Introduction and notation

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$ \def\cpop{c_{\rm pop}} \def\Vpop{V_{\rm pop}} \def\iparam{l} \newcommand{\trcov}[1]{#1} \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\aref{a^\star} \def\kref{k^\star} \def\model{M} \def\hmodel{m} \def\mmodel{\mu} \def\imodel{H} \def\thmle{\hat{\theta}} \def\ofim{I^{\rm obs}} \def\efim{I^{\star}} \def\Imax{\rm Imax} \def\probit{\rm probit} \def\vt{t} \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\pmacro{\text{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\pcpsiiyi{\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\pcetaiyi{\pmacro} \def\pypsiij{\pmacro} \def\pyipsiONE{\pmacro} \def\ptypsiij{\pmacro} \def\pcyzipsii{\pmacro} \def\pczipsii{\pmacro} \def\pcyizpsii{\pmacro} \def\pcyijzpsii{\pmacro} \def\pcyiONEzpsii{\pmacro} \def\pcypsiz{\pmacro} \def\pccypsiz{\pmacro} \def\pypsiz{\pmacro} \def\pcpsiz{\pmacro} \def\peps{\pmacro} \def\petai{\pmacro} \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\qx{p_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\qcpsiiyi{p_{\psi_i|y_i}} \def\qcetaiyi{p_{\eta_i|y_i}} \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\qetai{p_{\eta_i}} \def\neta{n_\eta} \def\ncov{M} \def\npsi{n_\psig} \def\beeta{\eta} \def\logit{\rm logit} \def\transy{u} \def\so{O} \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)} \newcommand{\Rset}{\mbox{$\mathbb{R}$}} \newcommand{\Yr}{\mbox{$\mathcal{Y}$}} \newcommand{\teps}{\varepsilon} \newcommand{\like}{\cal L} \newcommand{\logit}{\rm logit} \newcommand{\transy}{u} \newcommand{\repy}{y^{(r)}} \newcommand{\brepy}{\boldsymbol{y}^{(r)}} \newcommand{\vari}[3]{#1_{#2}^{{#3}}} \newcommand{\dA}[2]{\dot{#1}_{#2}(t)} \newcommand{\nitc}{N} \newcommand{\itc}{I} \newcommand{\vl}{V} \newcommand{tstart}{t_{start}} \newcommand{tstop}{t_{stop}} \newcommand{\one}{\mathbb{1}} \newcommand{\hazard}{h} \newcommand{\cumhaz}{H} \newcommand{\std}[1]{\mbox{sd}\left(#1\right)} \newcommand{\eqdef}{\mathop{=}\limits^{\mathrm{def}}} \def\mlxtran{\text{MLXtran}} \def\monolix{\text{Monolix}} $

Different representations of the same model

A description of the model involves several kinds of variables, including observations, individual parameters, population parameters, covariates,...

The tasks to execute concern these variables (estimation of the individual parameters, of the population parameters, ...). The algorithms used to perform these tasks may use different parametrizations, that is, different mathematical representations of the same model. We will see that, according to the task (estimation, simulation, likelihood calculation,...) some mathematical representations are more suitable than others.

There exist for the modeler a natural parametrization which involves a vector of individual parameters $\psi_i$ which have a physical or biological meaning (rate, volume, bioavailability,...). We will call {\it $\psi$-representation} the mathematical representation of the model which uses $\psi_i$.

\( \pyipsii(y_i , \psi_i ; \theta) = \pcyipsii(y_i | \psi_i)\ppsii( \psi_i ; \theta). \)
 (1)

When there exists a transformation $h: \Rset^d \to \Rset^d$ such that $\phi_i=h(\psi_i)$ is a Gaussian vector, we can use equivalently the $\phi$-representation which involves the transformed parameters (log-rate, log-volume, logit-bioavailability,...) and which represents the joint distribution of $y_i$ and $\phi_i$:

\( \pyiphii(y_i , \phi_i ; \theta, c_i) = \pcyiphii(y_i | \phi_i)\pphii( \phi_i ; \theta, c_i). \)
 (2)

where $ \phi_i =h(\psi_i) \sim {\cal N}( \mu(\beta,c_i) , \Omega)$ and $\theta=(\beta,\Omega)$.

