in several threads I have seen people ask about changing the CFA default of using the 1st indicator of the model as a marker indicator to something else.
I was wondering: What difference does it make whether I use the marker indicator approach or the fixing the variance of the latent variable to (e.g.) 1 to establish the metric of the latent variable? In which situations would you use the former and in which the latter?
I have tried to find out a rule for this, but could not find anything on this issue. Books & articles just tell me that both methods exist but not when to use which.
I don't think there is any advantage of one method over the other. In all cases, model fit and the standardized results are the same. IRT tends to fix the factor variance to one. In a conditional model, the factor residual variance will be fixed to one. This may not be what is intended. I think it is most common to fix one factor indicator to one.
The main difference is related to multiple group invariance testing. When you specify the baseline configural invariance model, fixing the marker indicator to 1 in both groups already assumes its invariance. I prefer to fix the variance in both group at this step and when the loadings are specified as invariant to relax the variance in all groups but one.
It's a matter of taste of how to think about it, but the results won't be different. Alexandre's approach for these 2 steps give the same number of parameters and same fit as the default approach we take in Mplus (fixing the first loading at 1). For the configural invariance model (zeros in the same places across groups) Alexandre says "fixing the marker indicator to 1 in both groups already assumes its invariance", but given the same chi-square fit as his approach it is clear that this does not assume anything more - it is just moving a parameter from variance to loading.
Shi Yu posted on Wednesday, February 12, 2020 - 8:17 pm
When performing invariance tests that need to constrain loadings between groups (e.g., metric, scalar), I noticed that these two ways of identification yield different results. I have also noticed that generally the fixing marker loading @1 approach is used (for example, such is the method used in the built-in invariance testing program in Mplus). Could you explain why the fixed factor variance approach of model identification is not used in invariance testing? Could you provide a reference?
Those two alternatives should give the same answer. Check that you get the same number of parameters for the two.
If this doesn't help, send the two outputs to Support along with your license number.
Shi Yu posted on Friday, February 14, 2020 - 4:41 am
Dear Dr Muthen,
Thank you for your reply. But I do not understand why these two models have the same number of parameters. As an example, I have a 4-item single-factor model, with 2 groups. For the marker indicator approach, in total I estimate 3 loadings and 2 factor variances and 8 residuals, which is 13 parameters. For the fixing variance to 1 approach, in total I estimate 4 loadings, 0 factor variances, and 8 residuals, which is 12 parameters. Please let me know if I got anything wrong.
In the second approach, you want to free the factor variance in the second group. Otherwise you are saying they are the same.
Shi Yu posted on Saturday, February 15, 2020 - 7:11 pm
Dear Dr Muthen,
Thank you very much! The model results are now the same. I was wondering if you have a reference for this special treatment? Normally one would expect to fix the factor variance in each group to make it identified, but that appears to be wrong. I did not find the correct instructions on how to do this (invariance testing with fixed factor variances) in my factor analysis textbook or the Mplus user manual. It would be very helpful if you are aware of a text to refer to.
The metric (and scalar) assumption of equal loadings makes it possible to identify a free factor variance in one of the two groups. With configural it is of course not identified. It is a common way to parameterize - you may ask on semnet for a reference.