# nonlinear mixed-effects model

• The platform of the nonlinear mixed effect models can be extended to consider the spatial association by incorporating the geostatistical processes such as Gaussian process
on the second stage of the model as follows: where • is a function that models the mean time-profile of log-scaled oil production rate whose shape is determined by the parameters .

•  The mixed-model approach allows modeling of both population level and individual differences in effects that have a nonlinear effect on the observed outcomes, for example
the rate at which a compound is being metabolized or distributed in the body.

•  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 where • is the number of groups/subjects, • is the number of observations for the th group/subject, • is a real-valued differentiable function of a group-specific
parameter vector and a covariate vector , • is modeled as a linear mixed-effects model where is a vector of fixed effects and is a vector of random effects associated with group , and • 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.

•  A research cycle using the Bayesian nonlinear mixed-effects model comprises two steps: (a) standard research cycle and (b) Bayesian-specific workflow.

• 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.

• Example: COVID-19 epidemiological modeling The platform of the nonlinear mixed effect models can be used to describe infection trajectories of subjects and understand
some common features shared across the subjects.

• A central task in the application of the Bayesian nonlinear mixed-effect models is to evaluate the posterior density: The panel on the right displays Bayesian research cycle
using Bayesian nonlinear mixed-effects model.

• 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.

• If Stage 3: Prior is not considered, then the model reduces to a frequentist nonlinear mixed-effect model.

• A popular approach is the Lindstrom-Bates algorithm 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.

• A basic version of the Bayesian nonlinear mixed-effects models is represented as the following three-stage: Stage 1: Individual-Level Model Stage 2: Population Model Stage
3: Prior Here, denotes the continuous response of the -th subject at the time point , and is the -th covariate of the -th subject.

• The Gaussian process regressions used on the latent level (the second stage) eventually produce kriging predictors for the curve parameters that dictate the shape of the mean
curve on the date level (the first level).

• The resulting posterior inference can be used to start a new research cycle.

• where the patient is along the nonlinear mean curve) can be included in the model.

• Therefore, a latent time variable that describe individual disease stage (i.e.

Works Cited

[‘Pinheiro, J; Bates, DM (2006). Mixed-effects models in S and S-PLUS. Statistics and Computing. New York: Springer Science & Business Media. doi:10.1007/b98882. ISBN 0-387-98957-9.
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9. ^ Lee, Se Yoon; Lei, Bowen; Mallick, Bani (2020). “Estimation of COVID-19 spread curves integrating global data and borrowing information”. PLOS ONE. 15 (7): e0236860. arXiv:2005.00662. doi:10.1371/journal.pone.0236860. PMC 7390340.
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10. ^ Lee, Se Yoon; Mallick, Bani (2021). “Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas”. Sankhya B. 84: 1–43. doi:10.1007/s13571-020-00245-8.
11. ^ Lee, Se Yoon (2022).
“Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications”. Mathematics. 10 (6): 898. arXiv:2201.12430. doi:10.3390/math10060898.
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Photo credit: https://www.flickr.com/photos/michaeljohnbutton/8838914676/’] 