A sequential reduction method for inference in generalized linear mixed models

Generalized linear mixed models are a natural and widely used class of models, but one in which the likelihood often involves an integral of very high dimension. Because of this intractability, many alternative methods have been developed for inference in these models. One class of approaches involves replacing the likelihood with some approximation, for example using Laplace's method or importance sampling. However, these approximations can fail in cases where the structure of the model is sparse, in that only a small amount of information is available on each random effect, especially when the data are binary. If there are n random effects in total, the likelihood may always be written as an n-dimensional integral over these random effects. If there are a large number of random effects, then it will be computationally infeasible to obtain an accurate approximation to this n-dimensional integral by direct numerical integration.