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# Model for the individual parameters: introduction

A model for observations depend on a vector of individual parameters ψ $\psi_i$. As we want to work with a population approach, we now suppose that $\psi_i$ comes from some probability distribution $p_{{\psi_i}}$.

In this section, we are interested in the implementation of individual parameter distributions $(p_{{\psi_i}}, 1\leq i \leq N)$. Generally speaking, we assume that individuals are independent. This means that in the following analysis, it is sufficient to take a closer look at the distribution $p_{{\psi_i}}$ of a unique individual i. The distribution $p_{{\psi_i}}$ plays a fundamental role since it describes the inter-individual variability of the individual parameter $\psi_i$.
In Monolix, we consider that some transformation of the individual parameters is normally distributed and is a linear function of the covariates:

$h(\psi_i) = h(\psi_{\rm pop})+ \beta \cdot ({c}_i - {c}_{\rm pop}) + \eta_i \,, \quad \eta_i \sim {\cal N}(0,\Omega).$

This model gives a clear and easily interpreted decomposition of the variability of $h(\psi_i)$ around $h(\psi_{\rm pop})$, i.e., of $\psi_i$ around $\psi_{\rm pop}$:

The component $\beta \cdot ({c}_i - {c}_{\rm pop})$ describes part of this variability by way of covariates ${c}_i$ that fluctuate around a typical value ${c}_{\rm pop}$.
The random component $\eta_i$ describes the remaining variability, i.e., variability between subjects that have the same covariate values.
By definition, a mixed effects model combines these two components: fixed and random effects. In linear covariate models, these two effects combine additively. In the present context, the vector of population parameters to estimate is $\theta = (\psi_{\rm pop},\beta,\Omega)$. Several extensions of this basic model are possible:

We can suppose for instance that the individual parameters of a given individual can fluctuate over time. Assuming that the parameter values remain constant over some periods of time called \emph{occasions}, the model needs to be able to describe the inter-occasion variability (IOV) of the individual parameters.
If we assume that the population consists of several homogeneous subpopulations, a straightforward extension of mixed effects models is a finite mixture of mixed effects models, assuming for instance that the distribution $p_{{\psi_i}}$ is a mixture of distributions.