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

TL;DR

This paper investigates the theoretical properties of structured latent factor models in large-scale data, establishing conditions for identifiability and proposing a consistent, computationally efficient estimator with practical implications for measurement design.

Contribution

It provides the first comprehensive analysis of identifiability and estimability in large-scale structured latent factor models, along with a scalable estimation method.

Findings

Necessary and sufficient conditions for structural identifiability are established.

A consistent estimator for latent factors is proposed and validated.

Simulation studies confirm the estimator's effectiveness and practical relevance.

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, 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.