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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} \tag{1} \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} \tag{2} \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.


Individual1.png

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