#### Eksplorerende strukturmodellering

##### Av Ronny Klæboe Dato: 15.8.2009

Jamilia—

The Browne (2001) reference is:

Browne, Michael (2001), “An Overview of Analytic Rotation in Exploratory Factor Analysis,” Multivariate Behavioral Research, 36, 111-150.

In this paper, Browne notes the sub-optimal nature of the process where

researchers start by applying overly constrained CFA models to data, then launch a process of modification using diagnostics. You find very few

authors who will endorse such an approach. Browne certainly does not.

Asparouhov and Muthen (2009) cite this paper in their introduction to “exploratory SEM:”

Asparouhov, Tihomer, and Bengt Muthen (2009), “Exploratory Structural Equation Modeling,” Structural Equation Modeling, 16, 397-438.

Their approach brings the generality of Mplus to a technique pioneered

by Bill Rozeboom in his HyBall package (still available free online):

http://web.psych.ualberta.ca/~rozeboom/

Let me also re-send Herbert Marsh’s message on this topic, from earlier

this year (below).

—Ed Rigdon

Dear Colleagues:

I wanted to share with you my excitement about the work I am doing with ESEM—a new Mplus procedure that combines the best features of exploratory and confirmatory factor analysis. It is very powerful, EFA

with: full mean structure tests of multigroup invariance, including differential item functioning; invariance over time with a priori correlated uniquenesses; growth modeling; MIMIC models; combining CFA factors and EFA factors in the same model etc.

At least in applied psychological work, the traditional Independent Cluster Model (ICM; each indicator loading on only one factor with no

cross-loadings) is rarely defensible. There are some very good instruments with a well-defined EFA factor structure that cannot be fit with an ICM-CFA structure. The typical strategies to deal with this have been to resort to dubious tricks to hide the mis-fit (e.g., parcels or scale scores) or to forgo the many advantages of CFA over EFA. With ESEM you can have your cake and eat it too.

In the Marsh et al paper we consider a very large database based on my SEEQ (student evaluation) data. It has a very robust, well-defined 9-factor EFA structure with much support for its construct validity, but has been criticized because ICM-CFA models do not fit the data very well. In this study, I again showed that the ICM-CFA structure did not fit very well and the average factor correlation was high (mdn r = was.72). The ESEM solution fit well, all nine factors were well-defined, and the average correlation dropped to .34 (because it allowed cross-loading that would otherwise inflate factor corrs).

Particularly in terms of being able to distinguish between multiple factors (and their discriminant validity) this is very important for applied research.

We developed a taxonomy of 13 models to evaluate multiple group mean structure and applied it to the SEEQ data. Results showed strong support for complete invariance (including factor loadings, uniquenesses, factor variance/covariances, item intercepts, and latent mean differences).

We then used MIMIC models to show that relations with background variables (workload/difficulty, class size, prior subject interest, expected grades) were small in size and varied systematically for different ESEM SET factors, supporting a construct validity interpretation of the relations and the discriminant validity of the latent factors.

Based on ESEM methodology, substantively important questions were addressed that could not be appropriately addressed with a traditional CFA approach. There are so many psychological instruments that cannot be fit with ICM-CFA because of complex cross-loadings (that inflate correlations when ignored) that I think ESEM will be used a lot. It probably will replace EFA in many applications but will also supplant the traditional ICM-CFA model in many applied studies. Indeed, the hugely inflated correlations among factors seems a serious problem with many CFA applications.

Two key studies will be published in SEM and prepublication versions are available on the Mplus website.

http://www.statmodel.com/esem.shtml

Marsh, H.W., Muthén, B., Asparouhov, A., Lüdtke, O., Robitzsch, A., Morin, A.J.S., & Trautwein, U. (2009). Exploratory Structural Equation Modeling, Integrating CFA and EFA: Application to Students’

Evaluations of University Teaching. Forthcoming in Structural Equation Modeling.

http://www.statmodel.com/download/ESEM%20SETs%20Final.pdf

Asparouhov, T. & Muthén, B. (2008). Exploratory structural equation modeling. Accepted for publication in Structural Equation Modeling.

http://www.statmodel.com/download/EFACFA810.pdf

I would welcome discussion on the potential of ESEM - particularly my suggestion that it is preferable to ICM-CFA in many applied studies and the usefulness of the 13-model taxonomy of mean structure invariance.