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 Joreskog (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 overestimated 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.
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