1.Monolix Documentation
Version 2023
This documentation is for Monolix starting from 2018 version.
©Lixoft
Monolix
Monolix (Nonlinear mixedeffects models or “MOdèles NOn LInéaires à effets miXtes” in French) is a platform of reference for model based drug development. It combines the most advanced algorithms with unique ease of use. Pharmacometricians of preclinical and clinical groups can rely on Monolix for population analysis and to model PK/PD and other complex biochemical and physiological processes. Monolix is an easy, fast and powerful tool for parameter estimation in nonlinear mixed effect models, model diagnosis and assessment, and advanced graphical representation. Monolix is the result of a ten years research program in statistics and modeling, led by Inria (Institut National de la Recherche en Informatique et Automatique) on nonlinear mixed effect models for advanced population analysis, PK/PD, preclinical and clinical trial modeling & simulation.
Objectives
The objectives of Monolix are to perform:
 Parameter estimation for nonlinear mixed effects models
 estimating the maximum likelihood estimator of the population parameters, without any approximation of the model (linearization, quadrature approximation, …), using the Stochastic Approximation Expectation Maximization (SAEM) algorithm,
 computing the conditional modes, sample from the conditional distribution to compute the conditional means and the conditional standard deviations of the individual parameters, using the HastingsMetropolis algorithm
 estimating standard errors for the maximum likelihood estimator
 Model selection and diagnosis
 comparing several models using some information criteria (AIC, BIC)
 testing parameters using the Wald Test
 testing correlation using Pearson’s correlation test
 testing normality of distribution using Shapiro’s test.
 Easy description of pharmacometric models (PK, PKPD, discrete data) with the Mlxtran language
 Goodness of fit plots
An interface for ease of use
Monolix can be used either via a graphical user interface (GUI) or a commandline interface (CLI) for powerful scripting. This means less programming and more focus on exploring models and pharmacology to deliver in time. The interface is depicted as follows:
The GUI consists of 7 tabs.
 Welcome
 Data
 Structural model
 Initial estimates
 Tasks and statistical model
 Results
 Plots
Each of these tabs refer to a specific section on this website. An advanced description of available plots is also provided.
2.Data and models
In the following, all demos of Monolix are presented. They were built to explore all functionalities of Monolix in terms of model creations, continuous and non continuous outcomes management, joint models for multivariate outcomes, models for the individual parameters, pharmacokinetic models, and some extensions.
Creating and using models
 Libraries of models: learn how to use the Monolix libraries of PKPD models and create your own libraries.
 Outputs and Tables: learn how to define outputs and create tables with selected outputs of the model.
Models for continuous outcomes
 Residual error model: learn how to use the predefined residual error models.
 Handling censored data: learn how to handle easily and properly censored data, i.e. data below (resp. above) a lower (resp.upper) limit of quantification (LOQ) or detection (LOD).
 Mixture of structural models: learn how to implement between subject mixture models (BSMM) and within subject mixture models (WSMM).
Models for non continuous outcomes
 Timetoevent data model: learn how to implement a model for (repeated) timetoevent data.
 Count data model: learn how to implement a model for count data, including hidden Markov model.
 Categorical data model: learn how to implement a model for categorical data, assuming either independence or a Markovian dependence between observations.
Joint models for multivariate outcomes
 Continuous PKPD model: learn how to implement a joint model for continuous pharmacokineticspharmacodynamics (PKPD) data.
 Joint continuous and non continuous data model: learn how to implement a joint model for continuous and non continuous data, including count, categorical and timetoevent data.
Models for the individual parameters
 Probability distribution of the individual parameters: learn how to define the probability distribution and the correlation structure of the individual parameters.
 Model for individual covariates: learn how to implement a model for continuous and/or categorical covariates.
 Inter occasion variability: learn how to take into account inter occasion variability (IOV).
 Mixture of distributions: learn how to implement a mixture of distributions for the individual parameters.
Pharmacokinetic models
 Single route of administration: learn how to define and use a PK model for a single route of administration.
 Multiple routes of administration: learn how to define and use a PK model for multiple routes of administration.
 From multiple doses to steadystate: learn how to define and use a PK model with multiple doses or assuming steadystate.
Some extensions
 Using regression variables: learn how to define and use regression variables (time varying covariates).
 Bayesian estimation: learn how to combine maximum likelihood estimation and Bayesian estimation of the population parameters.
 Delayed differential equations : learn how to implement a model with delayed differential equations (DDE).
2.1.Defining a data set
To start a new Monolix project, you need to define a dataset by loading a csv file in the Data tab, and load a model in the structural model tab. The project can be saved only after defining the data and the model.
The data set format expected in the Data tab is the same as for the entire MonolixSuite, to allow smooth transitions between applications. The columns available in this format and example datasets are detailed on this page. Briefly:
 Each line corresponds to one individual and one time point
 Each line can include a single measurement (also called observation), or a dose amount (or both a measurement and a dose amount)
 Dosing information should be indicated for each individual, even if it is identical for all.
