A model for observations depends on a vector of individual parameters \(\psi_i\). 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. 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 *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.