Exploratory Structural Equation Modeling

Abstract

Exploratory factor analysis (EFA) is a frequently used multivariate analysis technique in statistics. Jennrich and Sampson (1966) solved a significant EFA factor loading matrix rotation problem by deriving the direct Quartimin rotation. Jennrich was also the first to develop standard errors for rotated solutions, although these have still not made their way into most statistical software programs. This is perhaps because Jennrich's achievements were partly overshadowed by the subsequent development of confirmatory factor analysis (CFA) by Jöreskog (1969). The strict requirement of zero cross-loadings in CFA, however, often does not fit the data well and has led to a tendency to rely on extensive model modification to find a well-fitting model. In such cases, searching for a well-fitting measurement model may be better carried out by EFA (Browne, 2001). Furthermore, misspecification of zero loadings usually leads to distorted factors with over-estimated factor correlations and subsequent distorted structural relations. This article describes an EFA-SEM (ESEM) approach, where in addition to or instead of a CFA measurement model, an EFA measurement model with rotations can be used in a structural equation model. The ESEM approach has recently been implemented in the Mplus program. ESEM gives access to all the usual SEM parameters and the loading rotation gives a transformation of structural coefficients as well. Standard errors and overall tests of model fit are obtained. Geomin and Target rotations are discussed. Examples of ESEM models include multiple-group EFA with measurement and structural invariance testing, test–retest (longitudinal) EFA, EFA with covariates and direct effects, and EFA with correlated residuals. Testing strategies with sequences of EFA and CFA models are discussed. Simulated and real data are used to illustrate the points.

Notes

¹Examples of ESEM models illustrating structural invariance testing, EFA with covariates and direct effects, and EFA with correlated residuals are available in Bengt Muthén's multimedia presentation on this topic available at http://www.ats.ucla.edu/stat/mplus/seminars/whatsnewinmplus51/default.htm

²Note again, however, that Mplus will automatically use RowStandardization=Covariance, so that differences across groups in the residual variances Θ do not cause differences in the rotated solutions (see Appendix B).

³Using again RowStandardization=Covariance the estimated unrotated solution with equality of the loadings across groups and all Ψ = I leads to a rotated solution with equality in the rotated loadings as well as in the Ψ matrix (see Appendix B).

⁴All of these rotation criteria are implemented in Mplus.

⁵The Geomin rotation is now the default rotation criterion in Mplus.

⁶The Mplus default for ε for two factors is 0.0001, for three factors is 0.001, and for four or more factors it is 0.01.

⁷Mplus checks these conditions. If they fail, Mplus will automatically suggest alternative targets.

⁸This uses the Mplus MLR estimator.

⁹This uses the Mplus Type = Complex feature.

¹⁰The χ² difference testing using MLR is done as shown at www.statmodel.com/chidiff.shtml

¹¹Mplus also provides a standardized solution. This results in different loadings across groups due to different group variances for items and factors

¹²A tutorial on Mplus simulation studies with ESEM is available in Mplus 5.1, Examples Addendum available at www.statmodel.com/ugexcerpts.shtml. In addition, all Mplus input and outputs for the simulation studies presented in this article are available by e-mail from the second author ([email protected]).

¹³Note that Θ also influences the rotation through the correlation standardization.

¹⁴In Mplus the population-level rotations are obtained by generating a large sample, such as a sample with 1,000,000 observations. In such a large sample the estimated parameters are nearly identical with the population parameters.

¹⁵Future version of Mplus will include tools for resolving this problem.

¹⁶Mplus will automatically run 30 random starting values with the Geomin rotation. More random starting values can be requested using the rstarts= command. In addition the different rotation values are presented in regular EFA, as well as the loading structures for the different local minima. The ESEM output in Mplus 5.1 presents only the Geomin solution with lowest rotation function value.

¹⁷The standardization option is controlled in Mplus by the RowStandardization= command and the three options described earlier are RowStandardization= Correlation, Kaiser, or Covariance.

¹⁸In simulation studies for SEM models Mplus uses user-specified starting values to ensure that the order of the factors is the same across the replications. However, ESEM and EFA analysis in Mplus do not use user-specified starting values.

¹⁹Mplus will use the constraints in Equations D1 and D3 even for real data analysis, so the factors and their signs are always uniquely determined by Mplus.