The most likely cause is that the papers that do not show measurement invariance are not published. There are probably many failed analyses out there that you are not reading about.
Another issue has to do with whether the authors are looking at only factor loadings or both factor loadings and intercepts. In our work, we often find measurement invariance for factor loadings but not for intercepts.
finnigan posted on Friday, March 02, 2007 - 4:44 am
Thanks for this. Is there any process in MPLUS to identify non invariant and invariant indicators?
One way of testing for measurement invariance in Mplus is described in Chapter 13 at the end of the discussion of multiple group analysis. This is done by the testing of nested models. One can also use the MIMIC model which is shown in Chapter 5.
finnigan posted on Friday, March 02, 2007 - 8:36 am
Thanks for this.
As far as I understand from chp 13 on MGA, Mplus by default standardises the scale by selecting a referent item. How does MPLUS recognise that the referent is invariant across groups?
You can free the factor loading that is fixed for identification purposes and instead achieve identification by fixing the factor variance to one. How to do this is described in Chapter 16 under the BY option.
Vanessa posted on Tuesday, July 06, 2010 - 10:24 pm
In testing for longitudinal measurement invariance, is it still OK to fix the factor variances to 1, instead of a reference indicator?
Do you know any reasons why people would suggest not to fix the variances instead of an reference indicator?
I ask because in papers that I have read that discuss how to determine whether the reference indicator's loading is invariant or not, no-one suggests this route but it seems like the logical way?? Thanks!
The factor variances may change over time so fixing them all to one would distort that. You can fix the factor variance at one for the first time point and let them be free for the other time points, assuming that you hold loadings equal across time.
Vanessa posted on Wednesday, July 07, 2010 - 10:17 pm
Thanks very much for your reply. Am I correct in thinking that in such a longitudinal model I also need to over-ride the default of the latent means being fixed to zero?
When intercepts are held equal across time, one factor mean must be fixed to zero and the others can be free. When the intercepts are not held equal across time, all factor means must be fixed at zero.
Thanks Linda. Is that for identification purposes that you say all factor means must be fixed at zero when the intercepts are not held equal across time? Could there be alternate models whereby other parameters are fixed instead eg. the intercepts are not held equal across time (when testing for invariance of loadings first), but the 1st factor mean is 0, variance is 1, and the loading and intercept of the 1st indicator is held equal across time?
You can use alternative parametrizations but will gain no new information by doing this. When intercepts are free, comparing factor means is meaningless because the factors have different meanings. It is only when intercepts are constrained that means can be compared.
Vanessa posted on Tuesday, July 13, 2010 - 5:20 pm
Thank you. Is the preceding weaker form of invariance a prerequisite to test the next level of invariance? For instance, if not all my loadings are invariant, does this mean that I can't test for invariance of the intercepts for the indicators that don't have invariant loadings?
I think you might find our Topic 1 Video and course handout of interest. We go through testing measurement invariance in detail and address issues of partial measurement invariance.
Vanessa posted on Tuesday, July 13, 2010 - 6:16 pm
Thanks. Yes I have read the relevant sections in the Topic 1 handout and they were very useful - I asked only because it seems in the example for testing group invariance, on p 219, two intercepts are freely estimated, yet all residual variances are constrained... yet I have read that if you have partial invariance eg, for loadings, then the intercept for those indicators then shouldn't be constrained to invariance...
Vanessa posted on Tuesday, July 13, 2010 - 7:48 pm
I take it this is not the case then. I think I have read it explicitly elsewhere, but it seems to be suggested in Byrne et al. (1989), Testing for the equivalence of factor covariance and mean structures, Psych Bull (eg.p.457-458) or have I mis-read/misinterpreted this (entirely possible)?
I looked at Byrne et al. paper and she does seem to say that if you don't have invariance of either the intercept or slope of an item, you should not test for residual invariance. In many disciplines residual invariance is not examined because structural parameters do not require residual invariance for a valid comparison.