Difference between revisions of "Testing A joint probability"
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{{Equation1 | {{Equation1 | ||
|equation=<math> | |equation=<math> | ||
| − | \pypsi(\by,\bpsi) = \pcypsi(\by | + | \pypsi(\by,\bpsi) = \pcypsi(\by {{!}} \bpsi) \, \ppsi(\bpsi) .</math> }} |
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- 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 $\bt=(t_{ij} ,\ 1\leq i \leq N ,\ 1\leq j \leq n_i)$. | - 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 $\bt=(t_{ij} ,\ 1\leq i \leq N ,\ 1\leq j \leq n_i)$. | ||
| + | }} | ||
| + | |||
| + | |||
| + | {{Example | ||
| + | |title=Example | ||
| + | |text= Let us suppose $ N$ patients received the same treatment as the single patient did. We now have the same PK model [[#ex_proba1|(1.4)]] for each patient, except that each has its own individual PK parameters $ V_i$ and $ k_i$ and potentially its own residual error parameter $ a_i$: | ||
| + | |||
| + | {{EquationWithRef | ||
| + | |equation=<div id="ex_proba2a"><math> | ||
| + | y_{ij} \sim {\cal N} \left( \displaystyle{\frac{500}{V_i}e^{-k_i \, t_{ij} } } , a_i^2 \right). | ||
| + | </math></div> | ||
| + | |reference=(1.5) }} | ||
| + | |||
| + | Here, $\psi_i = (V_i,k_i,a_i)$. One possible model is then to assume the same residual error model for all patients, and log-normal distributions for $V$ and $k$: | ||
| + | |||
| + | {{Equation1 | ||
| + | |equation=<math>\begin{eqnarray} | ||
| + | a_i &=& a \end{eqnarray}</math> }} | ||
| + | {{EquationWithRef | ||
| + | |equation=<div id="ex_proba2b"><math>\begin{eqnarray} | ||
| + | \log(V_i) &\sim_{i.i.d.}& {\cal N}\left(\log(V_{\rm pop})+\beta\,\log(w_i/70),\ \omega_V^2\right) \end{eqnarray}</math></div> | ||
| + | |reference=(1.6) }} | ||
| + | {{Equation1 | ||
| + | |equation=<math>\begin{eqnarray} | ||
| + | \log(k_i) &\sim_{i.i.d.}& {\cal N}\left(\log(k_{\rm pop}),\ \omega_k^2\right). \end{eqnarray}</math> }} | ||
| + | |||
| + | where the only covariate we choose to consider, $w_i$, is the weight (in kg) of patient $i$. The model therefore consists of the conditional distribution of the concentrations defined in [[#ex_proba2a|(1.5)]] and the distribution of the individual PK parameters defined in [[#ex_proba2b|(1.6)]]. The inputs of the model are the population parameters $\theta = (V_{\rm pop},k_{\rm pop},\omega_V,\omega_k,\beta,a)$, the covariates (here, the weight) $(w_i, 1\leq i \leq N)$, and the design $\bt$. | ||
}} | }} | ||
Revision as of 15:24, 19 April 2013
