Iteratively reweighted least squares with random effects for maximum likelihood in generalized linear mixed effects models

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This article develops a new method called iteratively reweighted least squares with random effects (IRWLSR) for maximum likelihood in generalized linear mixed effects models (GLMMs). As normal distributions are used for random effects, the likelihood functions contain intractable integrals except when the responses are normal. This often induces computational difficulties in fitting GLMMs for non-normal responses. The proposed IRWLSR successfully overcomes the difficulties as it only needs computational methods for linear mixed effects models and can be applied to any GLMMs with arbitrary link functions. It can be used even when high-dimensional intractable integrals appear in the likelihood function. The simulation study shows that the results are comparable to and sometimes are more precise than those from the Laplace approximation in the case when the Laplace approximation can be applied. It can also be applied to the case when the Laplace approximation cannot be applied.