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Latent factor models are widely used to measure unobserved latent traits insocial and behavioral sciences, including psychology, education, and marketing.When used in a confirmatory manner, design information is incorporated,yielding structured (confirmatory) latent factor models. Motivated by theapplications of latent factor models to large-scale measurements which consistof many manifest variables (e.g. test items) and a large sample size, we studythe properties of structured latent factor models under an asymptotic settingwhere both the number of manifest variables and the sample size grow toinfinity. Specifically, under such an asymptotic regime, we provide adefinition of the structural identifiability of the latent factors andestablish necessary and sufficient conditions on the measurement design thatensure the structural identifiability under a general family of structuredlatent factor models. In addition, we propose an estimator that canconsistently recover the latent factors under mild conditions. This estimatorcan be efficiently computed through parallel computing. Our results shed lightson the design of large-scale measurement and have important implications onmeasurement validity. The properties of the proposed estimator are verifiedthrough simulation studies.

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