9.1 Nonlinear Mixed Models

A general form of a nonlinear mixed model is:

$$Y_{ij} = f(\mathbf{x}_{ij}, \boldsymbol{\theta}, \boldsymbol{\alpha}_i) + \epsilon_{ij}$$

for the $j$-th response from the $i$-th cluster (or subject), where:

• $i = 1, \ldots, n$ (number of clusters/subjects),
• $j = 1, \ldots, n_i$ (number of observations per cluster),
• $\boldsymbol{\theta}$ represents the fixed effects,
• $\boldsymbol{\alpha}_i$ are the random effects for cluster $i$,
• $\mathbf{x}_{ij}$ are the regressors or design variables,
• $f(\cdot)$ is a nonlinear mean response function,
• $\epsilon_{ij}$ represents the residual error, often assumed to be normally distributed with mean 0.

NLMMs are particularly useful when the relationship between predictors and the response cannot be adequately captured by a linear model.

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