  In the era of data explosion, statisticians have been developing
interpretable and computationally efficient statistical methods to measure
latent factors (e.g., skills, abilities, and personalities) using large-scale
assessment data. In addition to understanding the latent information, the
covariate effect on responses controlling for latent factors is also of great
scientific interest and has wide applications, such as evaluating the fairness
of educational testing, where the covariate effect reflects whether a test
question is biased toward certain individual characteristics (e.g., gender and
race) taking into account their latent abilities. However, the large sample
size, substantial covariate dimension, and great test length pose challenges to
developing efficient methods and drawing valid inferences. Moreover, to
accommodate the commonly encountered discrete types of responses, nonlinear
latent factor models are often assumed, bringing further complexity to the
problem. To address these challenges, we consider a covariate-adjusted
generalized factor model and develop novel and interpretable conditions to
address the identifiability issue. Based on the identifiability conditions, we
propose a joint maximum likelihood estimation method and establish estimation
consistency and asymptotic normality results for the covariate effects under a
practical yet challenging asymptotic regime. Furthermore, we derive estimation
and inference results for latent factors and the factor loadings. We illustrate
the finite sample performance of the proposed method through extensive
numerical studies and an application to an educational assessment dataset
obtained from the Programme for International Student Assessment (PISA).
