Nonlinear models capture relationships between predictor variable(s) and outcomes and can be useful in circumstances when a linear model does not provide a good fit for the predictor-outcome relationship. For example, one well-known model for early childhood growth is the Jenss-Bayley model^{[1] }(Figure 1). Mixed effects methodology incorporates variations in predictor variables that occur across a population, as well as repeated measures within individuals. This statistical approach provides a more flexible framework for integration of a wide range of geographically and socioeconomically diverse longitudinal data, while not being constrained to a linear shape.

FIGURE 1. Longitudinal weight growth trajectories from birth to 5 years by deciles of growth spurt (Ci) in the first months of life in Children of the EDEN Study, France 2003-2012^{[2]}

NLME models “accommodate individual variations through random effects but ties different individuals together through population level fixed effects.”^{[3] }

A non-linear model has model parameters which define the shape of the mean response. For the Jenss-Bayley model in Figure 1 the spurt of growth parameter (C_{W}) is displayed, and the model includes parameters for birth length (A_{W}), growth velocity (B_{W}), and curvature degree (D_{W}) with the subscript “i” for individual measures.

**Weight**_{i,j}**= exp(A**_{Wi}**) + exp(B**_{Wi}**) *t**_{i,j}**+ exp(C**_{Wi}**) * (1–exp(–exp (D**_{Wi}**) *t**_{i,j}**)) + e**_{i,j }

In the mixed effects modeling paradigm, each unit of observation (e.g., a child in a birth cohort study) has unit-specific parameters.

Model parameters are called “mixed” effects because they include fixed effects and random effects.^{[4] }

**Fixed Effects**are components that are assumed to be the same value for each individual in the population. For example, in the Jenss-Bayley model, the average birth length (A_{w}) and growth velocity (B_{W}) of a study population are fixed effects.- For example, in the Jenss-Bayley model, the average birth length (A
_{w}) and growth velocity (B_{W}) of a study population are fixed effects.

- For example, in the Jenss-Bayley model, the average birth length (A

**Random Effects**are individual-based components associated with how individuals differ from the population average effects.- For example, in the Jenss-Bayley model individual BMI trajectories incorporate individual weight parameters and deviation from average population weight parameters to calculate between-individual variability.

- Random effects are typically assumed to follow a parametric (e.g., Gaussian) distribution.

**Parametric Statistical Methods **assume a mathematical and statistical structure of a hypothesized relationship between outcome and predictor variables.

The statistical structure frequently involves assumptions about the distribution of both the random effects (e.g., how the individual linear growth rates vary around the mean growth rate) and the residual errors (e.g., how the observed data vary around the model predictions).

These assumptions allow estimation of both the fixed and random effects.

This assumes the data in your sample is reflective of the population from which it was sampled.

For example, sample data assessing the impact of maternal smoking on growth trajectories among children in France assumes the established relationship of increased risk of obesity in adulthood with maternal smoking during pregnancy.^{[2]}

**Interpretability****:**The mathematical equations for NLME models can be derived from biological/mechanistic understanding, meaning they can model a pre-determined biological relationship between variables in the dataset. These mechanistic models are designed to assess causality.- Compared to a polynomial model, an NLME derived from biological/mechanistic knowledge provides more reliable predictions for outcome variables outside of the observed range of the data.
Unlike linear models, NLME can more closely reflect the predictor-outcome relationship, because it is not dependent on a straight-line relationship.**Flexible modeling framework**:

A simulated function (at each time step for a repeated measure dataset) is required to produce a numerical estimate.

NLME is computationally intensive.

See the “Parametric statistical methods” bullet in the “What is a NLME model?” section above. This is the predominant statistical approach for NLME.

FREM incorporates random variation between multiple covariates and within individual observations of a single covariate (using variance and covariance estimates) to model the outcome as a function of the covariate(s).^{[4] }

FREM does not include fixed effects.

An advantage to FREM is its ability to conduct analysis with missing covariate values.

This model differs from other NLME models in that it takes a Bayesian approach, rather than a frequentist approach.

**The Frequentist Approach**** **utilizes assumptions (discussed above) of the predictor-outcome relationship between the data within your sample and the population overall. These parameters are fixed over repeated random samples.

**The Bayesian Approach**** **leverages prior knowledge of the model or its parameters, allowing parameters to vary based on the distribution of repeated random samples.

*Example: *Extensive developmental research and improvements in imaging technology have informed head circumference data, such that additional parameter information by gender and geography can predict estimated gestational age. Variations in the head circumference distribution based on gender and geography provide a better model fit for data.

- Slide Deck from North Carolina State University: An Introduction to Nonlinear Mixed Effects Models and PK/PD Analysis
- Mixed effect model, 14-minute video

- Jenss R, Bayley N. A Mathematical Method For Studying The Growth of A Child. Human Biology. 1937;9(4):556-63.
- Carles S, Charles M-A, Forhan A, et al. A Novel Method to Describe Early Offspring Body Mass Index (BMI) Trajectories and to Study Its Determinants. PloS one. 2016;11(6):e0157766.
- Pinheiro JC, Bates DM. Mixed- Effects Models in S and S-PLUS. New York: Springer; 2000.
- MathWorks. Nonlinear Mixed- Effects Modeling. 2017; https:// www.mathworks.com/help/ simbio/ug/what-is-nonlinear-mixed-effects-modeling.html. Accessed October 5, 2017.