Simultaneous or separate estimate of structural equation modeling? A typical SEM could be seen as a combination of a measurement model and a structural model, and all parameters from both models could be estimated simultaneously. An alternative is a two-step way, i.e. first run the measurement model, extract the factor scores (using some regression method, or Bayesian way), then use these factor scores as dependent or independent variables, together with other observed variable, to run a regression model.
I found in some literature that this two-step way could avoid the "interpretational confounding" problem. e.g. the factor loadings in the measurement model at the first step won’t be affected by the structural model at the second step, which is not the case in the simultaneous SEM.
So my question is: how popular is the two-stage way and what is the advantage/disadvantage of it, and when to choose this instead of the simultaneous estimate of SEM? I've checked some papers and books, but they are not very clear to me. Any comments, suggestions, recommended papers/books are appreciated!
 A: The two-stage way typically ignores that


*

*factor scores are measured with error, and

*factor scores contain sampling error through estimation of the measurement error loadings.


The first problem leads to attenuation bias (the coefficients are shrunk towards zero), and the second problem, to underestimation of the variability of the regression coefficients (the standard errors may be too small). Both problems can be corrected for, but the appropriate corrections get so cumbersome that basically you are not better off in the end than in running the simultaneous SEM.
Update: The estimates will inevitably change since you are using different estimation methods. The changes between say DWLS and MLE do not surprise you, do they? The differences in estimates between a simultaneous method and a two-stage method should not surprise you either. Interpretational confounding is a big set of words for these differences. If the model is correctly specified, joint/simultaneous estimation will give you more efficient estimates, and a test for the overall accuracy of the model. If the model is incorrect, then two-stage may actually sweep the problems under the carpet, as you have fewer diagnostic tools to work with.
To do things more rigorously, an econometrician would formulate a Hausman test between the two sets of estimates to see if they are significantly different from one another. This test should be taught to all statisticians and all quantitative social science methodologists.
