Generalized linear mixed model

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Generalized linear mixed models (GLMMs) extend generalized linear models by adding random effects to the usual fixed effects. They are useful for analyzing grouped data and non-normal outcomes, such as measurements taken over time on the same subjects.

In a GLMM, after conditioning on the random effects u, the response y follows a distribution from the exponential family. Its mean relates to the linear predictor Xβ + Zu through a link function g. Here Xβ contains the fixed effects, and Zu represents the random effects. A common special case is when the random effects u are normally distributed.

The full likelihood for GLMMs is difficult to compute because it requires integrating over the random effects; there is generally no closed-form solution, making estimation computationally intensive.

To fit GLMMs, researchers use approximations and numerical methods such as Gauss–Hermite quadrature, Laplace approximation, or penalized quasi-likelihood, and often employ maximum likelihood estimation. Model selection frequently uses the Akaike information criterion (AIC). Recent work has extended AIC calculations to GLMMs for certain distributions.