Understanding Composite-Based Structural Equation Modeling Methods From the Perspective of Regression Component Analysis

Abstract

Regression component analysis (RCA) replaces the factors in a factor analysis model with weighted composites of the model’s observed variables. The weight matrix may be calculated from the factor model’s parameter estimates. Thus, RCA parameter estimates can be obtained using factor model software, but RCA composites have determinate scores, rather than the indeterminate scores of factors. Analytically, RCA equates to modeling with “regression method” factor scores, except that, while those scores will be inconsistent with the original factor model, they are strictly consistent with the RCA model. When the original factor model is strictly correct in the population and the composites in RCA are standardized, RCA parameter estimates replicate those from regression-weighted forms of partial least squares (PLS) path modeling and generalized structured component analysis (GSCA)—affirming that those methods also equate to modeling with regression method factor scores under the same conditions. Parallel measurement allows RCA to replicate both correlation weight and regression weight versions of PLS and GSCA. These results suggest that RCA and regression-weighted forms of PLS and GSCA are all consistent approaches for modeling data that conforms to a factor model. All analytical methods are described using one consistent symbol palette. Complete R syntax is provided.

Keywords:

• Structural equation modeling
• partial least squares
• generalized structured component analysis
• regression component analysis
• composite-based methods

Open Scholarship

This article has earned the Center for Open Science badges for Open Data and Open Materials through Open Practices Disclosure. The data and materials are openly accessible at https://doi.org/10.17605/OSF.IO/PVYNF and https://doi.org/10.17605/OSF.IO/PVYNF. To obtain the author's disclosure form, please contact the Editor.

Notes

1 One highly accomplished researcher, who was leading a workshop on exploratory factor analysis, characterized factor indeterminacy as “a nothingburger.”.

2 An editor points out that Equation (5), by itself is only a decomposition and not a falsifiable model. Schönemann and Steiger (1976, pp. 185–186), with their Theorems 4 and 5, showed an equivalence between a correct factor model and a constraint on the covariance matrix of the error terms in RCA. The constraint specifies that the covariance matrix can be rescaled to a diagonal idempotent matrix with P – K 1’s on the diagonal and 0’s everywhere else. With this constraint in place, RCA is a falsifiable model, and will be falsified if and only if the original factor model is falsified.

3 For consistency in calculations, $\Sigma$ should be the model-implied covariance matrix from the factor analysis that produced the other parameter estimates. In the absence of sampling variance and factor model misspecification, the empirical covariance matrix can be used equivalently.

4 While this analysis could have been conducted at the level of population moments, as an editor points out, factor indeterminacy is a phenomenon at the level of individual observations (Schönemann & Steiger, 1976). Therefore, this analysis proceeds using individual observations.