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Section: Research Program

The mixed-effects models

Mixed-effects models are statistical models with both fixed effects and random effects. They are well-adapted to situations where repeated measurements are made on the same individual/statistical unit.

Consider first a single subject i of the population. Let yi=(yij,1jni) be the vector of observations for this subject. The model that describes the observations yi is assumed to be a parametric probabilistic model: let pY(yi;ψi) be the probability distribution of yi, where ψi is a vector of parameters.

In a population framework, the vector of parameters ψi is assumed to be drawn from a population distribution pΨ(ψi;θ) where θ is a vector of population parameters.

Then, the probabilistic model is the joint probability distribution

To define a model thus consists in defining precisely these two terms.

In most applications, the observed data yi are continuous longitudinal data. We then assume the following representation for yi:

Here, yij is the observation obtained from subject i at time tij. The residual errors (εij) are assumed to be standardized random variables (mean zero and variance 1). The residual error model is represented by function g in model (2).

Function f is usually the solution to a system of ordinary differential equations (pharmacokinetic/pharmacodynamic models, etc.) or a system of partial differential equations (tumor growth, respiratory system, etc.). This component is a fundamental component of the model since it defines the prediction of the observed kinetics for a given set of parameters.

The vector of individual parameters ψi is usually function of a vector of population parameters ψ pop , a vector of random effects ηi𝒩(0,Ω), a vector of individual covariates ci (weight, age, gender, ...) and some fixed effects β.

The joint model of y and ψ depends then on a vector of parameters θ=(ψ pop ,β,Ω).