Test

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 <math>\bpsi_i<math> depends on a vector of individual covariates <math>\bc_i<math>: \begin{equation} \label{prior2} \bpsi_i \sim \ppsi(\bpsi_i | \bc_i ; \theta) \end{equation} A mixed effects model assumes here that $\bpsi_i$ can be decomposed as follows: \begin{equation} \label{prior3} \bpsi_i=H(\psipop,\fmu,\covariate_i,\eta_i) \end{equation} where \begin{itemize} \item '"`UNIQ-MathJax3-QINU`"' is a ``typical'' value of '"`UNIQ-MathJax4-QINU`"' in the population, \item '"`UNIQ-MathJax5-QINU`"' is a vector of covariates, \item '"`UNIQ-MathJax6-QINU`"' is a vector of {\it fixed effects}, \item '"`UNIQ-MathJax7-QINU`"' is a vector of {\it random effects}, {\it i.e.} '"`UNIQ-MathJax8-QINU`"' are random vectors with mean 0. \end{itemize}

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.