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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.

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