Simple structure estimation via prenet penalization

TL;DR

The paper introduces prenet, a novel penalization method for factor analysis that encourages simple, interpretable loading matrices, bridging the gap between perfect simple structures and traditional rotation techniques.

Contribution

It proposes a new prenet penalty that achieves perfect simple structure estimation and generalizes k-means clustering and quartimin rotation in factor analysis.

Findings

Prenet effectively produces simple, interpretable loading matrices.

It generalizes k-means clustering of variables.

It approximates quartimin rotation with mild penalization.

Abstract

We propose a $prenet$ ($pr$oduct $e$lastic $net$), which is a new penalization method for factor analysis models. The penalty is based on the product of a pair of elements in each row of the loading matrix. The prenet not only shrinks some of the factor loadings toward exactly zero, but also enhances the simplicity of the loading matrix, which plays an important role in the interpretation of the common factors. In particular, with a large amount of prenet penalization, the estimated loading matrix possesses a perfect simple structure, which is known as a desirable structure in terms of the simplicity of the loading matrix. Furthermore, the perfect simple structure estimation via the prenet turns out to be a generalization of the $k$-means clustering of variables. On the other hand, a mild amount of the penalization approximates a loading matrix estimated by the quartimin rotation, one of the most commonly used oblique rotation techniques. Thus, the proposed penalty bridges a gap between the perfect simple structure and the quartimin rotation. Monte Carlo simulation is conducted to investigate the performance of the proposed procedure. Three real data analyses are given to illustrate the usefulness of our penalty.