Your dataset may not be originally in this format, and you may want to add information on dose amounts, limits of quantification, units, or filter part of the dataset. To do so, you should proceed in this order:
 Formatting: If needed, format your data first by loading the dataset in the Data Formatting tab. Briefly, it allows to:
 to deal with several header lines
 merge several observation columns into one
 add censoring information based on tags in the observation column
 add treatment information manually or from external sources
 add more columns based on another file
 Loading a new data set: If the data is already in the right format, load it directly in the Data tab (otherwise use the formatted dataset created by data formatting).
 Observation types: Specify if the observation is of type continuous, count/categorical or event.
 Labeling: label the columns not recognized automatically to indicate their type and click on ACCEPT.
 Filtering: If needed, filter your dataset to use only part of it in the Filters tab
 Explore: The interpreted dataset is displayed in Data, and Plots and covariate statistics are generated.
If you have already defined a dataset in Datxplore or in PKanalix, you can skip all those steps in Monolix and create a new project by importing a project from Datxplore or PKanalix.
Loading a new data set
To load a new data set, you have to go to “Browse” your data set (green frame), tag all the columns (purple frame), define the observation types in Data Information (blue frame), and click on the blue button ACCEPT as on the following. If the dataset does not follow a formatting rule, the dataset will not be accepted, but errors will guide you to find what is missing and could be added by data formatting.
Observation types
There are three types of observations
 continuous: The observation is continuous with respect to time. For example, a concentration is a continuous observation.
 count/categorical: The observation values takes place in a finite categorical space. For example, the observation can be a categorical observation (an effect can be observed as low, medium, high) or a count observation over a defined time (the number of epileptic crisis in a defined time).
 event: The observation is an event, for example the occurring of an epileptic crisis.
The type of observations can be specified by the user in the interface.
Labeling
The column type suggested automatically by Monolix based on the headers in the data can be customized in the preferences. By clicking on Settings>Preferences, the following windows pops up.
In the DATA frame, you can add or remove preferences for each column.
To remove a preference, doubleclick on the preference you would like to remove. A confirmation window will be proposed.
To add a preference, click on the header type you consider, add a name in the header name and click on “ADD HEADER” as on the following figure.
Notice that all the preferences are shared between Monolix, Datxplore, and PKanalix.
Resulting plots and tables to explore the data
Once the dataset is accepted:
 Plots are automatically generated based on the interpreted dataset to help you proceed with a first data exploration before running any task.
 The interpreted dataset appears in Data tab, which incorporates all changes after formatting and filtering.
 Covariate Statistics appear in a section of the data tab.
All the covariates (if any) are displayed and a summary of the statistics is proposed. For continuous covariates, minimum, median and maximum values are proposed along with the first and third quartile, and the standard deviation. For categorical covariates, all the modalities are displayed along with the number of each. Note the “Copy table” button that allows to copy the table in Word and Excel. The format and the display of the table will be preserved.
Importing a project from Datxplore or PKanalix
It is possible to import a project from Datxplore or PKanalix. For that, go to Project>New project for Datxplore/PKanalix (as in the green box of the following figure). In that case, a new project will be created and all the DATA frame will already be filled by the information from the Datxplore or PKanalix project.
2.2.Creating and using models
2.2.1.Libraries of models
 The PK library
 The PD and PKPD libraries
 The PK double absorption library
 The TMDD library
 The TTE library
 The Count library
 The TGI library
 A stepbystep example with the PK library
Objectives: learn how to use the Monolix libraries of models and use your own models.
Projects: theophylline_project, PDsim_project, warfarinPK_project, TMDD_project, LungCancer_project, hcv_project
For the definition of the structural model, the user can either select a model from the available model libraries or write a model itself using the Mlxtran language.
Discover how to easily choose a model from the libraries via stepbystep selection of its characteristics. An enriched PK, a PD, a joint PKPD, a targetmediated drug disposition (TMDD), and a time toevent (TTE) library are now available.
Model libraries
Five different model libraries are available in Monolix, which we will detail below. To use a model from the libraries, in the Structural model
tab, click on Load from library
and select the desired library. A list of model files appear, as well as a menu to filter them. Use the filters and indications in the file name (parameters names) to select the model file you need.
The model files are simply text files that contain prewritten models in Mlxtran language. Once selected, the model appears in the Monolix GUI. Below we show the content of the (ka,V,Cl) model:
The PK library
 theophylline_project (data = ‘theophylline_data.txt’ , model=’lib:oral1_1cpt_kaVCl.txt’)
The PK library includes model with different administration routes (bolus, infusion, firstorder absorption, zeroorder absorption, with or without Tlag), different number of compartments (1, 2 or 3 compartments), and different types of eliminations (linear or MichaelisMenten). More details, including the full equations of each model, can be found on the PK model library wepage. The PK library models can be used with single or multiple doses data, and with two different types of administration in the same data set (oral and bolus for instance).
