
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].