  Objectives: This paper develops two algorithms to achieve federated
generalized linear mixed effect models (GLMM), and compares the developed
model's outcomes with each other, as well as that from the standard R package
(`lme4').
  Methods: The log-likelihood function of GLMM is approximated by two numerical
methods (Laplace approximation and Gaussian Hermite approximation), which
supports federated decomposition of GLMM to bring computation to data.
  Results: Our developed method can handle GLMM to accommodate hierarchical
data with multiple non-independent levels of observations in a federated
setting. The experiment results demonstrate comparable (Laplace) and superior
(Gaussian-Hermite) performances with simulated and real-world data.
  Conclusion: We developed and compared federated GLMMs with different
approximations, which can support researchers in analyzing biomedical data to
accommodate mixed effects and address non-independence due to hierarchical
structures (i.e., institutes, region, country, etc.).