The PD and PKPD libraries
 PDsim_project (data = ‘PDsim_data.txt’ , model=’lib:immed_Emax_const.txt’)
The PD model library contains direct response models such as Emax and Imax with various baseline models, and turnover response models. These models are PD models only and the drug concentration over time must be defined in the data set and passed as a regressor.
 warfarinPKPD_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_IndirectModelInhibitionKin_TlagkaVClR0koutImaxIC50.txt’)
The PKPD library contains joint PKPD models, which correspond to the combination of the models from the PK and from the PD library. These models contain two outputs, and thus require the definition of two observation identifiers (i.e two different values in the OBSERVATION ID column).
Complete description of the PD and PK/PD model libraries.
The PK double absorption library
The library of double absorption models contains all the combinations for two mixed absorptions, with different types and delays. The absorptions can be specified as simultaneous or sequential, and with a predefined or independent order. This library simplifies the selection and testing of different types of absorptions and delays. More details about the library and examples can be found on the dedicated PK double absorption documentation page.
The TMDD library
 TMDD_project (data = ‘TMDD_dataset.csv’ , model=’lib:bolus_2cpt_MM_VVmKmClQV2_outputL.txt’)
The TMDD library contains various models for molecules displaying targetmediated drug disposition (TMDD). It includes models with different administration routes (bolus, infusion, firstorder absorption, zeroorder absorption, bolus + firstorder absorption, with or without Tlag), different number of compartments (1, or 2 compartments), different types of TMDD models (full model, MM approximation, QE/QSS approximation, etc), and different types of output (free ligand or total free+bound ligand). More details about the library and guidelines to choose model can be found on the dedicated TMDD documentation page.
The TTE library
 LungCancer_project (data = ‘lung_cancer_survival.csv’ , model=’lib:gompertz_model_singleEvent.txt’)
The TTE library contains typical parametric models for timetoevent (TTE) data. TTE models are defined via the hazard function, in the library we provide exponential, Weibull, loglogistic, uniform, Gompertz, gamma and generalized gamma models, for data with single (e.g death) and multiple events (e.g seizure) per individual. More details and modeling guidelines can be found on the TTE dedicated webpage, along with case studies.
The Count library
The Count library contains the typical parametric distributions to describe count data. More details can be found on the Count dedicated webpage, with a short introduction on count data, the different ways to model this kind of data, and typical models.
The tumor growth inhibition (TGI) library
A wide range of models for tumour growth (TG) and tumour growth inhibition (TGI) is available in the literature and correspond to different hypotheses on the tumor or treatment dynamics. In MonolixSuite2020, we provide a modular TG/TGI model library that combines sets of frequently used basic models and possible additional features. This library permits to easily test and combine different hypotheses for the tumor growth kinetics and effect of a treatment, allowing to fit a large variety of tumor size data.
Complete description of the TGI model library.
Stepbystep example with the PK library
 theophylline_project (data = ‘theophylline_data.txt’ , model=’lib:oral1_1cpt_kaVCl.txt’)
We would like to set up a one compartment PK model with first order absorption and linear elimination for the theophylline data set. We start by creating a new Monolix project. Next, the Data
tab, click browse, and select the theophylline data set (which can be downloaded from the data set documentation webpage). In this example, all columns are already automatically tagged, based on the header names. We click ACCEPT
and NEXT
and arrive on the Structural model
tab, click on LOAD FROM LIBRARY
to choose a model from the Monolix libraries. The menu at the top allow to filter the list of models: after selecting an oral/extravascular administration, no delay, firstorder absorption, one compartment and a linear elimination, two models remain in the list (ka,V,Cl) and (ka,V,k). Click on the oral1_1cpt_kaVCl.txt
file to select it.
After this step, the GUI moves to the Initial Estimates
tab, but it is possible to go back to the Structural model
tab to see the content of the file:
[LONGITUDINAL] input = {ka, V, Cl} EQUATION: Cc = pkmodel(ka, V, Cl) OUTPUT: output = Cc
Back to the Initial Estimates
tab, the initial values of the population parameters can be adjusted by comparing the model prediction using the chosen population parameters and the individual data. Click on SET AS INITIAL VALUES
when you are done.
In the next tab, the Statistical model & Tasks
tab, we propose by default:
 A combined error observation model
 Lognormal distributions for all parameters (ka, V and Cl)
At this stage, the monolix project should be saved. This creates a human readable text file with extension .mlxtran, which contains all the information defined via the GUI. In particular, the name of the model appears in the section [LONGITUDINAL]
of the saved project file:
<MODEL> [INDIVIDUAL] input = {ka_pop, omega_ka, V_pop, omega_V, Cl_pop, omega_Cl} DEFINITION: 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} [LONGITUDINAL] input = {a, b} file = 'lib:oral1_1cpt_kaVCl.txt' DEFINITION: CONC = {distribution=normal, prediction=Cc, errorModel=combined1(a,b)}
2.2.2.Outputs and Tables
Objectives: learn how to define outputs and create tables from the outputs of the model.
