Likelihood ratio tests with marginal maximum likelihood using Laplace approximations and adaptive quadrature

Estimation of generalized linear latent variable models is typically done with marginal maximum likelihood (MML) estimation which requires approximation methods to compute integrals without a closed form solution. In applied settings, likelihood ratio tests are often used to test hypotheses regarding parameter restrictions. This study examines the properties of likelihood ratio tests when using approximate MML estimation. We consider first- and second-order Laplace approximations, along with adaptive Gauss-Hermite quadrature with varying numbers of quadrature points. In two simulation studies we perform likelihood ratio tests, varying data types (binary, ordinal, and count data), sample sizes, and the number of observed variables. The first simulation examines slope parameter equivalence, while the second focuses on group restrictions in multiple group models. Results indicate adaptive quadrature and second-order Laplace approximations provide reliable inference across most settings. Meanwhile, the first-order Laplace approximation generally works well except in some binary data cases.