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# [INDIVIDUAL]

### Description

The [INDIVIDUAL] section is used to define a probability distribution for model parameters. It is used to model inter-individual variability for given parameters.

### Scope

The [INDIVIDUAL] section is used in Mlxtran models for simulation with Mlxplore or Simulx. It is only needed for models that have parameters with inter-individual variability. Mlxtran models for Monolix do not need this section because the parameter distributions are defined via the user interface.

### Inputs

The inputs for the [INDIVIDUAL] section are the parameters that are declared in the input = { } list of the [INDIVIDUAL] section. These parameters are obtained from the section or from the executing program that can be Mlxplore or an R-script in the case of Simulx.

### Outputs

Every parameter that has been defined in the [INDIVIDUAL] section can be an output. Outputs from the [INDIVIDUAL] section are always an input for the [LONGITUDINAL] section. [INDIVIDUAL] output to [LONGITUDINAL] input matching is made by matching parameter names in the [INDIVIDUAL] section with parameters in the inputs = { } list of the [LONGITUDINAL] section.

### Usage

The definition of probability distribution for a model parameter is done with the EQUATION: and DEFINITION: blocks. The EQUATION: block contains mathematical equations and the DEFINITION: block is used to definite probability distributions. The following syntax applies to define a probability distribution for the random variable X:

DEFINITION:
X = {distribution= distributionType, parameter1 = Var1, covariate = c, coefficient = beta, parameter2 = Var2}

The arguments to the probability distribution definition are

• distributionType: is one of the following reserved keywords: normal, lognormal, logitnormal, or probitnormal. For use in Simulx, the keyword uniform is also accepted.
• parameter1: is one of the following reserved keywords: mean or typical
• parameter2: is one of the following reserved keywords: sd or var
• Var1: is a double number or a parameter
• Var2: is a double number or a parameter
• c = {..}: is a list of strings referring to the covariates
• beta = {…}: is a list of strings referring to the covariate coefficients

The reserved keywords meanings are

• normal: normal distribution: $h(X)=X$
• lognormal: log-normal distribution: $h(X)=\log(X)$
• logitnormal: logit-normal distribution:  $h(X)=\frac{1}{1+e^{-X}}$
• probitnormal: probit-normal distribution: $h(X)=\Psi^{-1}(X)$, where $\Psi$ is the cumulative distribution function of the ${\cal N}(0, 1)$ distribution.
• mean: is the mean of the normal distribution
• typical: is the transformed mean
• sd: is the standard deviation of the normal distribution. If no variability is required, the user can set “no-variability” and no-variability will be computed
• var: is the variance of the normal distribution
• min: the lower bound of the interval for logitnormal or probitnormal distributions (default is 0)
• max: the upper bound of the interval for logitnormal or probitnormal distributions (default is 1)

These probability distribution are all Gaussian probability distributions that are defined through the existence of a monotonic transformation $h$ such that $h(X)$ is normally distributed. Notice that the mean, standard deviation, and variance refer to the normal distributed variable. In pharmacometrics it is more common to use the typical value of the distribution. This is achieved by using the keyword typical instead of mean in the definition of the random variable. The relationship between the mean value and the typical value is the following:

$\phantom{abc} X = \{ \textrm{distribution}=\textrm{lognormal}, \textrm{typical}=X_{pop}, \textrm{sd}=\sigma\} \\ \Leftrightarrow X = \{ \textrm{distribution}=\textrm{lognormal}, \textrm{mean}=\mu_{pop}, \textrm{sd}=\sigma\} \\ \Leftrightarrow \log(X) \sim {\cal N}(\mu_{pop},\sigma^2)$

where $\mu_{pop}=\log(X_{pop})$. Thus, typical is in the variable referential, while mean is in the transformed referential.

#### Examples

• The parameter ka below is defined with a log-normal distribution, a typical value ka_pop and a standard deviation for the random effect omega_ka:

ka = {distribution=logNormal, typical=ka_pop, sd=omega_ka}
• The parameter F below is defined with a logit-normal distribution in the interval [0,1], a typical value F_pop and no random effect.