$ \DeclareMathOperator{\argmin}{arg\,min} \DeclareMathOperator{\argmax}{arg\,max} \newcommand{\nominal}[1]{#1^{\star}} \newcommand{\psis}{\psi{^\star}} \newcommand{\phis}{\phi{^\star}} \newcommand{\hpsi}{\hat{\psi}} \newcommand{\hphi}{\hat{\phi}} \newcommand{\teps}{\varepsilon} \newcommand{\limite}[2]{\mathop{\longrightarrow}\limits_{\mathrm{#1}}^{\mathrm{#2}}} \newcommand{\DDt}[1]{\partial^2_\theta #1} \def\aref{a^\star} \def\kref{k^\star} \def\model{M} \def\hmodel{m} \def\mmodel{\mu} \def\imodel{H} \def\Imax{\text{\it Imax}} \def\id{ {\rm Id}} \def\teta{\tilde{\eta}} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \newcommand{\deriv}[1]{\frac{d}{dt}#1(t)} \newcommand{\pred}[1]{\tilde{#1}} \def\phis{\phi{^\star}} \def\hphi{\tilde{\phi}} \def\hw{\tilde{w}} \def\hpsi{\tilde{\psi}} \def\hatpsi{\hat{\psi}} \def\hatphi{\hat{\phi}} \def\psis{\psi{^\star}} \def\transy{u} \def\psipop{\psi_{\rm pop}} \newcommand{\psigr}[1]{\hat{\bpsi}_{#1}} \newcommand{\Vgr}[1]{\hat{V}_{#1}} \def\psig{\psi} \def\psigprime{\psig^{\prime}} \def\psigiprime{\psig_i^{\prime}} \def\psigk{\psig^{(k)}} \def\psigki{ {\psig_i^{(k)}}} \def\psigkun{\psig^{(k+1)}} \def\psigkuni{\psig_i^{(k+1)}} \def\psigi{ {\psig_i}} \def\psigil{ {\psig_{i,\ell}}} \def\phig{ {\phi}} \def\phigi{ {\phig_i}} \def\phigil{ {\phig_{i,\ell}}} \def\etagi{ {\eta_i}} \def\IIV{ {\Omega}} \def\thetag{ {\theta}} \def\thetagk{ {\theta_k}} \def\thetagkun{ {\theta_{k+1}}} \def\thetagkunm{\theta_{k-1}} \def\sgk{s_{k}} \def\sgkun{s_{k+1}} \def\yg{y} \def\xg{x} \def\qy{p_{_y}} \def\qt{p_{_t}} \def\qc{p_{_c}} \def\qu{p_{_u}} \def\qyi{p_{_{y_i}}} \def\qyj{p_{_{y_j}}} \def\qpsi{p_{_{\psi}}} \def\qpsii{p_{_{\psi_i}}} \def\qcpsith{p_{_{\psi|\theta}}} \def\qth{p_{_{\theta}}} \def\qypsi{p_{_{y,\psi}}} \def\qcypsi{p_{_{y|\psi}}} \def\qpsic{p_{_{\psi,c}}} \def\qcpsic{p_{_{\psi|c}}} \def\qypsic{p_{_{y,\psi,c}}} \def\qypsit{p_{_{y,\psi,t}}} \def\qcypsit{p_{_{y|\psi,t}}} \def\qypsiu{p_{_{y,\psi,u}}} \def\qcypsiu{p_{_{y|\psi,u}}} \def\qypsith{p_{_{y,\psi,\theta}}} \def\qypsithcut{p_{_{y,\psi,\theta,c,u,t}}} \def\qypsithc{p_{_{y,\psi,\theta,c}}} \def\qcypsiut{p_{_{y|\psi,u,t}}} \def\qcpsithc{p_{_{\psi|\theta,c}}} \def\qcthy{p_{_{\theta | y}}} \def\qyth{p_{_{y,\theta}}} \def\qcpsiy{p_{_{\psi|y}}} \def\qz{p_{_z}} \def\qw{p_{_w}} \def\qcwz{p_{_{w|z}}} \def\qw{p_{_w}} \def\qcyipsii{p_{_{y_i|\psi_i}}} \def\qyipsii{p_{_{y_i,\psi_i}}} \def\qypsiij{p_{_{y_{ij}|\psi_{i}}}} \def\qyipsi1{p_{_{y_{i1}|\psi_{i}}}} \def\qtypsiij{p_{_{\transy(y_{ij})|\psi_{i}}}} \def\qcyzipsii{p_{_{z_i,y_i|\psi_i}}} \def\qczipsii{p_{_{z_i|\psi_i}}} \def\qcyizpsii{p_{_{y_i|z_i,\psi_i}}} \def\qcyijzpsii{p_{_{y_{ij}|z_{ij},\psi_i}}} \def\qcyi1zpsii{p_{_{y_{i1}|z_{i1},\psi_i}}} \def\qcypsiz{p_{_{y,\psi|z}}} \def\qccypsiz{p_{_{y|\psi,z}}} \def\qypsiz{p_{_{y,\psi,z}}} \def\qcpsiz{p_{_{\psi|z}}} \def\qeps{p_{_{\teps}}} \def\neta{ {n_\eta}} \def\ncov{M} \def\npsi{n_\psig} \def\bu{\boldsymbol{u}} \def\bt{\boldsymbol{t}} \def\bT{\boldsymbol{T}} \def\by{\boldsymbol{y}} \def\bx{\boldsymbol{x}} \def\bc{\boldsymbol{c}} \def\bw{\boldsymbol{w}} \def\bz{\boldsymbol{z}} \def\bpsi{\boldsymbol{\psi}} \def\bbeta{\beta} \def\beeta{\eta} \def\logit{\rm logit} \def\transy{u} \def\so{O} \def\one{\mathbb 1} \newcommand{\prob}[1]{ \mathbb{P}\!