9.3 Relationship Between NLMMs and GLMMs

NLMMs can be viewed as a special case of GLMMs when the inverse link function corresponds to a nonlinear transformation of the linear predictor:

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

Here, \(g^{-1}(\cdot)\) represents the inverse link function, corresponding 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 (\(\epsilon_i\)) and random effects (\(\alpha_i\)) are both normal.

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