Projects: tgi_project, tgiWithTable_project
About the OUTPUT block
 tgi_project (data = ‘tgi_data.txt’ , model=’tgi_model.txt’)
We use the Tumor Growth Inhibition (TGI) model proposed by Ribba et al. in this example (Ribba, B., Kaloshi, G., Peyre, M., Ricard, D., Calvez, V., Tod, M., . & Ducray, F., A tumor growth inhibition model for lowgrade glioma treated with chemotherapy or radiotherapy. Clinical Cancer Research, 18(18), 50715080, 2012.)
DESCRIPTION: Tumor Growth Inhibition (TGI) model proposed by Ribba et al A tumor growth inhibition model for lowgrade glioma treated with chemotherapy or radiotherapy. Clinical Cancer Research, 18(18), 50715080, 2012. Variables  PT: proliferative equiescent tissue  QT: nonproliferative equiescent tissue  QP: damaged quiescent cells  C: concentration of a virtual drug encompassing the 3 chemotherapeutic components of the PCV regimen Parameters  K : maximal tumor size (should be fixed a priori)  KDE : the rate constant for the decay of the PCV concentration in plasma  kPQ : the rate constant for transition from proliferation to quiescence  kQpP : the rate constant for transfer from damaged quiescent tissue to proliferative tissue  lambdaP: the rate constant of growth for the proliferative tissue  gamma : the rate of damages in proliferative and quiescent tissue  deltaQP: the rate constant for elimination of the damaged quiescent tissue  PT0 : initial proliferative equiescent tissue  QT0 : initial nonproliferative equiescent tissue [LONGITUDINAL] input = {K, KDE, kPQ, kQpP, lambdaP, gamma, deltaQP, PT0, QT0} PK: depot(target=C) EQUATION: ; Initial conditions t0 = 0 C_0 = 0 PT_0 = PT0 QT_0 = QT0 QP_0 = 0 ; Dynamical model PSTAR = PT + QT + QP ddt_C = KDE*C ddt_PT = lambdaP*PT*(1PSTAR/K) + kQpP*QP  kPQ*PT  gamma*KDE*PT*C ddt_QT = kPQ*PT  gamma*KDE*QT*C ddt_QP = gamma*KDE*QT*C  kQpP*QP  deltaQP*QP OUTPUT: output = PSTAR
PSTAR is the tumor size predicted by the model. It is therefore used as a prediction for the observations in the project.
At the end of the scenario or of SAEM, individual predictions of the tumor size PSTAR are computed using the individual parameters available. Thus, individual predictions of the tumor size PSTAR are computed using both the conditional modes (indPred_mode), the conditional mean (indPred_mean), and the conditional means estimated during the last iterations of SAEM (indPred_SAEM) and saved in the table predictions.txt.
Notice that the population prediction is also proposed.
Remark: the same model file tgi_model.txt can be used with different tools, including Mlxplore or Simulx (see this Shiny application for instance).
Add additional outputs in tables
 tgiWithTable_project (data = ‘tgi_data.txt’ , model=’tgiWithTable_model.txt’)
We can save in the tables additional variables defined in the model, such as PT, Q and QP for instance, by adding a block OUTPUT: in the model file:
OUTPUT: output = PSTAR table = {PT, QT, QP}
An additional file tables.txt now includes the predicted values of these variables for each individual (columns PT_mean, QT_mean, QP_mean, PT_mode, QT_mode, QP_mode, PT_popPred, QT_popPred, QP_popPred, PT_popPred_medianCOV, QT_popPred_medianCOV, QP_popPred_medianCOV, PT_SAEM, QT_SAEM, and QP_SAEM.
Notice that only continuous variable are possible for variable in table.
Good to know: it is not allowed to do calculations directly in the output or table statement. The following example is not possible:
; not allowed: OUTPUT: output = {Cser+Ccsf}
It has to be replaced by:
EQUATION: Ctot = Cser+Ccsf OUTPUT: output = {Ctot}
2.3.Models for continuous outcomes
2.3.1.Residual error model
 Introduction
 Defining the residual error model from the Monolix GUI
 Some basic residual error models
 Residual error models for bounded data
 Autocorrelated residuals
 Using different error models per group/study
Objectives: learn how to use the predefined residual error models.
