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.