way the correlations are kept to a minimum, while allowing for the fact that everything has some degree of correlation, which often enhances the chance for getting a good model fit.
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