Chapter 9 Nonlinear and Generalized Linear Mixed Models

  • NLMMs extend the nonlinear model to include both fixed effects and random effects
  • GLMMs extend the generalized linear model to include both fixed effects and random effects.

A nonlinear mixed model has the form of

\[ Y_{ij} = f(\mathbf{x_{ij} , \theta, \alpha_i}) + \epsilon_{ij} \]

for the j-th response from cluster (or sujbect) i (\(i = 1,...,n\)), where

  • \(j = 1,...,n_i\)
  • \(\mathbf{\theta}\) are the fixed effects
  • \(\mathbf{\alpha}_i\) are the random effects for cluster i
  • \(\mathbf{x}_{ij}\) are the regressors or design variables
  • \(f(.)\) is nonlinear mean response function

A GLMM can be written as:

we assume

\[ y_i |\alpha_i \sim \text{indep } f(y_i | \alpha) \]

and \(f(y_i | \mathbf{\alpha})\) is an exponential family distribution,

\[ f(y_i | \alpha) = \exp [\frac{y_i \theta_i - b(\theta_i)}{a(\phi)} - c(y_i, \phi)] \]

The conditional mean of \(y_i\) is related to \(\theta_i\)

\[ \mu_i = \frac{\partial b(\theta_i)}{\partial \theta_i} \]

The transformation of this mean will give us the desired linear model to model both the fixed and random effects.

\[ E(y_i |\alpha) = \mu_i \\ g(\mu_i) = \mathbf{x_i' \beta + z'_i \alpha} \]

where \(g()\) is a known link function and \(\mu_i\) is the conditional mean. We can see similarity to GLM

We also have to specify the random effects distribution

\[ \alpha \sim f(\alpha) \]

which is similar to the specification for mixed models.

Moreover, law of large number applies to fixed effects so that you know it is a normal distribution. But here, you can specify \(\alpha\) subjectively.

Hence, we can show NLMM is a special case of the GLMM

\[ \mathbf{Y}_i = \mathbf{f}(\mathbf{x}_i, \mathbf{\theta, \alpha}_i) + \mathbf{\epsilon}_i \\ \mathbf{Y}_i = \mathbf{g}^{-1} (\mathbf{x}_i' \beta + \mathbf{z}_i' \mathbf{\alpha}_i) + \mathbf{\epsilon}_i \]

where the inverse link function corresponds to a nonlinear transformation of the fixed and random effects.

Note:

  • we can’t derive the analytical formulation of the marginal distribution because nonlinear combination of normal variables is not normally distributed, even in the case of additive error (\(e_i\)) and random effects (\(\alpha_i\)) are both normal.

Consequences of having random effects

The marginal mean of \(y_i\) is

\[ E(y_i) = E_\alpha(E(y_i | \alpha)) = E_\alpha (\mu_i) = E(g^{-1}(\mathbf{x_i' \beta + z_i' \alpha})) \]

Because \(g^{-1}()\) is nonlinear, this is the most simplified version we can go for.

In special cases such as log link (\(g(\mu) = \log \mu\) or \(g^{-1}() = \exp()\)) then

\[ E(y_i) = E(\exp(\mathbf{x_i' \beta + z_i' \alpha})) = \exp(\mathbf{x'_i \beta})E(\exp(\mathbf{z}_i'\alpha)) \]

which is the moment generating function of \(\alpha\) evaluated at \(\mathbf{z}_i\)


Marginal variance of \(y_i\)

\[ \begin{aligned} var(y_i) &= var_\alpha (E(y_i | \alpha)) + E_\alpha (var(y_i | \alpha)) \\ &= var(\mu_i) + E(a(\phi) V(\mu_i)) \\ &= var(g^{-1} (\mathbf{x'_i \beta + z'_i \alpha})) + E(a(\phi)V(g^{-1} (\mathbf{x'_i \beta + z'_i \alpha}))) \end{aligned} \]

Without specific assumption about \(g()\) and/or the conditional distribution of \(\mathbf{y}\), this is the most simplified version.


Marginal covariance of \(\mathbf{y}\)

In a linear mixed model, random effects introduce a dependence among observations which share any random effect in common

\[ \begin{aligned} cov(y_i, y_j) &= cov_{\alpha}(E(y_i | \mathbf{\alpha}),E(y_j | \mathbf{\alpha})) + E_{\alpha}(cov(y_i, y_j | \mathbf{\alpha})) \\ &= cov(\mu_i, \mu_j) + E(0) \\ &= cov(g^{-1}(\mathbf{x}_i' \beta + \mathbf{z}_i' \mathbf{\alpha}), g^{-1}(\mathbf{x}'_j \beta + \mathbf{z}_j' \mathbf{\alpha})) \end{aligned} \]

  • Important: conditioning to induce the covariability

Example:

Repeated measurements on the subjects. Let \(y_{ij}\) be the j-th count taken on the i-th subject.

then, the model is \(y_{ij} | \mathbf{\alpha} \sim \text{indep } Pois(\mu_{ij})\). Here

\[ \log(\mu_{ij}) = \mathbf{x}_{ij}' \beta + \alpha_i \]

where \(\alpha_i \sim iid N(0,\sigma^2_{\alpha})\)

which is a log-link with a random patient effect.