Should Regularization Replace Simple Structure Rotation in Exploratory Factor Analysis?

Abstract

Exploratory factor analysis (EFA) is an important tool when the measurement structure of psychological constructs is uncertain. Typically, factor rotation is applied to obtain interpretable results resembling a simple structure. However, an overwhelming multitude of rotation techniques is available of which none is unequivocally superior. Recently, regularization has been suggested as an alternative to factor rotation. In two simulation studies, we addressed the question if regularized EFA is a suitable alternative for rotated EFA. We compared their performance in recovering predefined factor loading patterns with varying amounts of cross-loadings. Elastic net regularized EFA yielded estimates comparable to rotated EFA. For complex loading patterns, both rotated and regularized EFA tended to underestimate cross-loadings and inflate factor correlations, but regularized EFA was able to recover loading patterns as long as a subset of items followed a simple structure. We conclude that regularization is a suitable alternative to factor rotation for psychometric applications.

Keywords:

• factor rotation
• regularization
• penalized maximum likelihood
• exploratory factor analysis
• structural equation modelling

Acknowledgments

We thank Jana Pförtner for her assistance in preparing the simulation study and for proof-reading an earlier draft of the manuscript. We also thank Richard Rau for his valuable comments on an earlier draft of this manuscript.

Supplementary material

Supplemental data for this article can be accessed here.

Notes

¹ For the sake of completeness, it should be noted that $\theta$ must not contain factor loadings and variable residuals at the same time because their strong relationship (the higher the factor loadings, the smaller the variable residuals) would lead to severe estimation problems (Jacobucci et al., 2016).

² Some readers may wonder if the ridge estimates were simply over-shrunken, explaining the low congruencies. This was not the case as indicated by low Pearson correlations between the ridge estimates and the population pattern ($0.65<r_{Pearson}<0.9$).