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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 ${y}_{i}=\left({y}_{ij},1\le j\le {n}_{i}\right)$ be the vector of observations for this subject. The model that describes the observations ${y}_{i}$ is assumed to be a parametric probabilistic model: let ${p}_{Y}\left({y}_{i};{\psi }_{i}\right)$ be the probability distribution of ${y}_{i}$, where ${\psi }_{i}$ is a vector of parameters.

In a population framework, the vector of parameters ${\psi }_{i}$ is assumed to be drawn from a population distribution ${p}_{\Psi }\left({\psi }_{i};\theta \right)$ where $\theta$ 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 ${y}_{i}$ are continuous longitudinal data. We then assume the following representation for ${y}_{i}$:

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 ${\psi }_{i}$ is usually function of a vector of population parameters ${\psi }_{\mathrm{pop}}$, a vector of random effects ${\eta }_{i}\sim 𝒩\left(0,\Omega \right)$, a vector of individual covariates ${c}_{i}$ (weight, age, gender, ...) and some fixed effects $\beta$.

The joint model of $y$ and $\psi$ depends then on a vector of parameters $\theta =\left({\psi }_{\mathrm{pop}},\beta ,\Omega \right)$.