Accuracy of Laplace approximation for discrete response mixed models

The Laplace approximation is amongst the computational methods used for estimation in generalized linear mixed models. It is computationally the fastest, but there hasn't been a clear analysis of when its accuracy is adequate. In this paper, for a few factors we do calculations for a variety of mixed models to show patterns in the asymptotic bias of the estimator based on the maximum of the Laplace approximation of the log-likelihood. The biggest factor for asymptotic bias is the amount of discreteness in the response variable; there is more bias for binary and ordinal responses than for a count response, and more bias for a count response when its support is mainly near 0. When there is bias, the bias decreases as the cluster size increases. Often, the Laplace approximation is adequate even for small cluster sizes. Even with bias, the Laplace approximation may be adequate for quick assessment of competing mixed models with different random effects and covariates.