Towards more interpretable factor analysis: a focus on sparsity and uncertainty

Exploratory Factor Analysis (EFA) is a statistical technique for uncovering latent structures in multivariate data by modelling observed variables as linear combinations of unobserved factors. For interpretability, estimated loading matrices are often rotated to achieve sparsity, but existing rotation methods may lack sufficient accuracy or computational efficiency. This thesis introduces a new family of rotation criteria for recovering loading matrices with varying sparsity in EFA, based on component‑wise Lp loss functions. These criteria measure the sum of the pth powers of the absolute values of all factor loadings to encourage sparsity in the factor structure. To address the nonsmooth nature of this objective, we develop an iteratively reweighted gradient projection algorithm that achieves high accuracy with significantly reduced computational cost compared to penalised estimation techniques. We further establish novel identification conditions for the Lp rotation estimator, allowing for a small proportion of non‑simple items in the true loading matrix. Empirical results confirm that the Lp rotation criterion consistently outperforms classical rotation methods when the underlying factor structure is sparse. To support valid inference, we also propose a methodology for computing p‑values for factor loadings under the Lp framework. Building on these p‑values, we incorporate False Discovery Rate (FDR) control procedures—such as the Benjamini‑Yekutieli (BY) and e‑value‑based Benjamini‑Hochberg (eBH) methods—to guide variable selection while controlling the expected proportion of false discoveries. These procedures are demonstrated to remain valid across various experiments. The proposed Lp rotation framework has been implemented in the R package GPArotation, with functions lpT and lpQ available for orthogonal and oblique solutions, respectively. Overall, this thesis offers a unified approach to estimation, identification, and inference in EFA, contributing both theoretical insights and practical tools for sparse and interpretable factor analysis.