Continuous data models

$ \newcommand{\pcyipsii}{p_{y_i|\psi_i}} \newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bu}{\boldsymbol{u}} \newcommand{\bx}{\boldsymbol{x}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\hazard}{h} \newcommand{\prob}[1]{ \mathbb{P}\!\left(#1\right)} \newcommand{\probs}[2]{ \mathbb{P}_{#1}\!\left(#2\right)} $

Introduction

We focus in this section on the model for the observations $\by=(\by_i, \ 1\leq i \leq N)$, i.e., the conditional probability distributions $\{\pcyipsii(\by_i | \bpsi_i), \ 1\leq i \leq N\}$, where

• $N$ is the number of subjects.
• $\by_i = (y_{ij}, \ 1\leq j \leq n_i)$ are the $n_i$ observations for individual $i$. Here, $y_{ij}$ is the measurement made on individual $i$ at time $t_{ij}$.
• $\bpsi_i$ 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 $\bx_i = (x_{ij}, \ 1\leq j \leq n_i)$. 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 {\it input terms} $\bu_i$. 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, see the Dynamical systems driven by ODEs chapter for more details.

In our framework, observations $\by$ are longitudinal. So, for a given individual $i$, the model has to describe the change in $\by_i=(y_{ij})$ 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 $\by_i$, while the fact it depends on the vector of individual parameters $\psi_i$ helps us to describe the inter-individual variability in $\by_i$.

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:

• Continuous data with a normal distribution:

\( y_{ij} \sim {\cal N}\left(f(t_{ij},\psi_i),\, g^2(t_{ij},\psi_i)\right) \)

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 $y_{ij}$.

• Categorical data with a Bernoulli distribution:

\( y_{ij} \sim {\cal B}\left(\lambda(t_{ij},\psi_i)\right) \)

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

• Count data with a Poisson distribution:

\( y_{ij} \sim {\cal P}\left(\lambda(t_{ij},\psi_i)\right) \)

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

• Time-to-event data:

\( \begin{array}{rcl} \prob{y_{i} >t} &= & S( t,\psi_i) \\[6pt] -\frac{d}{dt} \log S(t,\psi_i) &= & \hazard(t,\psi_i) \end{array} \)

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.