Exploratory Hierarchical Factor Analysis with an Application to Psychological Measurement

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 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. 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.