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Probability distribution of the individual parameters


Objectives: learn how to define the probability distribution and the correlation structure of the individual parameters.


Projects: warfarin_distribution1_project, warfarin_distribution2_project, warfarin_distribution3_project, warfarin_distribution4_project


Introduction

One way to extend the use of Gaussian distributions is to consider that some transformation of the parameters in which we are interested is Gaussian, i.e., assume the existence of a monotonic function h such that \(h(\psi)\) is normally distributed. Then, there exists some \(\omega\) such that, for each individual i:

\(h(\psi_i) \sim {\cal N}(h(\bar{\psi}_i), \omega^2)\)

where \(\bar{\psi}_i\) is the predicted value of \(\psi_i\). In this section, we consider models for the individual parameters without any covariate. Then, the predicted value of \(\psi_i\) is the \(\bar{\psi}_i = \psi_{\rm pop}\) and

\(h(\psi_i) \sim {\cal N}(h(\psi_{pop}), \omega^2)\)

Transformation h defines the distribution of (\psi_i). Some predefined distributions/transformations are available in Monolix:

  • Normal distribution:

In that case, \(h(\psi_i) = \psi_i\).
Remark: the two mathematical representations for normal distributions are equivalent:

\( \psi_i \sim {\cal N}(\bar{\psi}_{i}, \omega^2) ~~\Leftrightarrow~~ \psi_i = \bar{\psi}_i + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2).\)

  • Log-normal distribution:

In that case, \(h(\psi_i) = log(\psi_i)\). A log-normally random variable takes positive values only. A log-normal distribution looks like a normal distribution for a small variance \(\omega^2\). On the other hand, the asymmetry of the distribution increases when \(\omega^2\) increases.
Remark: the two mathematical representations for log-normal distributions are equivalent:

\(\log(\psi_i) \sim {\cal N}(\log(\bar{\psi}_{i}), \omega^2) ~~\Leftrightarrow~~ \psi_i = \bar{\psi}_i e^{\eta_i}, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2).\)

  • Logit-normal distribution:

In that case, \(h(\psi_i) = log\left(\frac{\psi_i}{1-\psi_i}\right)\). A random variable \(\psi_i\) with a logit-normal distribution takes its values in ]0,1[. The logit of \(\psi_i\) is normally distributed, i.e.,

\(\text{logit}(\psi_i) = \log \left(\frac{\psi_i}{1-\psi_i}\right) \ \sim \ \ {\cal N}( \text{logit}(\bar{\psi}_i), \omega^2).\)

  • Probit-normal distribution:

The probit function is the inverse cumulative distribution function (quantile function) \(\Phi^{-1}\) associated with the standard normal distribution \({\cal N}(0,1)\). A random variable \(\psi\) with a probit-normal distribution also takes its values in ]0,1[.

\(\text{probit}(\psi_i) = \Phi^{-1}(\psi_i) \ \sim \ {\cal N}( \Phi^{-1}(\bar{\psi}_i), \omega^2) .\)

To chose one of these distribution in the GUI, click on the distribution corresponding to the parameter you want to change in the individual model part and choose the corresponding distribution.

Remarks:

  1. If you change your distribution and your population parameter is not valid, then an error message is thrown. Typically, when you want to change your distribution to a logit or a probit distribution, typically for a bio-availability, make sure the associated population parameter is between 0 and 1 strictly.
  2. When creating a project, the default proposed distribution is lognormal.
  3. Logit and probit transformations can be generalized to any interval (a,b) by setting \( \psi_{(a,b)} = a + (b-a)\psi_{(0,1)}\) where \(\psi_{(0,1)}\) is a random variable that takes values in (0,1) with a logit-normal (or probit-normal) distribution. Thus, if you need to have bounds between a and b, you need to modify your structural model to reshape a parameter between 0 and 1 and use a logit or a probit distribution.

