When using CFA models to examine measurement invariance, one needs to identify these models by assigning a scale to the latent constructs. Two ways of doing this seems to appear frequently in the literature: (1) fixing the factor (latent) means to zero and the factor variances to one; and (2) fixing one item's (the reference item's) intercept to zero and that item's factor loading to one.
Both ways have their pros and cons. Regarding the first, it might be unrealistic to assume a latent mean of zero for all groups (or a factor variance of 1). Regarding the second, by constraining the factor loading(s) of the reference item(s), one assumes invariance, while in fact this item’s factor loading might be variant across groups. (Though this might be solved by running the models several times with different reference items.)
What is your opinion about this? What would be the best way to identify your model?
(In our case, we have 4 latent factors with 3 indicators each, and we would like to examine MI across four groups.)
Marlies, I guess you mixing two things. The first is how to set the scale for the latent. This is either fixing the variance of the latent or its marker to 1 (the intercepts and means have nothing to do with scaling. The common practice is to fix the marker. In the case of invariance testing, a noninvariant marker across groups will cause misfit when you set the equality constraint on ALL loadings (which afterwards can be detected by the modification indices). If the marker is the problem, change the marker and relax the constraint for the previous marker.
Fixing intercepts and means have somthing to do with invariance testing of the latent means (see my Methodology paper and the book chapter on mean testing).
HTH
Holger
Marlies, it's safe to use (2).
Within factor within group, the loadings on the other two items are relative to the unit loading. So invariance is assessed by the constraint that the *other two* loadings are identical across groups, which implies that the three indicators have the same relations to each other across groups -- invariance.
The only real caution I can think of is that this can get unstable if the item with the unit loading is a weak indicator of the latent, but that's not really specific to invariance testing.
By the way, if you wanted to use (1), you could do that as well, but you should constrain the factor variance to 1 only in one group, and let the others be freely estimated, resulting in estimates of variability relative to that in the reference group. This would be identified in the metric-invariant model (equated loadings).
Pat
Marlies, I guess you mixing two things. The first is how to set the scale for the latent. This is either fixing the variance of the latent or its marker to 1 (the intercepts and means have nothing to do with scaling. The common practice is to fix the marker. In the case of invariance testing, a noninvariant marker across groups will cause misfit when you set the equality constraint on ALL loadings (which afterwards can be detected by the modification indices). If the marker is the problem, change the marker and relax the constraint for the previous marker.
Fixing intercepts and means have somthing to do with invariance testing of the latent means (see my Methodology paper and the book chapter on mean testing).
HTH
Holger
Dear Marlies,
I completely agree with Holger, but it is important to say that when you fixed 0 mean and 1 variance for the latent variables, in fact you are assuming the normal distribution for the latent scale. Other question is the invariance testing, that you can consider by constraining parameters strategy.
All the best.
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