Projects: warfarinPK_project, bandModel_project, autocorrelation_project, errorGroup_project
Introduction
For continuous data, we are going to consider scalar outcomes (\(y_{ij} \in \mathbb{R}\)) and assume the following general model:
$$y_{ij}=f(t_{ij},\psi_i)+ g(t_{ij},\psi_i,\xi)\varepsilon_{ij}$$
for i from 1 to N, and j from 1 to \(\text{n}_{i}\), where \(\psi_i\) is the parameter vector of the structural model f for individual i. The residual error model is defined by the function g which depends on some additional vector of parameters \(\xi\). The residual errors \((\varepsilon_{ij})\) are standardized Gaussian random variables (mean 0 and standard deviation 1). In this case, it is clear that \(f(t_{ij}, \psi_i)\) and \(g(t_{ij}, \psi_i, \xi)\) are the conditional mean and standard deviation of \(y_{ij}\), i.e.,
$$\mathbb{E}(y_{ij}  \psi_i) = f(t_{ij}, \psi_i)~~\textrm{and}~~\textrm{sd}(y_{ij}  \psi_i)= g(t_{ij}, \psi_i, \xi)$$
Available error models
In Monolix
, we only consider the function g to be a function of the structural model f, i.e. \(g(t_{ij}, \psi_i, \xi)= g(f(t_{ij}, \psi_i), \xi)\) leading to an expression of the observation model of the form
$$y_{ij}=f(t_{ij},\psi_i)+ g(f(t_{ij}, \psi_i), \xi)\varepsilon_{ij}$$
The following error models are available:
 constant : \(y = f + a \varepsilon\). The function g is constant, and the additional parameter is \(\xi=a\)
 proportional : \(y = f + bf^c \varepsilon\). The function g is proportional to the structural model f, and the additional parameters are \(\xi = (b,c)\). By default, the parameter c is fixed at 1 and the additional parameter is .
 combined1 : \(y = f + (a+ bf^c) \varepsilon\). The function g is a linear combination of a constant term and a term proportional to the structural model f, and the additional parameters are \(\xi = (a, b)\) (by default, the parameter c is fixed at 1).
 combined2 : \(y = f + \sqrt{a^2+ b^2(f^c)^2} \varepsilon\). The function g is a combination of a constant term and a term proportional to the structural model f (g = bf^c), and the additional parameters are \(\xi = (a, b)\) (by default, the parameter c is fixed at 1).
Notice that the parameter c is fixed to 1 by default. However, it can be unfixed and estimated.
The assumption that the distribution of any observation \(y_{ij}\) is symmetrical around its predicted value is a very strong one. If this assumption does not hold, we may want to transform the data to make it more symmetric around its (transformed) predicted value. In other cases, constraints on the values that observations can take may also lead us to transform the data.
Available transformations
The model can be extended to include a transformation of the data:
$$u(y_{ij})=u(f(t_{ij},\psi_i)) + g(u(f(t_{ij},\psi_i)) ,\xi) $$
As we can see, both the data \(y_{ij}\) and the structural model f are transformed by the function u so that \(f(t_{ij}, \psi_i)\) remains the prediction of \(y_{ij}\). Classical distributions are proposed as transformation:
 normal: u(y) = y. This is equivalent to no transformation.
 lognormal: u(y) = log(y). Thus, for a combined error model for example, the corresponding observation model writes \(\log(y) = \log(f) + (a + b\log(f)) \varepsilon\). It assumes that all observations are strictly positive. Otherwise, an error message is thrown. In case of censored data with a limit, the limit has to be strictly positive too.
 logitnormal: u(y) = log(y/(1y)). Thus, for a combined error model for example, the corresponding observation model writes \(\log(y/(1y)) = \log(f/(1f)) + (a + b\log(f/(1f)))\varepsilon\). It assumes that all observations are strictly between 0 and 1. It is also possible to modify these bounds and not “impose” them to be 0 and 1, i.e. to define the logit function between a minimum and a maximum: the function u becomes u(y) = log((yy_min)/(y_maxy)). Again, in case of censored data with a limit, the limits too must belong strictly to the defined interval.
Any interrogation on what is the formula behind your observation model? There is a button FORMULA
on the interface as on the figure below where the observation model is described linking the observation (named CONC in that case) and the prediction (named Cc in that case). Note that \(\epsilon\) is noted e here.
Remarks: In previous Monolix version, only the error was available. Thus, what happens to the errors that are not proposed anymore? Is it possible to have “exponential”, “logit”, “band(0,10)”, and “band(0,100)”? Yes, in this version, we choose to split the observation model between its error model and its distribution. The purpose is to have a more unified vision of models and increase the number of possibilities. Thus, here is how to configure new projects with the previous error model definition.
 “exponential” is an observation model with a constant error model and a lognormal distribution.
 “logit” is an observation model with a constant error model and a logitnormal distribution.
 “band(0,10)” is an observation model with a constant error model and a logitnormal distribution with min and max at 0 and 10 respectively.
 “band(0,100)” is an observation model with a constant error model and a logitnormal distribution with min and max at 0 and 100 respectively.