 

Marginal distributions of the individual parameters

  • warfarin_distribution1_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

We use the warfarin PK example here. The four PK parameters Tlag, ka, V and Cl are log-normally distributed. LOGNORMAL distribution is then used for these four log-normal distributions in the main Monolix graphical user interface:

The distribution of the 4 PK parameters defined in the MonolixGUI is automatically translated into Mlxtran in the project file:

[INDIVIDUAL]
input = {Tlag_pop, omega_Tlag, ka_pop, omega_ka, V_pop, omega_V, Cl_pop, omega_Cl}
DEFINITION:
Tlag = {distribution=lognormal, typical=Tlag_pop, sd=omega_Tlag}
ka = {distribution=lognormal, typical=ka_pop, sd=omega_ka}
V = {distribution=lognormal, typical=V_pop, sd=omega_V}
Cl = {distribution=lognormal, typical=Cl_pop, sd=omega_Cl}

Estimated parameters are the parameters of the 4 log-normal distributions and the parameters of the residual error model:


Here, \(V_{\rm pop} = 7.94\) and \(\omega_V=0.326\) means that the estimated population distribution for the volume is: \(\log(V_i) \sim {\cal N}(\log(7.94) , 0.326^2)\) or, equivalently, \(V_i = 7.94 e^{\eta_i}\) where \(\eta_i \sim {\cal N}(0,0.326^2)\).

Remarks:

  • \(V_{\rm pop} = 7.94\) is not the population mean of the distribution of \(V_i\), but the median of this distribution (in that case, the mean value is 7.985). The four probability distribution functions are displayed figure Parameter distributions:
  • \(V_{\rm pop}\) is not the population mean of the distribution of \(V_i\), but the median of this distribution. The same property holds for the 3 other distributions which are not Gaussian.
  • Here, standard deviations \(\omega_{Tlag}\), \(\omega_{ka}\), \(\omega_V\) and \(\omega_{Cl}\) are approximately the coefficients of variation (CV) of Tlag, ka, V and Cl since these 4 parameters are log-normally distributed with variances < 1.
  • warfarin_distribution2_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

Other distributions for the PK parameters are used in this project:

  • NORMAL for Tlag, we fix the population value \(Tlag_{\text{pop}}\) to 1.5 and the standard deviation \(\omega_{\rm Tlag}\) to 1:
  • NORMAL for ka,
  • NORMAL for V,
  • and LOGNORMAL for Cl

Estimated parameters are the parameters of the 4 transformed normal distributions and the parameters of the residual error model:

Here, \( Tlag_{\rm pop} = 1.5\) and \(\omega_{Tlag}=1\) means that \(Tlag_i \sim {\cal N}(1.5, 1^2)\) while \(Cl_{\rm pop} = .133\) and \(\omega_{Cl}=..29\) means that \(log(Cl_i) \sim {\cal N}(log(.133), .29^2)\).
The four probability distribution functions are displayed Figure Parameter distributions:

 

 

Correlation structure of the random effects






  • warfarin_distribution3_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

Correlation between the random effects can be introduced in the model. For that, you can define correlation groups. Thus, you can define a correlation group between \(\eta_{V,i}\) and \(\eta_{Cl,i}\) (#1 in that case) and between \(\eta_{Tlag,i}\) and \(\eta_{ka,i}\) in an other group as you can see in the following figure. It is important to mention that the estimated correlations are not the correlation between the individual parameters (between \(Tlag_i\) and \(ka_i\), and between \(V_i\) and \(Cl_i\)) but the (linear) correlation between the random effects (between \(\eta_{Tlag,i}\) and \(\eta_{ka,i}\), and between \(\eta_{V,i}\) and \(\eta_{Cl,i}\)  respectively).

To define a correlation between the random effects of V and Cl, you just have to click on the check boxes of the correlation for those two parameters. If you want to define a correlation between the random effects ka and Tlag independently of the first correlation group, click on the + next to CORRELATION to define a second group and click on the check boxes corresponding to the parameters ka and Tlag under the correlation group #2. Notice, that as the random effects of Cl and V are already in the correlation group #1, these random effects can not be used in another correlation group.
Remarks

  • If the box is greyed, it means that the associated random effects can not be used in a correlation group, as in the following cases
    • when the parameter has no random effects
    • when the random effect of the parameter is already used in another correlation group
  • There are no limitation in terms of number of parameters in a correlation group
  • You can have a look in the FORMULA to have a recap of all correlations
  • In case of inter-occasion variability, you can define the correlation group for each level of variability independantly.

Estimated population parameters now include these 2 correlations:

  • warfarin_distribution4_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

In this example, \(Tlag_i\) does not vary in the population, which means that \(\eta_{Tlag,i}=0\) for all subjects i, while the three other random effects are correlated:

Estimated population parameters now include the 3 correlations between \(\eta_{ka,i}\), \(\eta_{V,i}\) and \(\eta_{Cl,i}\) :