Modeling the observations
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 $i$.
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:
- 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 $y_{ij}$.
- Here, $\lambda(t_{ij},\psi_i)$ is the probability that $y_{ij}$ takes the value 1.
- Here, $\lambda(t_{ij},\psi_i)$ is the Poisson parameter, i.e., the expected value of $y_{ij}$.
- 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.