When conducting a multiple group CFA of categorical variables, under what conditions are the scale parameters in the second group identified? I cannot identify them if the factor loadings are not equal in the two groups, which seems reasonable but leads to another question.
If Model 1 has factor loadings free in two groups with scale factors set to 1 in both groups, and model two equates the loadings but frees the scale factors, are the models equivalent? They have the same number of estimated parameters, it's just that they are loadings in one model and scale parameters in the other.
Or is the time to fix the scale parameters when the item intercepts are equated between groups?