F = {distribution=logitNormal, typical=F_pop, no-variability}
• The parameter V below is defined with a logit-normal distribution in the interval [0.2,5], a typical value V_pop and a standard deviation for the random effect omega_V:
V = {distribution=logitNormal, min=0.2, max=5, typical=V_pop, sd=omega_V}
• The parameter fu below is defined with a uniform distribution in the interval [0.2,0.6] (not accepted in Monolix):

fu = {distribution=uniform, min=0.2, max=0.6}

### General probability distributions and inclusion of covariates

#### Linear Gaussian models with covariates

A linear Gaussian statistical model for the variable $X$ assumes that there exists a transformation $h$, a typical value $X_{\rm pop}$, a vector of individual covariates $(c_{1}, \ldots c_{L})$, a vector of coefficients $(\beta_1, \ldots, \beta_L)$ and a random variable $\eta$ normally distributed such that

$h(X) = h(X_{pop}) + \sum_{\ell=1}^L \beta_\ell \, c_\ell + \eta$

This model can be implemented with Mlxtran, using the keywords typical, covariate and coefficient.

input = {Xpop, beta1, beta2, c1, c2, omega}
DEFINITION:
X = {distribution=lognormal, typical=Xpop, covariate={c1,c2}, coefficient={beta1,beta2}, sd=omega}


The keyword covariate is used to define the name of the covariates used in the correlation, and the coefficient keyword is used to complete the equation. Obviously, the number of parameters in the coefficient is equal to the number of covariates.

#### Non linear Gaussian model with covariates

A nonlinear Gaussian statistical model for the variable $X$ assumes that there exists a transformation $h$, a vector of individual covariates $(c_{1}, \ldots c_{L})$, a vector of coefficients $(\beta_1, \ldots, \beta_M)$, a function $\mu$ and a random variable $\eta$ normally distributed such that

$h(X) = \mu(c_{1}, \ldots c_{L},\beta_1, \ldots, \beta_M) + \eta$

The mean of $h(X)$ can be defined in a block DEFINITION:, with for example $\mu(\beta_1,\beta_2,c_1,c_2)=\frac{\beta_1c_1}{\beta_2+c_2}$

input = {beta1, beta2, c1, c2, omega}
EQUATION:
mu = beta1*c1/(beta2 + c2)

DEFINITION:
X = {distribution=lognormal, mean=mu, sd=omega}


#### Non Gaussian model with covariates

Non Gaussian model for $X$ can be defined, at the condition that $X$ can be defined as a nonlinear function of normally distributed random variables. For example, let

$X = \frac{\beta_1 + \eta_1}{1+ \beta_2 \, e^{\eta_2}}$

It is not possible to express explicitly the distribution of $X$ as a transformation of a normal distribution. We therefore need a block EQUATION: for implementing this model:

input = {beta1, beta2, omega1, omega2}
DEFINITION:
eta1 = {distribution=normal, mean=0, sd=omega1}
eta2 = {distribution=normal, mean=0, sd=omega2}

EQUATION:
X = (beta1 + eta1)/(1+beta2*exp(eta2))


### Rules

• When defining a distribution with covariate, one can not define numerically the coefficients. For example, if we consider $X=X_{pop}+c+\eta$, one should write
input = {c}
EQUATION:
beta = 1
DEFINITION :
X = {distribution=normal, typical=Xpop, covariate=c, coefficient=beta, sd=omega}


and define $\beta = 1$ in the section <PARAMETER> or define it in an EQUATION: block. Otherwise, putting directly 1 instead of $\beta$ in the distribution definition will lead to an error.

• We strongly advise to define the distribution in the more synthetic way. If for example, you want to define a log-normally distributed volume with a dependence w.r.t. the weight $V=V_{pop}(w/70)^{\beta}$, we encourage you not to define a lot of equations but to summarize it in the definition as for example
input = {Vpop, w, beta}
EQUATION:
cov = w/70

DEFINITION:
V = {distribution=normal, typical=Vpop, covariate=cov, coefficient=beta, sd=omega}