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.