Non Invariance
Message/Author
 Evgenia  posted on Friday, January 18, 2013 - 5:01 am
I have a general question about Factor Mixture models. Having a 2-Class, 1 Factor (IRT model)
and allowing factor loading and thresholds different at each class implies that I have in each class a different factor? What does this difference mean? Difference in mean and variance , i.e., I have fj~N(mu_j, var_j).
Thanks alot
 Bengt O. Muthen posted on Friday, January 18, 2013 - 2:58 pm
Yes, then you have a different factor in each group and the factors can't be compared. Still, it relaxes the conditional independence assumption and it says that the within-class item correlations are different in the different classes. Or using other words, the "severity" dimension is defined differently in the different classes.
 Evgenia  posted on Thursday, January 24, 2013 - 1:39 am
I want to ask you one more question.
Having a 2-Class, 1 Factor (IRT model)
and assuming non invariant thresholds at each class and invariant loadings, do you have any guidance how to check if I have to assume a latent factor with equal variance at two classes or different variances at the two latent classes for my data, except AIC measure?
Assumming different variance means that I have one latent factor with the same meaning , -"severity"- but that there are different amounts of "severity" within each class?
Thanks alot
 Bengt O. Muthen posted on Thursday, January 24, 2013 - 10:09 am
I think class-varying factor variances is a good model - it is more parsimonious than having class-varying loadings.

To answer your question, I think you can test variance equality using a likelihood ratio chi-square test, so working with 2 times the loglikelihood difference.
 Evgenia  posted on Friday, March 01, 2013 - 10:01 pm
Having a Factor Mixture models, 2-Class, 1 Factor (IRT model) , with measurement invariance and mean and variance of factor allowed to be different, for identification of the model I fix mean (0) and variance (1) of one class and freely estimate them at the other one. (The alternative scenario is to fix mean (0) and the first factor loading in one group to 1). Are these the only necessary coefficients that I have to fix in order the model to be identified?
Thanks alot
 Linda K. Muthen posted on Saturday, March 02, 2013 - 2:31 pm
Yes.