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Nonlinear mixed-effects models constitute a class of statistical models generalizing linear mixed-effects models. Like linear mixed-effects models, they are particularly useful in settings where there are multiple measurements within the same statistical units or when there are dependencies between measurements on related statistical units. Nonlinear mixed-effects models are applied in many fields including medicine, public health, pharmacology, and ecology.^[1][2]

Definition

While any statistical model containing both fixed effects and random effects is an example of a nonlinear mixed-effects model, the most commonly used models are members of the class of nonlinear mixed-effects models for repeated measures^[1]

${\displaystyle {y}_{ij}=f(\phi _{ij},{v}_{ij})+\epsilon _{ij},\quad i=1,\ldots ,M,\,j=1,\ldots ,n_{i}}$

where

• ${\displaystyle M}$ is the number of groups/subjects,
• ${\displaystyle n_{i}}$ is the number of observations for the ${\displaystyle i}$th group/subject,
• ${\displaystyle f}$ is a real-valued differentiable function of a group-specific parameter vector ${\displaystyle \phi _{ij}}$ and a covariate vector ${\displaystyle v_{ij}}$,
• ${\displaystyle \phi _{ij}}$ is modeled as a linear mixed-effects model ${\displaystyle \phi _{ij}={\boldsymbol {A}}_{ij}\beta +{\boldsymbol {B}}_{ij}{\boldsymbol {b}}_{i},}$ where ${\displaystyle \beta }$ is a vector of fixed effects and ${\displaystyle {\boldsymbol {b}}_{i}}$ is a vector of random effects associated with group ${\displaystyle i}$, and
• ${\displaystyle \epsilon _{ij}}$ is a random variable describing additive noise.

Estimation

When the model is only nonlinear in fixed effects and the random effects are Gaussian, maximum-likelihood estimation can be done using nonlinear least squares methods, although asymptotic properties of estimators and test statistics may differ from the conventional general linear model. In the more general setting, there exist several methods for doing maximum-likelihood estimation or maximum a posteriori estimation in certain classes of nonlinear mixed-effects models – typically under the assumption of normally distributed random variables. A popular approach is the Lindstrom-Bates algorithm^[3] which relies on iteratively optimizing a nonlinear problem, locally linearizing the model around this optimum and then employing conventional methods from linear mixed-effects models to do maximum likelihood estimation. Stochastic approximation of the expectation-maximization algorithm gives an alternative approach for doing maximum-likelihood estimation.^[4]

Applications

Example: Disease progression modeling

Nonlinear mixed-effects models have been used for modeling progression of disease.^[5] In progressive disease, the temporal patterns of progression on outcome variables may follow a nonlinear temporal shape that is similar between patients. However, the stage of disease of an individual may not be known or only partially known from what can be measured. Therefore, a latent time variable that describe individual disease stage (i.e. where the patient is along the nonlinear mean curve) can be included in the model.

Example: Modeling cognitive decline in Alzheimer's disease

Alzheimer's disease is characterized by a progressive cognitive deterioration. However, patients may differ widely in cognitive ability and reserve, so cognitive testing at a single time point can often only be used to coarsely group individuals in different stages of disease. Now suppose we have a set of longitudinal cognitive data ${\displaystyle (y_{i1},\ldots ,y_{in_{i}})}$ from ${\displaystyle i=1,\ldots ,M}$ individuals that are each categorized as having either normal cognition (CN), mild cognitive impairment (MCI) or dementia (DEM) at the baseline visit (time ${\displaystyle t_{i1}=0}$ corresponding to measurement ${\displaystyle y_{i1}}$). These longitudinal trajectories can be modeled using a nonlinear mixed effects model that allows differences in disease state based on baseline categorization:

${\displaystyle {y}_{ij}=f_{\tilde {\beta }}(t_{ij}+A_{i}^{MCI}\beta ^{MCI}+A_{i}^{DEM}\beta ^{DEM}+b_{i})+\epsilon _{ij},\quad i=1,\ldots ,M,\,j=1,\ldots ,n_{i}}$

where

• ${\displaystyle f_{\tilde {\beta }}}$ is a function that models the mean time-profile of cognitive decline whose shape is determined by the parameters ${\displaystyle {\tilde {\beta }}}$,
• ${\displaystyle t_{ij}}$ represents observation time (e.g. time since baseline in the study),
• ${\displaystyle A_{i}^{MCI}}$ and ${\displaystyle A_{i}^{DEM}}$ are dummy variables that are 1 if individual ${\displaystyle i}$ has MCI or dementia at baseline and 0 otherwise,
• ${\displaystyle \beta ^{MCI}}$ and ${\displaystyle \beta ^{DEM}}$ are parameters that model the difference in disease progression of the MCI and dementia groups relative to the cognitively normal,
• ${\displaystyle b_{i}}$ is the difference in disease stage of individual ${\displaystyle i}$ relative to his/her baseline category, and
• ${\displaystyle \epsilon _{ij}}$ is a random variable describing additive noise.

Zdroj:https://en.wikipedia.org?pojem=Nonlinear_mixed-effects_model
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