# Model for the individual parameters: introduction

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.