Structured latent factor analysis for large-scale data: identifiability, estimability, and their implications

Latent factor models are widely used to measure unobserved latent traits in so- cial and behavioral sciences, including psychology, education, and marketing. When used in a conrmatory manner, design information is incorporated as zero constraints on corresponding parameters, yielding structured (conrmatory) latent factor models. In this paper, we study how such design information aects the identiability and the estimation of a structured latent factor model. Insights are gained through both asymptotic and non-asymptotic analyses. Our asymptotic results are established under a regime where both the number of manifest variables and the sample size diverge, mo- tivated by applications to large-scale data. Under this regime, we dene the structural identiability of the latent factors and establish necessary and sucient conditions that ensure structural identiability. In addition, we propose an estimator which is shown to be consistent and rate optimal when structural identiability holds. Finally, a non-asymptotic error bound is derived for this estimator, through which the eect of design information is further quantied. Our results shed lights on the design of 1 large-scale measurement in education and psychology and have important implications on measurement validity and reliability.