  Latent class models with covariates are widely used for psychological,
social, and educational research. Yet the fundamental identifiability issue of
these models has not been fully addressed. Among the previous research on the
identifiability of latent class models with covariates, Huang and Bandeen-Roche
(2004, Psychometrika, 69:5-32) studied the local identifiability conditions.
However, motivated by recent advances in the identifiability of the restricted
latent class models, particularly Cognitive Diagnosis Models (CDMs), we show in
this work that the conditions in Huang and Bandeen-Roche (2004) are only
necessary but not sufficient to determine the local identifiability of the
model parameters. To address the open identifiability issue for latent class
models with covariates, this work establishes conditions to ensure the global
identifiability of the model parameters in both strict and generic senses.
Moreover, our results extend to the polytomous-response CDMs with covariates,
which generalizes the existing identifiability results for CDMs.
