Accessibility setting

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

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 $y_i$:

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 ${\eta }_i\sim 𝒩(0,\Omega )$, 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 $\theta =({\psi }_{\mathrm{pop}},\beta ,\Omega )$.