Another mathematical representation uses the vector of random effects $\eta_i$ to represent the model of the individual parameters:

\(\begin{eqnarray} \phi_i &=& \mu(\beta,c_i) + \eta_i \end{eqnarray}\)

where $\eta_i \sim {\cal N}( 0 , \Omega)$. The $\eta$-representation then represents the joint distribution of $y_i$ and $\eta_i$:

\( \pyietai(y_i , \eta_i ; \theta, c_i) = \pcyietai(y_i | \eta_i;\beta,c_i)\petai( \eta_i ; \Omega). \)
 (3)

One can see that the fixed effects $\beta$ now appear in the conditional distribution of the observations. This will have a strong impact for tasks such as estimation of the population parameters since a sufficient statistic derived from this representation for estimating $\beta$ will be a function of the observations $\by$, contrary to the other representations for which the sufficient statistic is a function of the individual parameters $\bpsi$ (or equivalently $\bphi$).

In the $\psi$-representation (1), if model $\ppsii( \psi_i ; \theta, c_i)$ is not a regular statistical model (some components of $\psi_i$ may have no variability for instance, or more generally $\Omega$ may not be positive definite), there does not exist any sufficient statistic $S(\psi_i)$ for estimating $\theta$. Thus, algorithms for estimation will not use the representation as in (1), but another decomposition into regular statistical models.


Man02.jpg
Some examples


1. Consider the following model for continuous data with a constant error model:

\(\begin{eqnarray} y_{ij} &\sim& {\cal N}(f(t_{ij},\phi_i) ,a_i^2) \\ \phi_i &\sim& {\cal N}(\beta, \Omega) \\ a_i &\sim& p_a(\, \cdot \, ; \theta_a) \end{eqnarray}\)

Here, the variance of the residual error is a random variable. The vector of individual parameters is $(\phi_i, a_i)$ and the vector of population parameters is $\theta=(\beta,\Omega,\theta_a)$. Assuming that $\Omega$ is positive definite, the joint model of $y_i$ and $(\psi_i , a_i)$ can be decomposed as a product of three regular models:

\( \pyiphii(y_i , \phi_i, a_i ; \theta) = \pcyiphii(y_i | \phi_i ,a_i)\pphii( \phi_i ; \beta, \Omega)p_a(a_i ; \theta_a). \)


2. Assume now that the variance of the residual error is fixed in the population

\(\begin{eqnarray} y_{ij} &\sim& {\cal N}(f(t_{ij},\phi_i) ,a^2) \end{eqnarray}\)

The vector of population parameters is $\theta=(\beta,\Omega,a)$ and the joint model of $y_i$ and $\phi_i$ can be decomposed as

\(\pyiphii(y_i , \phi_i ; \theta) = \pcyiphii(y_i | \phi_i ; a)\pphii( \phi_i ; \beta, \Omega). \)


3. Assume now that some components of $\phi_i$ have no inter-individual variability. More precisely, let $\phi_i=(\phi_i^{(1)},\phi_i^{(0)})$ and $\beta=(\beta_1,\beta_0)$ such that

\(\begin{eqnarray} \phi_i^{(1)} &\sim& {\cal N}(\beta_1, \Omega_1) \\ \phi_i^{(0)} &=& \beta_0 \end{eqnarray}\)

where $\Omega_1$ is positive definite. Here, $\theta=(\beta_1,\beta_0,\Omega,a)$ and

\( \pyiphii(y_i , \phi_i^{(1)} ; \theta) = \pcyiphii(y_i | \phi_i^{(1)} ; \beta_0, a)\pphii( \phi_i^{(1)} ; \beta_1, \Omega_1). \)


4. Assume that $\phi_i = (\phi_{i,1}, \phi_{i,2})$ such that

\(\begin{eqnarray} \phi_{i,1} &=& \beta_1 + \omega_1\eta_i \\ \phi_{i,2} &=& \beta_2 + \omega_2\eta_i \end{eqnarray}\)

where $\eta_i \sim {\cal N}(0,1)$. A regular model is the joint distribution of $y_i$ and $\eta_i$. One can then use the following $\eta$-representation for instance:

\(\pyietai(y_i , \eta_i ; \theta) = \pcyietai(y_i | \eta_i ;\theta)\petai( \eta_i). \)

where $\theta= (\beta_1,\beta_2, \omega_1,\omega_2,a)$.



Some notations

Let $\theta$ be the set of population parameters. We assume that $\theta$ takes its values in $\Theta$, an open subset of $\Rset^m$.

Let $f : \Theta \to \Rset$ be a twice differentiable function of $\theta$. We will denote $\Dt{f(\theta)} = (\partial f(\theta)/\partial \theta_j, 1 \leq j \leq m) $ the gradient of $f$ (i.e. the vector of partial derivatives of $f$) and $\DDt{f(\theta)} = = (\partial^2 f(\theta)/\partial \theta_j\partial \theta_k, 1 \leq j,k \leq m) $ the Hessian of $f$ (i.e. the square matrix of second-order partial derivatives of $f$).

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