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Exploratory Structural Equation Modeling

Exploratory structural equation modeling (ESEM) is a data analytic framework developed in 2009 by Tihomir Asparouhov and Bengt Muthén and implemented in the Mplus statistical software. ESEM extends the structural equation modeling (SEM) framework to incorporate latent factors defined according to exploratory factor analysis (EFA) specifications. The ESEM framework can thus incorporate different sets of EFA factors (a set corresponding to a series of indicators related to a series of factors with all cross-loadings freely estimated within this set, but not between sets), confirmatory factor analysis (CFA) factors (where factor indicators are used to define their a priori factors without the incorporation of all possible cross-loadings), and observed variables measured using any combination of continuous and categorical measurement scales. These factors and observed variables can be correlated with one another and/or related via regressions and incorporate a variety of methodological controls (e.g., method factors, correlated uniquenesses) in a way that can be extended to multiple-group analyses, longitudinal analyses, or a combination of both.

ESEM makes available for EFA factors all of the statistical advances traditionally associated with CFA/SEM: (a) multiple-group or longitudinal tests of measurement invariance, (b) goodness-of-fit, (c) predictions among latent factors corrected for measurement error, (d) bifactor models, (e) a priori specification (i.e., confirmatory) using target rotation, (f) methodological controls, and (g) longitudinal analyses. This entry provides a review of how EFA and ESEM have been revived and answers questions about why it remains useful compared to CFA. Construct-relevant psychometric multidimensionality is then considered, followed by a concluding section on limitations of ESEM.

Reviving EFA

EFA, then referred to as factor analysis, was developed at the start of the 20th century by the pioneering work of psychologists, such as Charles Spearman, interested in understanding the structure of intelligence. EFA quickly became the approach of choice to study the underlying structure of the unobservable entities, referred to as psychological constructs, that form the core of psychological research. Many years later, in the 1970s, Karl Jöreskog developed an alternative approach to factor analyses, CFA, which allowed researchers to explicitly rely on a priori expectations to define factors and to obtain goodness-of-fit information regarding the ability of this representation to appropriately reflect the underlying structure of the data.

By merging path analytic methods with CFA, Jöreskog created a way to estimate predictive relations between CFA factors corrected for measurement errors, which came to be known as SEM. This new analytic framework rapidly superseded EFA, relegating its use to preliminary analyses of new measures for which a priori expectations were unclear, always with the caveat that these analyses should be replicated using CFA. ESEM revives EFA by making all of the advances traditionally reserved to CFA available to researchers interested in adopting an EFA approach. However, the apparent superiority of CFA is so well established that some questions regarding the true usefulness of EFA, and thus ESEM, remain.

Cross-Loadings and Parsimony

In CFA, all indicators are typically related to one, and only one, factor. However, research evidence has been accumulating for years that some well-established measures with a well-replicated EFA structure systematically fail to be supported using CFA. Statistical research has also demonstrated that whenever cross-loadings exist, CFA tends to produce inflated estimates of factor correlations, whereas the unnecessary incorporation of cross-loadings does not result in biased estimates. These inflated factor correlations carry the risk of creating unnecessary multicollinearity, leading to biased estimates of relations among constructs and to an underestimation of their construct validity. It remains true that adding all possible cross-loadings to a model reduces parsimony, which is why best-practice recommendations still favor CFA when both models have a comparable level of fit (particularly considering parsimony-adjusted indices) and factor correlations remain unchanged.

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