Interpretability Indices and Soft Constraints for Factor Models

Factor analysis is a way to characterize the relationships between many (observable) variables in terms of a smaller number of unobservable random variables which are called factors. However, the application of factor models and its success can be subjective or difficult to gauge, since infinitely many factor models that produce the same correlation matrix can be fit given sample data. Thus, there is a need to operationalize a criterion that measures how meaningful or "interpretable" a factor model is in order to select the best among many factor models. While there are already techniques that aim to measure and enhance interpretability, new indices, as well as rotation methods via mathematical optimization based on them, are proposed to measure interpretability. The proposed methods directly incorporate semantics with the help of natural language processing and are generalized to incorporate any "prior information". Moreover, the indices allow for complete or partial specification of relationships at a pairwise level. Aside from these, two other main benefits of the proposed methods are that they do not require the estimation of factor scores, which avoids the factor score indeterminacy problem, and that no additional explanatory variables are necessary. The implementation of the proposed methods is written in Python 3 and is made available together with several helper functions through the package interpretablefa on the Python Package Index. The methods' application is demonstrated here using data on the Experiences in Close Relationships Scale, obtained from the Open-Source Psychometrics Project.