9.2 Generalized Linear Mixed Models

GLMMs extend GLMs by incorporating random effects, which allows for modeling data with hierarchical or clustered structures.

The conditional distribution of $y_i$ given the random effects $\boldsymbol{\alpha}_i$ is:

$$y_i \mid \boldsymbol{\alpha}_i \sim \text{independent } f(y_i \mid \boldsymbol{\alpha})$$

where $f(y_i \mid \boldsymbol{\alpha})$ belongs to the exponential family of distributions:

$$f(y_i \mid \boldsymbol{\alpha}) = \exp \left( \frac{y_i \theta_i - b(\theta_i)}{a(\phi)} - c(y_i, \phi) \right)$$

• $\theta_i$ is the canonical parameter,
• $a(\phi)$ is a dispersion parameter,
• $b(\theta_i)$ and $c(y_i, \phi)$ are specific functions defining the exponential family.

The conditional mean of $y_i$ is related to $\theta_i$ by:

$$\mu_i = \frac{\partial b(\theta_i)}{\partial \theta_i}$$

Applying a link function $g(\cdot)$, we relate the mean response to both fixed and random effects:

$$\begin{aligned} E(y_i \mid \boldsymbol{\alpha}) &= \mu_i \\ g(\mu_i) &= \mathbf{x}_i' \boldsymbol{\beta} + \mathbf{z}_i' \boldsymbol{\alpha} \end{aligned}$$

• $g(\cdot)$ is a known link function,
• $\mathbf{x}_i$ and $\mathbf{z}_i$ are design matrices for fixed and random effects, respectively,
• $\boldsymbol{\beta}$ represents fixed effects, and $\boldsymbol{\alpha}$ represents random effects.

We also specify the distribution of the random effects:

$$\boldsymbol{\alpha} \sim f(\boldsymbol{\alpha})$$

This distribution is often assumed to be multivariate normal (Law of large Number applies to fixed effects) but can be chosen (subjectively) based on the context.

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