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 Judite Gonçalves posted on Monday, December 03, 2012 - 3:20 am
Dear Linda and Bengt Muthen,

I am running multiple-group CFA to assess measurement invariance of a scale with 12 categorical items (1-4 likerts). I use WLSMV. I have two questions:

(A) Following chapter 14 of the user guide, I got the impression that with WLSMV we (1) first test configural invariance and (2) then test for metric and scalar invariance simultaneously (constraining loadings and thresholds to be equal across groups at the same time). That is, I got the impression that we cannot first test for metric invariance, and secondly, separately, for scalar invariance, like one does when items are continuous. Searching for applied papers that do MGCFA using WLSMV, I have not yet found one that constrains loadings and thresholds at the same time, but I have found many that do the usual three steps: configural -> metric -> scalar. My question is: is the later correct? Or I should indeed be constraining loadings and thresholds at the same time, following chapter 14, p.485 (v7)?

(continues)
 Judite Gonçalves posted on Monday, December 03, 2012 - 3:21 am
(continued)

(B) Assuming that I should be constraining loadings and thresholds to be equal across groups at the same time, suppose I do not have full configural invariance, but rather for group 'g2' I need to introduce a cross-loading for item 'u3' on factor 'f2' in addition to its loading on factor 'f1'. In the next step (metric-scalar invariance), 2 doubts:
(1) I need to introduce, under 'model g2', not only 'f2 by u3' (to introduce the cross-loading), but also 'f1 by u3', correct?
(2) I should simultaneously free the thresholds of item 'u3' [u3$1 u3$2 u3$3] in group 'g2', correct?

Thank you very much,

Judite
 Linda K. Muthen posted on Monday, December 03, 2012 - 10:27 am
The three steps should be used only with polytomous items. The two steps should be used with binary items. See Example 5.16 for partial measurement invariance.

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