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# Population parameters

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

#### 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

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_Pop and mlx_PopInit are vectors and are common 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.
This element cannot be modified nor duplicated.