**2.5 Multilevel generalized linear models**

Multilevel generalized linear models (MGLM) are an extension of generalized linear models. What makes both models different is that the former assumes dependence in the observations of the dependent variable and the latter assumes independence in the observations.

A three-level MGLM of the dependent variable *Y* conditioned in the random effects *ν* and *u* is

$$\lg[E(Y|\nu,\ \mu)]=\eta = \mathcal{X}\beta + \mathcal{W}\mu + \mathcal{Z}\nu \tag{23}$$

where *g*ð Þ� is the link function, which is a known monotonic, differentiable function, and *η* is the linear predictor. As in multilevel models, the random effects are

assumed to have a normal distribution with zero mean vector and variance/covariance matrixes *D* and *G*, respectively. The multilevel models described in Sections 2.1 and 2.2 are a particular case of MGLM with *g*, the identity function.
