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

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.