## 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 ${y}_{i}=({y}_{ij},1\le j\le {n}_{i})$ 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 ${p}_{Y}({y}_{i};{\psi}_{i})$ 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 ${p}_{\Psi}({\psi}_{i};\theta )$ where $\theta $ is a vector of population parameters.

Then, the probabilistic model is the joint probability distribution

$p({y}_{i},{\psi}_{i};\theta )={p}_{Y}\left({y}_{i}\right|{\psi}_{i}){p}_{\Psi}({\psi}_{i};\theta )$ | (1) |

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 ${y}_{i}$:

${y}_{ij}=f({t}_{ij},{\psi}_{i})+g({t}_{ij},{\psi}_{i}){\epsilon}_{ij}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}1\le i\le N\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1\le j\le {n}_{i}.$ | (2) |

Here, ${y}_{ij}$ is the observation obtained from subject $i$ at time ${t}_{ij}$. The residual errors $\left({\epsilon}_{ij}\right)$ 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 ${\psi}_{i}$ is usually function of a vector of population parameters ${\psi}_{\mathrm{pop}}$, a vector of random effects ${\eta}_{i}\sim \mathcal{N}(0,\Omega )$, 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 $\theta =({\psi}_{\mathrm{pop}},\beta ,\Omega )$.