Identification of Factor Scores by Regression with External Variables in Exploratory Factor Analysis

Factor score indeterminacy is a characteristic property of factor analysis (FA) models. This research introduces a novel procedure, regression-based factor score exploration (RFE), which uniquely determines factor scores and simultaneously estimates other parameters of the FA model. RFE uniquely determines factor scores by minimizing a loss function that balances FA and multivariate regression, regulated by a tuning parameter. Theoretical aspects of RFE, including the uniqueness of factor scores, the relationship between observed and latent variables, and rotational indeterminacy, are examined. Additionally, clustering-based factor exploration (CFE) is presented as a variant of RFE, derived by generalizing the penalty term to enable the clustering of factor scores. It is demonstrated that CFE creates cluster structures more accurately than the existing method. A simulation study shows that the proposed procedures accurately recover true parameter matrices even in the presence of error-contaminated data, with lower computational demand compared to existing methods. Real data examples illustrate that the proposed procedures provide interpretable results, demonstrating high relevance to the factor scores obtained by existing methods.