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 often challenging.

Therefore, an exploratory analysis is needed to learn the hierarchical factor structure from data. Unfortunately, there does not exist an identifiability theory for

the learnability of this hierarchical structure, nor 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. 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

Information

Preprint No.SS-2025-0196
Manuscript IDSS-2025-0196
Complete AuthorsJiawei Qiao, Yunxiao Chen, Zhiliang Ying
Corresponding AuthorsYunxiao Chen
Emailsy.chen186@lse.ac.uk

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Acknowledgments

The authors would like to thank the editor, the associate editor, and the reviewers for their

constructive and valuable comments, which have substantially improved the manuscript.