Latent variable models have been widely applied in different fields of research in which the constructs of interest are not directly observable, so that one or more latent variables are required to reduce the complexity of the data. In these cases, problems related to the integration of the likelihood function of the model arise since analytical solutions do not exist. In the recent literature, a numerical technique that has been extensively applied to estimate latent variable models is the adaptive Gauss-Hermite quadrature. It provides a good approximation of the integral, and it is more feasible than classical numerical techniques in presence of many latent variables and/or random effects. In this paper, we formally investigate the properties of maximum likelihood estimators based on adaptive quadratures used to perform inference in generalized linear latent variable models.