# 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!

• What books did you read? If these were Bollen (1989), Skrondal and Rabe-Hesketh (2004) and Bartholomew and Knott (1999), you cannot get any better treatments! Jun 26, 2013 at 13:26
• For example Kline (2011), chapter 10; Hair et al (2009, Multivariate data analysis 7th edition). I dont see a clear preference to any of the two methods. Jun 26, 2013 at 14:39

The two-stage way typically ignores that

1. factor scores are measured with error, and
2. 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.

• If the factor is used as dependent variable in the second stage, then the measurement error is taken into account, but not the sampling error, correct? Also, should we worry about the "interpretational confounding" in the simultaneous SEM? I found in my case that the loadings changed somewhat when I changed the structural model in SEM. Jun 26, 2013 at 13:47
• If you have just one factor score, and it is a dependent variable in the structural regression, then what you have is a MIMIC model... which is a narrow subclass of general SEMs. In other circumstances, you will have to utilize the factor scores as explanatory variables leading to the measurement error issue. Jun 26, 2013 at 20:37
• But I can also run the MIMIC model in two steps, right? The reason is that I dont want the loadings change with the structural model which in my case is quite a complex multivariate regression model (I have 3 factors, all are treated as dependent variables in the structural part). Btw, I dont think there is any limitation for the number of factors in MIMIC model. Jun 26, 2013 at 21:19
• If you have that sort of a multi-factor MIMIC with exogenous observed variables used in the structural part as regressors, then you indeed don't have that first issue. That's a somewhat unusual model, that's all I wanted to point out -- most people just use factor scores in regression without a second thought. Jun 27, 2013 at 18:08
• Thanks very much. One more question, is there a reference for the two points you mentioned? especially the second point about the sampling error. Jul 4, 2013 at 15:52