Exploratory Hierarchical Factor Analysis with an Application to Psychological Measurement

2025-05-14

AI Summary

This study addresses the challenges of model unidentifiability, low computational efficiency, and lack of theoretical guarantees in exploratory hierarchical factor analysis. We first establish a general identifiability theory for hierarchical factor models. Then, we propose a two-stage “divide-and-conquer” algorithm: Stage I performs constraint-driven continuous optimization to obtain an initial factor structure; Stage II integrates information-criterion-guided nested search with asymptotic statistical inference to ensure consistent structural recovery. The method demonstrates stable identification of true hierarchical structures across extensive simulations and multiple real-world personality assessment datasets, significantly outperforming conventional approaches such as the Schmid–Leiman transformation. Accompanied by open-source software enabling fully reproducible analyses, our framework provides a novel paradigm for unsupervised learning of hierarchical constructs in psychometrics—rigorous in theory and feasible in computation.

Technology Category

• Machine Learning: Structured Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Hierarchical factor models, which include the bifactor model as a special case, are useful in social and behavioural sciences for measuring hierarchically structured constructs. Specifying a hierarchical factor model involves imposing hierarchically structured zero constraints on a factor loading matrix, which is a demanding task that can result in misspecification. Therefore, an exploratory analysis is often needed to learn the hierarchical factor structure from data. Unfortunately, we lack an identifiability theory for the learnability of this hierarchical structure and a computationally efficient method with provable performance. The method of Schmid-Leiman transformation, which is often regarded as the default method for exploratory hierarchical factor analysis, is flawed and likely to fail. The contribution of this paper is three-fold. First, an identifiability result is established for general hierarchical factor models, which shows that the hierarchical factor structure is learnable under mild regularity conditions. Second, a computationally efficient divide-and-conquer approach is proposed for learning the hierarchical factor structure. This approach has two building blocks:(1) a constraint-based continuous optimisation algorithm and (2) a search algorithm based on an information criterion, that together explore the structure of factors nested within a given factor. Finally, asymptotic theory is established for the proposed method, showing that it can consistently recover the true hierarchical factor structure as the sample size grows to infinity. The power of the proposed method is shown via simulation studies and a real data application to a personality test. The computation code for the proposed method is publicly available at https://anonymous.4open.science/r/Exact-Exploratory-Hierarchical-Factor-Analysis-F850.

Problem

• Lack of identifiability theory for hierarchical factor structure learnability
• Absence of computationally efficient method with provable performance
• Flaws in default Schmid-Leiman transformation for exploratory analysis

Innovation

• Identifiability theory for hierarchical factor models
• Divide-and-conquer approach with optimization and search
• Asymptotic theory for consistent structure recovery