A Profile-Based Framework for Factorial Similarity and the Congruence Coefficient

ABSTRACT

We present a novel profile-based framework for understanding factorial similarity in the context of exploratory factor analysis in general, and for understanding the congruence coefficient (a commonly used index of factor similarity) specifically. First, we introduce the profile-based framework articulating factorial similarity in terms of 3 intuitive components: general saturation similarity, differential saturation similarity, and configural similarity. We then articulate the congruence coefficient in terms of these components, along with 2 additional profile-based components, and we explain how these components resolve ambiguities that can be—and are—found when using the congruence coefficient. Finally, we present secondary analyses revealing that profile-based components of factorial are indeed linked to experts' actual evaluations of factorial similarity. Overall, the profile-based approach we present offers new insights into the ways in which researchers can examine factor similarity and holds the potential to enhance researchers' ability to understand the congruence coefficient.

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Acknowledgments

We would like to thank Urbano Lorenzo-Seva for his generosity in sharing his data set with us.

Funding

This project and publication was made possible through the support of a grant from the Templeton World Charity Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the Templeton World Charity Foundation.

Notes

¹ Factorial similarity is also frequently examined in the context of confirmatory factor analysis (CFA), and much attention has been given to those procedures in the form of multigroup CFA (e.g., Millsap & Meredith, 2007). Given the frequent use of EFA in many domains of psychological research, our current focus is on procedures mostly likely to be used by researchers conducting EFA, rather than CFA.

² These components are sometimes referred to as “elevation,” “scatter,” and “shape” in profile-based approaches (see Furr, 2010).

³ Differential (and general) saturation similarity can also be indexed in ways other than taking the difference in standard deviations (or means), including taking the ratio of the variances or the standard deviations. We use the difference method here for its relative simplicity.

⁴ However, it should be noted that there is some disagreement about the guidelines for interpreting the congruence coefficient, and there is relatively little research on what these guidelines are based on (see Lorenzo-Seva & ten Berge, 2006).

⁵ In addition, the arithmetic mean of the two differential saturations is held constant at .40. Similarly, Figure 1c (the effects of general saturation similarity) holds the arithmetic mean of the two general saturations constant at .50. We return to this later, in the context of joint differential saturation, joint general saturation, and the geometric mean.

⁶ As we noted earlier, although ϕ is the most widely used index of factorial similarity in EFA, there are others (R_V coefficient, KHB index, SVS index). Although it is beyond the scope of this article, such indexes also merit examination in terms of a profile-based framework.