Non-linear mixed effects models

  • Method

Why does Ki use non-linear mixed effects (NLME) models? Nonlinear models provide a more flexible framework for integration of a wide range of geographically and socioeconomically diverse longitudinal data. This approach captures 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.


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 (CW) is displayed, and the model includes parameters for birth length (AW), growth velocity (BW), and curvature degree (DW) with the subscript “i” for individual measures.

Weighti,j= exp(AWi) + exp(BWi) *ti,j+ exp(CWi) * (1–exp(–exp (DWi) *ti,j)) + ei,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 (Aw) and growth velocity (BW) of a study population are fixed effects.
    • For example, in the Jenss-Bayley model, the average birth length (Aw) and growth velocity (BW) of a study population are fixed effects.
  • 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.



Ki has uses NLME in various ways including parametric non-linear model, full random effects model (FREM), and Bayesian NLME model.

The parametric NLME approach is the predominant statistical approach found in the literature. Ki also has implemented more novel approaches in global health like FREM and Bayesian NLME in child growth and cognitive development applications.

Full Random Effects Model (FREM)

  1. 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]
  2. FREM does not include fixed effects.
  3. An advantage to FREM is its ability to conduct analysis with missing covariate values.

Bayesian non-linear mixed effects model

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

Advantages of NLME3

  • 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.
  • Flexible modeling framework: Unlike linear models, NLME can more closely reflect the predictor-outcome relationship, because it is not dependent on a straight-line relationship.

Disadvantages of NLME

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

The Frequentist Approach utilizes assumptions 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.


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]


  1. Jenss R, Bayley N. A Mathematical Method For Studying The Growth of A Child. Human Biology. 1937;9(4):556-63.
  2. 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.
  3. Pinheiro JC, Bates DM. Mixed- Effects Models in S and S-PLUS. New York: Springer; 2000.
  4. MathWorks. Nonlinear Mixed- Effects Modeling. 2017; https:// simbio/ug/what-is-nonlinear-mixed-effects-modeling.html. Accessed October 5, 2017.

Last Updated

October, 2020