  Fitting cross-classified multilevel models with binary response is
challenging. In this setting a promising method is Bayesian inference through
Integrated Nested Laplace Approximations (INLA), which performs well in several
latent variable models. Therefore we devise a systematic simulation study to
assess the performance of INLA with cross-classified logistic data under
different scenarios defined by the magnitude of the random effects variances,
the number of observations, the number of clusters, and the degree of
cross-classification. In the simulations INLA is systematically compared with
the popular method of Maximum Likelihood via Laplace Approximation. By an
application to the classical salamander mating data, we compare INLA with the
best performing methods. Given the computational speed and the generally good
performance, INLA turns out to be a valuable method for fitting the considered
cross-classified models.
