Definition of population parameters as simulation elements allows to simulate individual parameters from probability distributions. In the mlxtran model they are:

- [INDIVIDUAL] block input parameters that are neither inputs nor outputs of the block [COVARIATE].
- [LONGITUDINAL] block input parameters that are neither individual parameters nor regressors.

POP.PARAM section in the “Definition” tab is available only if the list of population parameters from the model is not empty.

**Demo projects:** 3.3. population parameters

#### Overview

In the “Population parameters” tab, elements created automatically after a “new project” or an “import from Monolix/PKanalix” and elements created by the user are shown.

The buttons on the right allow to do the following actions:

- VIEW: display the content of the element (the value of the population parameters)
- EDIT: modify the content of the element.
*Note:*the elements “mlx_PopUncertainSA”, “mlx_PopUncertainLin”, “mlx_TypicalUncertainSA”, and “mlx_TypicalUncertainLin” represent multidimentional distributions and cannot be edited. - DUPLICATE: open a pop-up window for a new element with the same content as the previous element.
*Note:*the elements “mlx_PopUncertainSA”, “mlx_PopUncertainLin”, “mlx_TypicalUncertainSA”, and “mlx_TypicalUncertainLin” cannot be duplicated. Elements based on external files can also not be duplicated. - DELETE: delete the element. When the deleted element is based on an external file located in <result folder>/ExternalFiles, the file itself is also deleted upon save.
- CREATE NEW INDIV: creates a new element “individual parameters” from the population parameters fixed effects (e.g V = V_pop). This option is available for population elements of type “manual” only.

#### New population parameters element

After loading a model, Simulx generates automatically a default population parameter element. It is a table with parameters names from the model and all entries equal to one. It helps to create a new element by just modifying the values. Clicking the button “plus” adds a new population parameter, which may be one of the three types.

**Manual:**It is a vector and has one value for each parameter.

**Distribution:**Parameters are a distribution law – normal, lognormal (default), logitnormal, uniform – with a typical value and a standard deviation. If the distribution is lognormal or logitnormal, sd is the standard deviation of the distribution in the Gaussian space. The typical value is the median of the population parameter distribution. In the case of a lognormal distribution, in order to get the sd \(s_G\) of the distribution in the Gaussian space given a typical value of the lognormally distributed covariate \(\mu\), or given its mean \(m\), and given its sd \(s\), you can use the following formulae:\[s_G = \sqrt{\ln\Big(1+\Big(\frac{s}{m}\Big)^2\Big)} = \sqrt{\ln\Bigg(1+\sqrt{1+4\Big(\frac{s}{\mu}\Big)^2}\Bigg) – \ln(2)}\]Both formulae are equivalent if \(\frac{s}{\mu}<<1\) (in that case \(\mu \approx m\)).**External:**It is a file with a table that has each population parameter in a separate column.

The external file can have only one column header different from the population parameters names. It indicates replicates (eg. from bootstrap or simpopmlx). All individual parameters of the same replicate come from the same population distribution.

#### Population parameters imported from Monolix

**Demo projects:** 1.overview/importFromMonolix … .smlx

Importing a Monolix project generates automatically a population parameter element with values from the Monolix result folder.

**mlx_Pop**contains population parameters estimated by Monolix if the POP.PARAM task results are available.**mlx_PopInit**contains initial estimates of the population parameters if the mlx_pop does not exist.**mlx_Typical**(NEW since 2023 version) is the same as mlx_Pop but with all standard deviations of random effects (omega parameters) set to 0. It enables to simulate a typical individual in terms of parameters (no unexplained variability), and still include non-typical covariates in the simulation. The variability in the sampled individual parameters will come only from sampling different covariate values. To see how this element can be used in practice, check the demo project 1.overview/importFromMonolix_CovEffectOnTypical.smlx.

mlx_Pop, mlx_PopInit and mlx_Typical are vectors (single set of population parameters). If replicates are used in the Simulation tab, with these elements, the same set of poplation parameters will be used for all replicates.

**mlx_PopUncertainSA**(resp.**mlx_PopUncertainLin**) enables to sample population parameters using the variance-covariance matrix of the estimates computed by Monolix if the Standard Error task (Estimation of the Fisher Information matrix) was performed by stochastic approximation (resp. by linearization). To sample several population parameter sets, this element needs to be used with replicates. In the interface, the element is displayed with a reminder of the estimated values and RSEs next to the variance-covariance matrix which is used to generate new samples.

The displayed variance-covariance matrix and the sampling is done in the gaussian space (i.e we sample {log(V_pop), log(Cl_pop), omega_V, omega_Cl, b} from a multivariate normal distribution, if V and Cl have a lognormal distribution). The sampled values are then converted to the non-gaussian space. For omega and error parameters, negative samples values are rejected, they are thus sampled from truncated normal distributions.

This element cannot be modified nor duplicated.

**mlx_TypicalUncertainSA**(resp.**mlx_PopUncertainLin,**NEW since 2023 version) is another population parameter element with the variance-covariance matrix of population parameter values estimated in Monolix. The only difference to the previous mlx_PopUncertainSA/Lin is that it has standard deviations of random effects (omega parameters) set to 0. It enables to simulate a typical individual with different typical parameter values for each replicate, such that the uncertainty of parameters is propagated to the predictions. To sample several parameter sets, this element needs to be used with replicates. To see how this element can be used in practice, check the demo project 1.overview/importFromMonolix_UncertaintyOnTypical.smlx.

In the following video we apply uncertainty on fixed effects only (with version 2021). With the 2023 version, the steps performed outside of the interface in this video can all be replaced by simply selecting the new element mlx_TypicalUncertain.