\left(#1\right)} \newcommand{\probs}[2]{ \mathbb{P}_{#1}\!\left(#2\right)} \newcommand{\esp}[1]{\mathbb{E}\left(#1\right)} \newcommand{\esps}[2]{\mathbb{E}_{#1}\left(#2\right)} \newcommand{\var}[1]{\mbox{Var}\left(#1\right)} \newcommand{\vars}[2]{\mbox{Var}_{#1}\left(#2\right)} \newcommand{\std}[1]{\mbox{sd}\left(#1\right)} \newcommand{\stds}[2]{\mbox{sd}_{#1}\left(#2\right)} \newcommand{\corr}[1]{\mbox{Corr}\left(#1\right)} \def\pmacro{\mathbf{p}} \def\py{\pmacro} \def\pt{\pmacro} \def\pc{\pmacro} \def\pu{\pmacro} \def\pyi{\pmacro} \def\pyj{\pmacro} \def\ppsi{\pmacro} \def\ppsii{\pmacro} \def\pcpsith{\pmacro} \def\pth{\pmacro} \def\pypsi{\pmacro} \def\pcypsi{\pmacro} \def\ppsic{\pmacro} \def\pcpsic{\pmacro} \def\pypsic{\pmacro} \def\pypsit{\pmacro} \def\pcypsit{\pmacro} \def\pypsiu{\pmacro} \def\pcypsiu{\pmacro} \def\pypsith{\pmacro} \def\pypsithcut{\pmacro} \def\pypsithc{\pmacro} \def\pcypsiut{\pmacro} \def\pcpsithc{\pmacro} \def\pcthy{\pmacro} \def\pyth{\pmacro} \def\pcpsiy{\pmacro} \def\pz{\pmacro} \def\pw{\pmacro} \def\pcwz{\pmacro} \def\pw{\pmacro} \def\pcyipsii{\pmacro} \def\pyipsii{\pmacro} \def\pypsiij{\pmacro} \def\pyipsi1{\pmacro} \def\ptypsiij{\pmacro} \def\pcyzipsii{\pmacro} \def\pczipsii{\pmacro} \def\pcyizpsii{\pmacro} \def\pcyijzpsii{\pmacro} \def\pcyi1zpsii{\pmacro} \def\pcypsiz{\pmacro} \def\pccypsiz{\pmacro} \def\pypsiz{\pmacro} \def\pcpsiz{\pmacro} \def\peps{\pmacro} \def\vt{ {t} } \def\mlxtran{\mathbb MLXtran} $
Contents
Introduction
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.
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.
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.
Tasks such as estimation, model selection, simulation and optimization can then be expressed as specific ways of using this probability distribution.
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.
An illustrative example
A model for the observations of a single individual
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 $\py$ the distribution (or pdf) of $y$. If we assume that $y$ is a parametric model, then there exists a vector of parameters $\psi$ that completely define $y$.
We can then explicitly represent this dependency with respect to $\bpsi$ by writing $\qy( \, \cdot \, ; \psi)$ for the pdf of $y$.
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.
By convention, the variables which are before the symbol ";" are random variables. Those that are after the ";" are non-random parameters or variables. When there is no risk of confusion, the non-random terms can be left out of the notation.
A model for several individuals
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:
If $\qpsi$ is a parametric distribution that depends on a vector $\theta$ of {\it population parameters} and a set of {\it 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 $\bpsi$.
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.
