Gaussian models

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The normal distribution

Gaussian models have several advantages, including the capacity of describing with ease both the predicted value of a random variable and its fluctuations around this value. Indeed, if we consider a Gaussian random variable $\psi$ with mean $\mu$ and standard deviation $\omega$, we can work with two entirely equivalent mathematical representations:

\( \begin{eqnarray} \psi &\sim& {\cal N}(\mu , \omega^2) \end{eqnarray}\)
(1)
\( \begin{eqnarray} \psi &=& \mu + \eta, \quad {\rm where }\ \quad \ \eta \sim {\cal N}(0,\omega^2) . \end{eqnarray}\)
(2)

The form (1) provides an explicit description of the distribution of $\psi$ from which we can deduce the pdf and other characteristics such as the median, mode and quantiles. The figure below shows the pdf of a normal distribution with mean $\mu$ and standard deviation $\omega$. Each vertical band contains 10% of the distribution.


{{{2}}}
The ${\cal N}(\mu,\omega^2)$ distribution


This type of graphical representation is powerful and helps us to better visualize the types of values the random variable can take and those values that are more likely than others.

Examples of normal distributions with various parameters are shown in the next figure.


{{{2}}}
Normal distributions


Representation (2) lets us separate the random and non-random components of $\psi$. If we define as the predicted value the value obtained in the absence of randomness ($\eta=0$), we get that $\hat{\psi}=\mu$. In the particular case of a normal distribution, this predicted value is the mean, median and mode of $\psi$. We can therefore rewrite equations (1) and (2) using $\hpsi$:

\(\begin{eqnarray} \psi &\sim& {\cal N}(\hpsi , \omega^2) \\ \psi &=& \hpsi + \eta, \quad {\rm where } \quad \ \ \eta \sim {\cal N}(0,\omega^2) . \end{eqnarray}\)



Extensions of the normal distribution

Clearly, not all distributions are Gaussian. To begin with, the normal distribution has the support $\Rset$, unlike many parameters that take values in precise ranges; some variables take only positive values (e.g., concentrations and volumes) and others are restricted to bounded intervals (e.g., bioavailability).

Furthermore, the Gaussian distribution is symmetric, which is not a property shared by all distributions. One way to extend the use of Gaussian distributions is to consider that some transform of the parameters we are interested in is Gaussian, i.e., assume the existence of a monotonic function $h$ such that $h(\psi)$ is normally distributed. Then, there exists some $\mu$ and $\omega$ such that $h(\psi) \sim {\cal N}(\mu , \omega^2)$.

For a given transformation $h$, we can parametrize using $\hat{\psi}$, the predicted value of $\psi$. Indeed, the predicted value of $h(\psi)$ is $\mu=h(\hat{\psi})$, and

\(\begin{eqnarray} h(\psi) &\sim& {\cal N}(h(\hat{\psi}) , \omega^2) \end{eqnarray}\)
(3)
\(\begin{eqnarray} h(\psi) &=& h(\hat{\psi}) + \eta , \quad {\rm where } \quad \ \eta \sim {\cal N}(0,\omega^2). \end{eqnarray}\)
(4)

It is possible to derive the pdf of $\psi$ from (4):

\( \ppsi(\psi)=\displaystyle{ \frac{h^\prime(\psi)}{\sqrt{2 \pi \omega^2} } } \ \exp\left\{-\displaystyle{ \frac{1}{2 \, \omega^2} } (h(\psi) - h(\hpsi))^2 \right\}. \)
(5)

Let us now see some examples of transformed normal pdfs:



Log-normal distribution

The log-normal distribution is widely used for describing the distribution of PK/PD parameters. This choice is usually justified by the fact that it ensures non-negative values, and rarely because it is shown to properly describe the population distribution of the parameter of interest.

Let $\psi$ be a log-normally distributed random variable with parameters $(\mu,\omega)$:

\(\log(\psi) \sim {\cal N}( \mu, \omega). \)

This distribution can be also parameterized with $(m,\omega)$, where $m = \mu = \hat{\psi}$. Then, $\log(\psi) \sim {\cal N}( \log(m), \omega)$ and

\( \ppsi(\psi)=\displaystyle{ \frac{1}{\psi \, \sqrt{2 \pi \omega^2} } }\ \exp\left\{- \displaystyle{\frac{1}{2 \, \omega^2} (\log(\psi) - \log(m))^2} \right\}. \)

We display below some log-normal pdfs obtained with different parameters $(m,\omega)$.


{{{2}}}
Log-normal distributions


We see that for a given standard deviation $\omega$, the pdfs obtained for different $m$ are simply rescaled. On the other hand, for a given $m$ the asymmetry of the distribution increases when the standard deviation $\omega$ increases.


