  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,
yielding structured (confirmatory) latent factor models. Motivated by the
applications of latent factor models to large-scale measurements which consist
of many manifest variables (e.g. test items) and a large sample size, we study
the properties of structured latent factor models under an asymptotic setting
where both the number of manifest variables and the sample size grow to
infinity. Specifically, under such an asymptotic regime, we provide a
definition of the structural identifiability of the latent factors and
establish necessary and sufficient conditions on the measurement design that
ensure the structural identifiability under a general family of structured
latent factor models. In addition, we propose an estimator that can
consistently recover the latent factors under mild conditions. This estimator
can be efficiently computed through parallel computing. Our results shed lights
on the design of large-scale measurement and have important implications on
measurement validity. The properties of the proposed estimator are verified
through simulation studies.
