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

In this section, we are interested in the implementation of individual parameter distributions . 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 of a unique individual *i*. The distribution plays a fundamental role since it describes the *inter-individual variability* of the individual parameter .

In Monolix, we consider that some transformation of the individual parameters is normally distributed and is a linear function of the covariates:

This model gives a clear and easily interpreted decomposition of the variability of around , i.e., of around :

The component describes part of this variability by way of covariates that fluctuate around a typical value .

The random component 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 . 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 is a mixture of distributions.