Remarks

Note that the log-normal distribution takes its values in $(0,+\infty)$. It is straightforward to define a rescaled distribution in $(a,+\infty)$ by shifting it:

\(\begin{eqnarray} \log(\psi-a) &\sim& {\cal N}( \log(m-a), \omega^2). \end{eqnarray}\)



Power-normal (or Box-Cox) distribution

This is the distribution of a random variable $\psi$ for which the Box-Cox transformation of $\psi$,

\(\begin{eqnarray} h(\psi) = \displaystyle{ \frac{\psi^\lambda -1}{\lambda} } \end{eqnarray}\)

(with $\lambda > 0$) follows a normal distribution ${\cal N}( \mu, \omega^2)$ truncated such that $h(\psi)>0$. It therefore takes its values in $(0,+\infty)$. The distribution converges to the log-normal distribution when $\lambda \to 0$ and a truncated normal distribution when $\lambda \to 1$. The main interest of a power-normal distribution is its ability to represent a distribution "between" the log-normal distribution and the normal distribution.

Here, $m = \hat{\psi} = (\lambda \mu + 1)^{1/\lambda}$. We display below several power-normal pdfs obtained with various parameter sets $(\lambda,m,\omega)$.


{{{2}}}
Power-normal distributions



Logit-normal and probit-normal distributions.

A random variable $\psi$ with a logit-normal distribution takes its values in $(0,1)$. The logit of $\psi$ is normally distributed, i.e.,

\(\begin{eqnarray} \logit(\psi) &= &\log \left(\displaystyle{ \frac{\psi}{1-\psi} }\right) \ \sim \ \ {\cal N}( \mu, \omega^2) \\ m &=& \displaystyle{ \frac{1}{1+e^{-\mu} } }. \end{eqnarray}\)

This means that $\mu=\logit(m)$.

A random variable $\psi$ with a probit-normal distribution also takes its values in $(0,1)$. Then, the probit of $\psi$ is normally distributed:

\(\begin{eqnarray} \probit(\psi) &= &\Phi^{-1}(\psi) \ \sim \ {\cal N}( \mu, \omega^2) \\ m &=& \Phi(\mu). \end{eqnarray}\)

This means that $\mu=\probit(m)$.

We can see in the figures below that the pdfs of the logit and probit distributions with the same $m$ and well-chosen $\omega$ are very similar. Thus, these two distributions can be used interchangeably for modeling the distribution of a parameter that takes its values in $(0,1)$.


{{{2}}}
Logit-normal and probit-normal distributions


Logit and probit transformations can be generalized to any interval $(a,b)$ by setting

\( \psi = a + (b-a)\tilde{\psi}, \)

where $\tilde{\psi}$ is a random variable that takes its values in $(0,1)$ with a logit (or probit) distribution.

Furthermore, it is easy to show that the probit-normal distribution with $m=0.5$ and $\omega=1$ is the uniform distribution on $(0,1)$. Thus, any uniform distribution can easily be derived from the probit-normal distribution.



Extension to transformed Student's $t$-distributions

These extensions (log-$t$, power-$t$, etc.) can be obtained simply by replacing the normal distribution of the random effects with a Student $t$-distribution. Such extensions can be useful for modeling heavy-tailed distributions. Several Student's $t$-distributions with different degrees of freedom (d.f.) are displayed below. The Student's $t$-distribution converges to the normal distribution as the d.f. increases, whereas heavy tails are obtained for small d.f.


{{{2}}}
Standardized normal and Student's $t$ probability distribution functions



$\mlxtran$ for the Gaussian model

Man02.jpg
Example


\(\begin{eqnarray} \logit(F_i) &\sim& {\cal N}(\logit(F_{\rm pop}), \omega_F^2) \\ \log(ka_i) &\sim& {\cal N}(\log(ka_{\rm pop}), \omega_{ka}^2) \\ V_i &\sim& {\cal N}(V_{\rm pop}, \omega_V^2) \\ \displaystyle{\frac{Cl_i^{\lambda_{Cl} } - 1}{\lambda_{Cl} } } &\sim& {\cal N}(\frac{Cl_{\rm pop}^{\lambda_{Cl} } - 1}{\lambda_{Cl} }, \omega_{Cl}^2) \end{eqnarray}\)
Monolix icon2.png
MLXTran
[INDIVIDUAL]
input={F_pop, ka_pop, V_pop, Cl_pop, lambda_Cl, 
       omega_F, omega_ka, omega_V, omega_Cl}

DEFINITION:
F  = {distribution=logitnormal,reference=F_pop,sd=omega_F}
ka = {distribution=lognormal,reference=ka_pop,sd=omega_ka}
V  = {distribution=normal,reference=V_pop,sd=omega_V}
Cl = {distribution=powernormal,
      reference=Cl_pop,power=lambda_Cl,sd=omega_Cl}


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