Defining the residual error model from the Monolix GUI
A menu in the frame Statistical modelTasks of the main GUI allows one to select both the error model and the distribution as on the following figure (in green and blue respectively)
A summary of the statistical model which includes the residual error model can be displayed by clicking on the button formula.
Some basic residual error models
 warfarinPKlibrary_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
The residual error model used with this project for fitting the PK of warfarin is a combined error model, i.e. \(y_{ij} = f(t_{ij}, \psi_i))+ (a+bf(t_{ij}, \psi_i)))\varepsilon_{ij}\)Several diagnosis plots can then be used for evaluating the error model. The observation versus prediction figure below seems ok.
Remarks:
 Figures showing the shape of the prediction interval for each observation model available in Monolix are displayed here.
 When the residual error model is defined in the GUI, a bloc DEFINITION: is then automatically added to the project file in the section [LONGITUDINAL] of <MODEL> when the project is saved:
DEFINITION: y1 = {distribution=normal, prediction=Cc, errorModel=combined1(a,b)}
Residual error models for bounded data
 bandModel_project (data = ‘bandModel_data.txt’, model = ‘lib:immed_Emax_null.txt’)
In this example, data are known to take their values between 0 and 100. We can use a constant error model and a logitnormal for the transformation with bounds (0,100) if we want to take this constraint into account.
In the Observation versus prediction plot, one can see that the error is smaller when the observations are close to 0 and 100 which is normal. To see the relevance of the predictions, one can look at the 90% prediction interval. Using a logitnormal distribution, we have a very different shape of this prediction interval to take that specificity into account.
VPCs obtained with this error model do not show any mispecification
This residual error model is implemented in Mlxtran
as follows:
DEFINITION: effect = {distribution=logitnormal, min=0, max=100, prediction=E, errorModel=constant(a)}
Autocorrelated residuals
For any subject i, the residual errors \((\varepsilon_{ij},1 \leq j \leq n_i)\) are usually assumed to be independent random variables. The extension to autocorrelated errors is possible by assuming, that \((\varepsilon_{ij})\) is a stationary autoregressive process of order 1, AR(1), which autocorrelation decreases exponentially:
$$ \textrm{corr}(\varepsilon_{ij},\varepsilon_{i,{j+1}}) = r_i^{(t_{i,j+1}t_{ij})}$$
where \(0 \leq r_i \leq 1\) for each individual i. If \(t_{ij}=j\) for any (i,j), then \(t_{i,j+1}t_{i,j}=1\) and the autocorrelation function \(\gamma_i\)for individual i is given by
$$\gamma_i(\tau) = \textrm{corr}(\varepsilon_{ij}, \varepsilon_{i,j+\tau}) = r_i^{\tau}$$
The residual errors are uncorrelated when \(r_i=0\).
 autocorrelation_project (data = ‘autocorrelation_data.txt’, model = ‘lib:infusion_1cpt_Vk.txt’)
Autocorrelation is estimated since the checkbox r is ticked in this project:Estimated population parameters now include the autocorrelation r:
Important remarks:
Monolix
accepts both regular and irregular time grids. For a proper estimationg of the autocorrelation structure of the residual errors, rich data is required (i.e. a large number of time points per individual) .
 To add autocorrelation, the user should either use the connectors, or write it directly in the Mlxtran
 add “autoCorrCoef=r” in definition “DV = {distribution=normal, prediction=Cc, errorModel=proportional(b), autoCorrCoef=r}” for example
 add “r” as an input parameter.
Using different error models per group/study
 errorGroup_project (data = ‘errorGroup_data.txt’, model = ‘errorGroup_model.txt’)
Data comes from 3 different studies in this example. We want to have the same structural model but use different error models for the 3 studies. A solution consists in defining the column STUDY with the reserved keyword OBSERVATION ID. It will then be possible to define one error model per outcome:
Here, we use the same PK model for the 3 studies:
[LONGITUDINAL] input = {V, k} PK: Cc1 = pkmodel(V, k) Cc2 = Cc1 Cc3 = Cc1 OUTPUT: output = {Cc1, Cc2, Cc3}
Since 3 outputs are defined in the structural model, one can now define 3 error models in the GUI:
Different residual error parameters are estimated for the 3 studies. One can remark than, even if 2 proportional error models are used for the 2 first studies, different parameters b1 and b2 are estimated:
2.3.2.Handling censored (BLQ) data
 Introduction
 Theory
 Censoring definition in a data set
 PK data below a lower limit of quantification
 PK data below a lower limit of quantification or below a limit of detection
 PK data below a lower limit of quantification and PD data above an upper limit of quantification
 Combination of interval censored PK and PD data
 Case studies
Objectives: learn how to handle easily and properly censored data, i.e. data below (resp. above) a lower (resp.upper) limit of quantification (LOQ) or below a limit of detection (LOD).
