https://wiki.inria.fr/wikis/popix/index.php?title=TestMarc6&feed=atom&action=historyTestMarc6 - Revision history2024-03-29T08:57:02ZRevision history for this page on the wikiMediaWiki 1.32.6https://wiki.inria.fr/wikis/popix/index.php?title=TestMarc6&diff=4780&oldid=prevAdmin: Page créée avec « <div style="font-size:12pt;font-family:Helvetica"> <!-- some LaTeX macros we want to use: --> $ \newcommand{\argmin}[1]{ \mathop{\rm arg} \mathop{\rm min}\limits_{#1} } \... »2013-04-26T11:31:20Z<p>Page créée avec « <div style="font-size:12pt;font-family:Helvetica"> <!-- some LaTeX macros we want to use: --> $ \newcommand{\argmin}[1]{ \mathop{\rm arg} \mathop{\rm min}\limits_{#1} } \... »</p>
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$<br />
<br />
==Introduction==<br />
<br />
A model built for real-world applications can involve various types of variable, such as measurements, individual and population parameters, covariates, design, etc. The model allows us to represent relationships between these variables.<br />
<br />
If we consider things from a probabilistic point of view, some of the variables will be random, so the model becomes a probabilistic one, representing the joint distribution of these random variables.<br />
<br />
Defining a model therefore means defining a joint distribution. The hierarchical structure of the model will then allow it to be decomposed into submodels, i.e., the joint distribution decomposed into a product of conditional distributions.<br />
<br />
Tasks such as estimation, model selection, simulation and optimization can then be expressed as specific ways of using this probability distribution.<br />
<br />
<br />
{{OutlineTextL<br />
|text= <br />
- A model is a joint probability distribution. <br />
<br />
- A submodel is a conditional distribution derived from this joint distribution. <br />
<br />
- A task is a specific use of this distribution. <br />
}}<br />
<br />
We will illustrate this approach starting with a very simple example that we will gradually make more sophisticated. Then we will see in various situations what can be defined as the model and what its inputs are.<br />
<br />
<br />
<br><br />
<br />
==An illustrative example==<br />
<br />
<br><br />
===A model for the observations of a single individual===<br />
Let $y=(y_j, 1\leq j \leq n)$ be a vector of ''observations'' obtained at times $\vt=(t_j, 1\leq j \leq n)$. We consider that the $y_j$ are random variables and we denote $\qy$ the distribution (or pdf) of $y$. If we assume a ''parametric model'', then there exists a vector of parameters $\psi$ that completely define $y$.<br />
<br />
We can then explicitly represent this dependency with respect to $\bpsi$ by writing $\qy( \, \cdot \, ; \psi)$ for the pdf of $y$.<br />
<br />
If we wish to be even more precise, we can even make it clear that this distribution is defined for a given design, i.e., a given vector of times $\vt$, and write $ \qy(\, \cdot \, ; \psi,\vt)$ instead.<br />
<br />
By convention, the variables which are before the symbol ";" are random variables. Those that are after the ";" are non-random parameters or variables.<br />
When there is no risk of confusion, the non-random terms can be left out of the notation.<br />
<br />
<br />
{{OutlineText<br />
|text= <br />
-In this context, the model is the distribution of the observations $\qy(\, \cdot \, ; \psi,\vt)$. <br><br />
-The inputs of the model are the parameters $\psi$ and the design $\vt$.<br />
}}<br />
<br />
<br />
{{Example<br />
|title=Example<br />
|text= 500 mg of a drug is given by intravenous bolus to a patient at time 0. We assume that the evolution of the plasmatic concentration of the drug over time is described by the pharmacokinetic (PK) model<br />
<br />
{{Equation1<br />
|equation=<math> f(t;V,k) = \displaystyle{ \frac{500}{V} }e^{-k \, t} , </math> }}<br />
<br />
where $V$ is the volume of distribution and $k$ the elimination rate constant. The concentration is measured at times $(t_j, 1\leq j \leq n)$ with additive residual errors:<br />
<br />
{{Equation1<br />
|equation=<math> y_j = f(t_j;V,k) + e_j , \quad 1 \leq j \leq n . </math> }}<br />
<br />
Assuming that the residual errors $(e_j)$ are independent and normally distributed with constant variance $a^2$, the observed values $(y_j)$ are also independent random variables and<br />
<br />
{{EquationWithRef<br />
|equation=<div id="ex_proba1" ><math><br />
y_j \sim {\cal N} \left( f(t_j ; V,k) , a^2 \right), \quad 1 \leq j \leq n. </math></div><br />
|reference=(1.4) }}<br />
<br />
Here, the vector of parameters $\psi$ is $(V,k,a)$. $V$ and $k$ are the PK parameters for the structural PK model and $a$ the residual error parameter.<br />
As the $y_j$ are independent, the joint distribution of $y$ is the product of their marginal distributions:<br />
<br />
{{Equation1<br />
|equation=<math> \py(y ; \psi,\vt) = \prod_{j=1}^n \pyj(y_j ; \psi,t_j) ,<br />
</math> }}<br />
<br />
where $\qyj$ is the normal distribution defined in [[#ex_proba1|(1.4)]]. <br />
}}<br />
<br />
=== A model for several individuals ===<br />
<br />
Now let us move to $N$ individuals. It is natural to suppose that each is represented by the same basic parametric model, but not necessarily the exact same parameter values. Thus, individual $i$ has parameters $\psi_i$. If we consider that individuals are randomly selected from the population, then we can treat the $\psi_i$ as if they were random vectors. As both $\by=(y_i , 1\leq i \leq N)$ and $\bpsi=(\psi_i , 1\leq i \leq N)$ are random, the model is now a joint distribution: $\qypsi$. Using basic probability, this can be written as:<br />
<br />
{{Equation1<br />
|equation=<math> <br />
\pypsi(\by,\bpsi) = \pcypsi(\by {{!}} \bpsi) \, \ppsi(\bpsi) .</math> }}<br />
<br />
If $\qpsi$ is a parametric distribution that depends on a vector $\theta$ of ''population parameters'' and a set of ''individual covariates'' $\bc=(c_i , 1\leq i \leq N)$, this dependence can be made explicit by writing $\qpsi(\, \cdot \,;\theta,\bc)$ for the pdf of $\bpsi$.<br />
Each $i$ has a potentially unique set of times $t_i=(t_{i1},\ldots,t_{i \ \!\!n_i})$ in the design, and $n_i$ can be different for each individual.<br />
<br />
<br />
{{OutlineTextL<br />
|text= <br />
- In this context, the model is the joint distribution of the observations and the individual parameters:<br />
<br />
{{Equation1<br />
|equation=<math> \pypsi(\by , \bpsi; \theta, \bc,\bt)=\pcypsi(\by {{!}} \bpsi;\bt) \, \ppsi(\bpsi;\theta,\bc) . </math>}}<br />
<br />
- The inputs of the model are the population parameters $\theta$, the individual covariates $\bc=(c_i , 1\leq i \leq N)$ and the measurement times <br />
:$\bt=(t_{ij} ,\ 1\leq i \leq N ,\ 1\leq j \leq n_i)$.<br />
}}</div>Admin