Choosing the Optimal Number of Factors in Exploratory Factor Analysis: A Model Selection Perspective

Abstract

A central problem in the application of exploratory factor analysis is deciding how many factors to retain (m). Although this is inherently a model selection problem, a model selection perspective is rarely adopted for this task. We suggest that Cudeck and Henly's (1991) framework can be applied to guide the selection process. Researchers must first identify the analytic goal: identifying the (approximately) correct m or identifying the most replicable m. Second, researchers must choose fit indices that are most congruent with their goal. Consistent with theory, a simulation study showed that different fit indices are best suited to different goals. Moreover, model selection with one goal in mind (e.g., identifying the approximately correct m) will not necessarily lead to the same number of factors as model selection with the other goal in mind (e.g., identifying the most replicable m). We recommend that researchers more thoroughly consider what they mean by “the right number of factors” before they choose fit indices.

Notes

¹Famous examples of incorrect but useful models are Newtonian physics and the Copernican theory of planetary motion.

²Model complexity is not to be confused with factor complexity, which is the number of factors for which a particular MV serves as an indicator (Bollen, 1989; Browne, 2001; Comrey & Lee, 1992; Thurstone, 1947; Wolfle, 1940). Our definition of complexity mirrors that commonly used in the mathematical modeling literature and other fields and is different from the restrictive definition sometimes seen in the factor-analytic literature in reference to a model's degrees of freedom (Mulaik, 2001; Mulaik et al., 1989). Model complexity and fitting propensity (Preacher, 2006) are identical concepts.

³The framework of Cudeck and Henly (1991) finds a close parallel in the framework presented by MacCallum and Tucker (1991) to identify sources of error in EFA.

⁴ MacCallum (2003) noted that OD should be regarded as an aspect or facet of verisimilitude. We consider OD more closely related to generalizability, which represents a model's balance of fit and complexity. It is possible for a model to have high verisimilitude and low generalizability (this situation frequently occurs when models are fit to small samples, as in the top right panel of Figure 1), but it is rare to find models with low verisimilitude and high generalizability. Because OD ≈ DA + DE we regard verisimilitude as an aspect of OD, not the reverse, and we consider verisimilitude a fixed quantity that is independent of sampling.

⁵ http://quantpsy.org

^a The 10-factor solution contained an extremely large negative unique variance for the fourth values on social love (VSL) item, compromising the admissibility and interpretability of this solution.

*Asterisks indicate the selected model (for AIC, BIC, RMSEA, and RMSEA.LB) or the model with the highest CV-lnL.