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Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications

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  • Yunxiao Chen
  • Xiaoou Li
  • Siliang Zhang

Abstract

Abstract–Latent factor models are widely used to measure unobserved latent traits in social and behavioral sciences, including psychology, education, and marketing. When used in a confirmatory manner, design information is incorporated as zero constraints on corresponding parameters, yielding structured (confirmatory) latent factor models. In this article, we study how such design information affects the identifiability and the estimation of a structured latent factor model. Insights are gained through both asymptotic and nonasymptotic analyses. Our asymptotic results are established under a regime where both the number of manifest variables and the sample size diverge, motivated by applications to large-scale data. Under this regime, we define the structural identifiability of the latent factors and establish necessary and sufficient conditions that ensure structural identifiability. In addition, we propose an estimator which is shown to be consistent and rate optimal when structural identifiability holds. Finally, a nonasymptotic error bound is derived for this estimator, through which the effect of design information is further quantified. Our results shed lights on the design of large-scale measurement in education and psychology and have important implications on measurement validity and reliability.

Suggested Citation

  • Yunxiao Chen & Xiaoou Li & Siliang Zhang, 2020. "Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(532), pages 1756-1770, December.
    • Handle: RePEc:taf:jnlasa:v:115:y:2020:i:532:p:1756-1770
    DOI: 10.1080/01621459.2019.1635485
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    Cited by:

    1. Cosimo Magazzino & Marco Mele, 2022. "A Dynamic Factor and Neural Networks Analysis of the Co-movement of Public Revenues in the EMU," Italian Economic Journal: A Continuation of Rivista Italiana degli Economisti and Giornale degli Economisti, Springer;Società Italiana degli Economisti (Italian Economic Association), vol. 8(2), pages 289-338, July.
    2. Liu, Xinyi Lin & Wallin, Gabriel & Chen, Yunxiao & Moustaki, Irini, 2023. "Rotation to sparse loadings using Lp losses and related inference problems," LSE Research Online Documents on Economics 118349, London School of Economics and Political Science, LSE Library.
    3. Siliang Zhang & Yunxiao Chen, 2022. "Computation for Latent Variable Model Estimation: A Unified Stochastic Proximal Framework," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1473-1502, December.
    4. Y Chen & X Li, 2022. "Determining the number of factors in high-dimensional generalized latent factor models [Eigenvalue ratio test for the number of factors]," Biometrika, Biometrika Trust, vol. 109(3), pages 769-782.
    5. Alexander Robitzsch, 2023. "Linking Error in the 2PL Model," J, MDPI, vol. 6(1), pages 1-27, January.
    6. Leeb, William, 2021. "A note on identifiability conditions in confirmatory factor analysis," Statistics & Probability Letters, Elsevier, vol. 178(C).
    7. Zhang, Siliang & Chen, Yunxiao, 2022. "Computation for latent variable model estimation: a unified stochastic proximal framework," LSE Research Online Documents on Economics 114489, London School of Economics and Political Science, LSE Library.
    8. Xinyi Liu & Gabriel Wallin & Yunxiao Chen & Irini Moustaki, 2023. "Rotation to Sparse Loadings Using $$L^p$$ L p Losses and Related Inference Problems," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 527-553, June.
    9. Chen, Yunxiao & Li, Xiaoou, 2022. "Determining the number of factors in high-dimensional generalized latent factor models," LSE Research Online Documents on Economics 111574, London School of Economics and Political Science, LSE Library.

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