Test

$ \newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)} $

The population distribution of the individual parameters

Introduction

How to mathematically represent a mixed effects model?

Assume that the distribution of the vector of individual parameters $\psi_i$ depends on a vector of individual covariates $\beta c_i$:

\begin{equation} \label{prior2} \psi_i \sim \psi(\psi_i | \beta c_i ; \theta) \end{equation}

A mixed effects model assumes here that $\psi_i$ can be decomposed as follows:

\begin{equation} \label{prior3} \psi_i = H(\psi_{pop},\beta, c_i,\eta_i) \end{equation}

where

• $\psi_{pop}$ is a ``typical value of $\psi$ in the population,
• $c_i$ is a vector of covariates,
• $\beta$ is a vector of fixed effects,
• $\eta_i$ is a vector of random effects, i.e. $(\eta_i, 1 \leq i \leq N)$ are random vectors with mean 0.

In this model, part of the inter-individual variability is explained by the covariates $c_i$. The random effects describe the part of IIV which is not explained by the covariates.