Projects: censoring1log_project, censoring1_project, censoring2_project, censoring3_project, censoring4_project
Introduction
Censoring occurs when the value of a measurement or observation is only partially known. For continuous data measurements in the longitudinal context, censoring refers to the values of the measurements, not the times at which they were taken. For example, the lower limit of detection (LLOD) is the lowest quantity of a substance that can be distinguished from its absence. Therefore, any time the quantity is below the LLOD, the “observation” is not a measurement but the information that the measured quantity is less than the LLOD. Similarly, in longitudinal studies of viral kinetics, measurements of the viral load below a certain limit, referred to as the lower limit of quantification (LLOQ), are so low that their reliability is considered suspect. A measuring device can also have an upper limit of quantification (ULOQ) such that any value above this limit cannot be measured and reported.
As hinted above, censored values are not typically reported as a number, but their existence is known, as well as the type of censoring. Thus, the observation (i.e., what is reported) is the measurement if not censored, and the type of censoring otherwise.
We usually distinguish between three types of censoring: left, right and interval. In each case, the SAEM algorithm implemented in Monolix
properly computes the maximum likelihood estimate of the population parameters, combining all the information provided by censored and non censored data.
Theory
In the presence of censored data, the conditional density function needs to be computed carefully. To cover all three types of censoring (left, right, interval), let be the (finite or infinite) censoring interval existing for individual i at time . Then,
$$\displaystyle p(y^{(r)}\psi)=\prod_{i=1}^{N}\prod_{j=1}^{n_i}p(y_{ij}\psi_i)^{1_{y_{ij}\notin I_{ij}}}\mathbb{P}(y_{ij}\in I_{ij}\psi_i)^{1_{y_{ij}\in I_{ij}}}$$
where
$$\displaystyle \mathbb{P}(y_{ij}\in I_{ij}\psi_i)=\int_{I_{ij}} p_{y_{ij}\psi_i} (u\psi_i)du$$
We see that if is not censored (i.e. ), its contribution to the likelihood is the usual , whereas if it is censored, the contribution is .
For the calculation of the likelihood, this is equivalent to the M3 method in NONMEM when only the CENSORING column is given, and to the M4 method when both a CENSORING column and a LIMIT column are given.
Censoring definition in a data set
In the dataset format used by Monolix and PKanalix, censored information is included in this way:
 The censored measurement should be in the OBSERVATION column.
 In an additional CENSORING column, put 0 if the observation is not censored, and 1 or – 1 depending if the measurement given in the observation column is a lower or an upper limit.
 Optionally, include a LIMIT column to set the other limit.
To quickly include censoring information to your dataset by using BLQ tags in the observation column, you can use data formatting.
Examples are provided below and here.
PK data below a lower limit of quantification
Left censored data
 censoring1log_project (data = ‘censored1log_data.txt’, model = ‘pklog_model.txt’)
PK data are logconcentration in this example. The limit of quantification of 1.8 mg/l for concentrations becomes log(1.8)=0.588 for logconcentrations. The column of observations (Y) contains either the LLOQ for data below the limit of quantification (BLQ data) or the measured logconcentrations for non BLQ data. Furthermore, Monolix
uses an additional column CENSORING to indicate if an observation is left censored (CENS=1) or not (CENS=0). In this example, subject 1 has two BLQ data at times 24h and 30h (the measured logconcentrations were below 0.588 at these times):
The plot of individual fits displays BLQ (red band) and non BLQ data (blue dots) together with the predicted logconcentrations (purple line) on the whole time interval:
Notice that the band goes from .8 to Infinity as no bound has been specified (no LIMIT column was proposed).
For diagnosis plots such as VPC, residuals of observations versus predictions, Monolix
samples the BLQ data from the conditional distribution
$$p(y^{BLQ}  y^{non BLQ}, \hat{\psi}, \hat{\theta})$$
where and are the estimated population and individual parameters. This is done by adding a residual error on top of the prediction, using a truncated normal distribution to make sure that the simulated BLQ remains within the censored interval. This is the most efficient way to take into account the complete information provided by the data and the model for diagnosis plots such as VPCs:
A strong bias appears if LLOQ is used instead for the BLQ data (if you choose LOQ instead of simulated in the display frame of the settings) :
Notice that ignoring the BLQ data entails a loss of information as can be seen below (if you choose no in the “Use BLQ” toggle):
As can be seen below, imputed BLQ data is also used for residuals (IWRES on the left) and for observations versus predictions (on the right)
More on these diagnosis plots
Impact of the BLQ in residuals and observations versus predictions plots
A strong bias appears if LLOQ is used instead for the BLQ data for these two diagnosis plots:
while ignoring the BLQ data entails a loss of information:
BLQ predictive checks
The BLQ predictive check is a diagnosis plot that displays the fraction of cumulative BLQ data (blue line) with a 90% prediction interval (blue area).
