# Chapter 29 Generalized Linear Mixef-Effects Model

Example: Consider the experiment designed to evaluate the effectiveness of an anti-fungal chemical on plants. A total of 60 plant leaves were randomly assigned to treatment with 0, 5, 10, 15, 20, or 25 units of the anti-fungal chemical, with 10 plant leaves for each amount of anti-fungal chemical. All leaves were infected with a fungus. Following a two-week period, the leaves were studied under a microscope, and the number of infected cells was counted and recorded for each leaf.

Let \(\ell_i \sim N(0, \sigma_\ell^2)\) denote a random effect for the \(i\)th leaf. Suppose \(\log(\lambda_i) = \beta_0 + \beta_1x_i + \ell_i\) and \(y_i\mid \lambda_i \sim \text{Poisson}(\lambda_i)\). Finally, suppose \(\ell_1, \ldots, \ell_n\) are independent and that \(y_1, \ldots, y_n\) are conditionally independent given \(\lambda_1, \ldots, \lambda_n\).

**Lognormal distribution**: if \(\log(v) \sim N(\mu, \sigma^2 )\), then \(v\) is said to have a lognormal distribution. The mean and variance of a lognormal distribution are \(E(v) = \exp(\mu + \sigma^2/2)\), \(Var(v) = \exp(2\mu + 2\sigma^2) - \exp(2\mu + \sigma^2)\).

Suppose \(\log(v) \sim N(\mu, \sigma^2)\) and \(u\mid v \sim \text{Poisson}(v)\). Then \(E(u) = E(v) = \exp(\mu + \sigma^2/2)\),
\[
\begin{aligned}
\operatorname{Var}(u) &=E(\operatorname{Var}(u \mid v))+\operatorname{Var}(E(u \mid v))=E(v)+\operatorname{Var}(v) \\
&=\exp \left(\mu+\sigma^{2} / 2\right)+\exp \left(2 \mu+2 \sigma^{2}\right)-\exp \left(2 \mu+\sigma^{2}\right) \\
&=\exp \left(\mu+\sigma^{2} / 2\right)+\left(\exp \left(\sigma^{2}\right)-1\right) \exp \left(2 \mu+\sigma^{2}\right) \\
&=E(u)+\left(\exp \left(\sigma^{2}\right)-1\right)[E(u)]^{2}
\end{aligned}
\]
So that \(E(y_i) = \exp(\beta_0 + \beta_1x_i + \sigma_\ell^2/2)\) and \(Var(y_i) = E(y_i) + (\exp(\sigma_\ell^2) - 1)[E(y_i)]^2\). Moreover, we have
\[
f_i(y) = P(y_i = y) = \int_{0}^{\infty} \frac{\lambda^{y} \exp (-\lambda)}{y !} \frac{1}{\lambda \sqrt{2 \pi \sigma_{\ell}^{2}}} \exp \left\{\frac{-\left(\log (\lambda)-x_{i}^{\prime} \boldsymbol{\beta}\right)^{2}}{2 \sigma_{\ell}^{2}}\right\} d \lambda.
\]
There is no close-form expression for \(f_i(y)\) so the integral must be approximated by numerical methods. The `glmer`

function in `lme4`

package uses *Laplace approximation* to approximate the integral, but `glmer`

also permits the use of the more general integral approximation method known as *adaptive Gauss-Hermite quadrature*.

```
library(lme4)
= glmer(y ~ x + (1|leaf), family = poisson(link = "log")) # nAGQ = 10
o summary(o)
```

we can set

`nAGQ = xx`

to choose the number of points per axis for Gaussian-Hermite Approximation. The default is 1 (Laplace Approximation). Larger`nAGQ`

increases the accuracy but reduces the speed.