Joel Nigg posted on Monday, February 02, 2009 - 9:57 pm
I am running a two group CFA with the goal of determining whether the factor loadings are equivalent in the 2 groups. The indicators are ordinal. (The fit is marginal with RMSEA at .096 on first run). Because I have missing data and type=complex (sibling and twin data), I cannot use the specification NOMEANSTRUCTURE. My two questions (1) because of this, am I requiring "too much" of my model by requiring both equivalent factor loadings and equivalent mean structure? (2) Is there any alternative to relax the model without beginning to relax path coefficients or factor loadings? Or am I mis-apprehending the analysis..(could be)
See pages 398-401 of the Mplus User's Guide to see our suggestions for testing measurement invariance. See also short course handouts for Topics 1 and 2.
Joel Nigg posted on Friday, February 06, 2009 - 1:29 am
Linda, Thank you--that is just what I needed to get going. I now see the options I think.
The theory I want to test is that the factor loadings are equal across two groups (population heterogeneity). I would like to relax the means (thresholds?) for the indicators and the factors, as the theory does not require those to be the same across groups (and even predicts they should differ).
I realize (p. 389) that the variances are already released for the latent variables (using delta paramaterization). However, I am not clear what it means to release the scale factor.
It seems like the reduced model for categorical indicators described on p. 399 (and illustrated in Ex 5.16) is releasing the factor loadings (which I don't want to release) as well as the thresholds, while holding constant the scale factors.
I realize that you point out in the manual that it is mathematically necessary to release these parameters together. However, I am not sure that this is testing my theory. Can you clarify that or point me to an alternative?
(my indicators are 3 level ordinal) Thank you as before.
With continuous outcomes, it makes sense to look at only factor loading invariance because means do not need to be included in the model. This is not the case with categorical outcomes where thresholds and factors loadings determine the basis of categorical data modeling, the item characteristic curves. With categorical outcomes, thresholds and factor loadings should be looked at in tandem. With categorical outcomes, you theory should be about item characteristics curves rather than factor loadings or factor loadings and intercepts.
The scale factor is one divided by the standard deviation of the latent response variable.
Yoon Oh posted on Thursday, June 18, 2015 - 12:46 am
Linda, I have a question about your previous post (Feb.06, 2009) about measurement invariance with categorical outcomes. You said that "with categorical outcomes, thresholds and factor loadings should be looked at in tandem." But I found many studies testing first the equality of factor loadings only and then testing the equality of factor loadings and thresholds. These studies used categorical variables. I'd like to hear your thoughts about this approach. Thanks.
With binary items, the model is not identified when factor loadings and thresholds are not freed in tandem. See the Version 7.1 language addendum on the website with the user's guide where we describe the models to use for testing for measurement invariance.