Modeling the observations

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We focus in this section on the model for the observations $\by=(y_i, \ 1\leq i \leq N)$ when the individual parameters $\bpsi=(\psi_i, \ 1\leq i \leq N)$ are given, i.e., the the conditional probability distributions $({p_{_{y_i|\psi_i}}}, \ 1\leq i \leq N)$, where


    • $N$ is the number of subjects.
    • $y_i = (y_{ij}, \ 1\leq j \leq n_i)$ are the $n_i$ observations for individual $i$. Here, $y_{ij}$ is the measurement made on individual $i$ at time $t_{ij}$.
    • $\psi_i$ is the vector of individual parameters for subject $i$.


Remarks

  • We suppose that the model we will use to describe the observations is a function of regression variables $x_i = (x_{ij}, \ 1\leq j \leq n_i)$. Each $x_{ij}$ is made up of the time $t_{ij}$ and perhaps other variables that vary with time. For example, a pharmacokinetic model can depend on time and weight: $x_{ij} = (t_{ij},w_{ij})$ where $w_{ij}$ is the weight of individual $i$ at time $t_{ij}$, whereas a pharmacodynamic model can depend on time and concentration: $x_{ij} = (t_{ij},c_{ij})$.


  • The model for individual $i$ can also depend on input terms $u_i$. For example, a pharmacokinetic model include the dose regimen administrated to the patients: $\bu_i$ is made up of the dose(s) given to patient $i$, the time(s) of administration, and their type (IV bolus, infusion, oral, etc.). If the structural model is a dynamical system (e.g., defined by a system of ODEs), the input terms $(\bu_i)$ are also called source terms.


In our framework, observations $\by$ are longitudinal. So, for a given individual $i$, the model has to describe the change in $y_i=(y_{ij})$ over time. To do this, we suppose that each observation $y_{ij}$ comes from a probability distribution, one that evolves with time. As we have decided to work with parametric models, we suppose that there exists a function $\lambda$ such that the distribution of $y_{ij}$ depends on $\lambda(t_{ij},\psi_i)$. Implicitly, this includes the time-varying variables $x_{ij}$ mentioned above.

The time-dependence in $\lambda$ helps us to describe the change with time of each $y_i$, while the fact it depends on the vector of individual parameters $\psi_i$ helps us to describe the inter-individual variability in $y_i$.

We will distinguish in the following between continuous data models, discrete data models (including categorical and count data) and time-to-event (or survival) models.

Here are some examples of these various types of data:


    \(y_{ij} \sim {\cal N}\left(f(t_{ij},\psi_i),\, g^2(t_{ij},\psi_i)\right)\)
    Here, $\lambda(t_{ij},\psi_i)=\left(f(t_{ij},\psi_i),\,g(t_{ij},\psi_i)\right)$, where $f(t_{ij},\psi_i)$ is the mean and $g(t_{ij},\psi_i)$ the standard deviation of $y_{ij}$.


    \( y_{ij} \sim {\cal B}\left(\lambda(t_{ij},\psi_i)\right) \)
    Here, $\lambda(t_{ij},\psi_i)$ is the probability that $y_{ij}$ takes the value 1.


    \( y_{ij} \sim {\cal P}\left(\lambda(t_{ij},\psi_i)\right) \)
    Here, $\lambda(t_{ij},\psi_i)$ is the Poisson parameter, i.e., the expected value of $y_{ij}$.


    \( \begin{eqnarray} \prob { y_{i} >t} & = & S( t,\psi_i) \\[6pt] - \displaystyle{\frac{d}{dt} } \log S(t,\psi_i) & = & \hazard(t,\psi_i) \end{eqnarray}\)
    Here, $\lambda(t,\psi_i) = \hazard(t,\psi_i)$ is known as the hazard function.


In summary, defining a model for the observations means choosing a (parametric) distribution. Then, a model must be chosen for the parameters of this distribution.

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