$$\newcommand{\esp}{\mathbb{E}\left(#1\right)} \newcommand{\var}{\mbox{Var}\left(#1\right)} \newcommand{\deriv}{\dot{#1}(t)} \newcommand{\prob}{ \mathbb{P}\!(#1)} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\bc}{\boldsymbol{c}} \newcommand{\bpsi}{\boldsymbol{\psi}} \def\pmacro{\texttt{p}} \def\like{{\cal L}} \def\llike{{\cal LL}} \def\logit{{\rm logit}} \def\probit{{\rm probit}} \def\one{{\rm 1\!I}} \def\iid{\mathop{\sim}_{\rm i.i.d.}} \def\simh0{\mathop{\sim}_{H_0}} \def\df{\texttt{df}} \def\res{e} \def\xomega{x} \newcommand{\argmin}{{\rm arg}\min_{#1}} \newcommand{\argmax}{{\rm arg}\max_{#1}} \newcommand{\Rset}{\mbox{\mathbb{R}}} \def\param{\theta} \def\setparam{\Theta} \def\xnew{x_{\rm new}} \def\fnew{f_{\rm new}} \def\ynew{y_{\rm new}} \def\nnew{n_{\rm new}} \def\enew{e_{\rm new}} \def\Xnew{X_{\rm new}} \def\hfnew{\widehat{\fnew}} \def\degree{m} \def\nbeta{d} \newcommand{\limite}{\mathop{\longrightarrow}\limits_{#1}} \def\ka{k{a}} \def\ska{k{\scriptscriptstyle a}} \def\kel{k{e}} \def\skel{k{\scriptscriptstyle e}} \def\cl{C{\small l}} \def\Tlag{T\hspace{-0.1em}{lag}} \def\sTlag{T\hspace{-0.07em}{\scriptscriptstyle lag}} \def\Tk{T\hspace{-0.1em}{k0}} \def\sTk{T\hspace{-0.07em}{\scriptscriptstyle k0}} \def\thalf{t{1/2}} \newcommand{\Dphi}{\partial_\pphi #1} \def\asigma{a} \def\pphi{\psi} \newcommand{\stheta}{{\theta^\star}} \newcommand{\htheta}{{\widehat{\theta}}}$$

The models we are interested with are mixed effects models (see also this web animation), i.e. hierarchical models that involves different types of variables:

• We call $$y_i=(y_{ij},1\leq j \leq n_i)$$ the set of longitudinal data recorded at times $$(t_{ij},1\leq j \leq n_i)$$ for subject $$i$$, and $$\by$$ the combined set of observations for all $$N$$ individuals: $$\by = (y_1,\ldots,y_N)$$.

• We write $$\psi_i$$ for the vector of individual parameters for individual $$i$$ and $$\bpsi$$ the set of individual parameters for all $$N$$ individuals: $$\bpsi = (\psi_1,\ldots,\psi_N)$$.

• The distribution of the individual parameters $$\psi_i$$ of subject $$i$$ may depend on a vector of individual covariates $$c_i$$. Then, let $$\bc = (c_1,c_2,\ldots , c_N)$$.

• In a population approach context, we call $$\theta$$ the vector of population parameters.

Considering these variables as random variables, the joint probability distribution of $$\by$$, $$\bpsi$$, $$\bc$$ and $$\theta$$ can be decomposed into a product of conditional distributions:

$\pmacro(\by,\bpsi,\bc,\theta) = \pmacro(\by|\bpsi,\theta)\pmacro(\bpsi|\bc,\theta)\pmacro(\bc|\theta)\pmacro(\theta)$

Mlxtran takes advantage of the hierarchical structure of this joint probability distribution by decomposing the joint model into several submodels. Then, each component of the model is implemented in a different section:

 [LONGITUDINAL]

[INDIVIDUAL]

[COVARIATE]

[POPULATION]

In each section, Mlxtran supports flexible equation-based descriptions implemented in a block EQUATION and explicit definition-based descriptions of probability distributions in a block DEFINITION.

Models for described in the section [LONGITUDINAL] include models for continuous data, categorical data, count data and survival data.

We will see how to define a model for the individual parameters in the section [INDIVIDUAL], for the individual covariates in the section [COVARIATES] and for the population parameters in the section [POPULATION].