Interval censored data
 censoring1_project (data = ‘censored1_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
We use the original concentrations in this project. Then, BLQ data should be treated as interval censored data since a concentration is know to be positive. In other word, a data reported as BLQ data means that the (non reported) measured concentration is between 0 and 1.8mg/l. The value in the observation column 1.8 indicates the value, the value in the CENSORING column indicates that the value in the observation column is the upper bound. An additional column LIMIT reports the lower limit of the censored interval (0 in this example):
Remarks
 if this column is missing, then BLQ data is assumed to be leftcensored data that can take any positive and negative value below LLOQ.
 the value of the limit can vary between observations of the same subject.
Monolix
will use this additional information to estimate the model parameters properly and to impute the BLQ data for the diagnosis plots.
Plot of individual fits now displays LLOD at 1.8 with a red band when a PK data is censored. We see that the band lower limit is at 0 as defined in the limit column.
PK data below a lower limit of quantification or below a limit of detection
 censoring2_project (data = ‘censored2_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
Plot of individual fits now displays LLOQ or LLOD with a red band when a PK data is censored. We see that the band lower limits depend on the observation.
PK data below a lower limit of quantification and PD data above an upper limit of quantification
 censoring3_project (data = ‘censored3_data.txt’, model = ‘pkpd_model.txt’)
We work with PK and PD data in this project and assume that the PD data may be right censored and that the upper limit of quantification is ULOQ=90. We use CENS=1 to indicate that an observation is right censored. In such case, the PD data can take any value above the upper limit reported in column Y (here the YTYPE column of type OBSERVED ID defines the type of observation, YTYPE=1 and YTYPE=2 are used respectively for PK and PD data):
Plot of individual fits for the PD data now displays ULOQ and the predicted PD profile:
We can display the cumulative fraction of censored data both for the PK and the PD data (on the left and right respectively):
Combination of interval censored PK and PD data
 censoring4_project (data = ‘censored4_data.txt’, model = ‘pkpd_model.txt’)
We assume in this example
 2 different censoring intervals(0,1) and (1.2, 1.8) for the PK,
 a censoring interval (80,90) and right censoring (>90) for the PD.
Combining columns CENS, LIMIT and Y allow us to combine efficiently these different censoring processes:
This coding of the data means that, for subject 1,
 PK data is between 0 and 1 at time 30h (second blue frame),
 PK data is between 1.2 and 1.8 at times 0.5h and 24h (first blue frame for time .5h),
 PD data is between 80 and 90 at times 12h and 16h (second green frame for time 12h),
 PD data is above 90 at times 4h and 8h (first green frame for time 4h).
Plot of individual fits for the PK and the PD data displays the different limits of these censoring intervals (PK on the left and PD on the right):
Other diagnosis plots, such as the plot of observations versus predictions, adequately use imputed censored PK and PD data:
Case studies
 8.case_studies/hiv_project (data = ‘hiv_data.txt’, model = ‘hivLatent_model.txt’)
 8.case_studies/hcv_project (data = ‘hcv_data.txt’, model = ‘hcvNeumann98_model_latent.txt’)
2.3.3.Mixture of structural models
 Introduction
 Between subject mixture models
 Supervised learning (with regressor)
 Unsupervised learning (with bsmm function)
 Within subject mixture models
Objectives: learn how to implement between subject mixture models (BSMM) and within subject mixture models (WSMM).
Projects: bsmm1_project, bsmm2_project, wsmm_project
Introduction
There are two approaches to define a mixture of models:
 defining a mixture of structural models (via a regressor or via the bsmm function). This approach is detailed here.
 introducing a categorical covariate (known or latent). –> click here to go to the page dedicated to this approach.
Several types of mixture models exist, they are useful in the context of mixed effects models. It may be necessary in some situations to introduce diversity into the structural models themselves:
 Betweensubject model mixtures (BSMM) assume that there exists subpopulations of individuals. Different structural models describe the response of each subpopulation, and each subject belongs to one of these subpopulations. One can imagine for example different structural models for responders, nonresponders and partial responders to a given treatment.
The easiest way to model a finite mixture model is to introduce a label sequence that takes its values in such that if subject i belongs to subpopulation m. is the probability for subject i to belong to subpopulation m. A BSMM assumes that the structural model is a mixture of M different structural models:
$$f\left(t_{ij}; \psi_i, z_i \right) = \sum_{m=1}^M 1_{z_i = m} f_m\left( t_{ij}; \psi_i \right) $$
In other word, each subpopulation has its own structural model: is the structural model for subpopulation m.
 Withinsubject model mixtures (WSMM) assume that there exist subpopulations (of cells, viruses, etc.) within each patient. In this case, different structural models can be used to describe the response of different subpopulations, but the proportion of each subpopulation depends on the patient.
Then, it makes sense to consider that the mixture of models happens within each individual. Such withinsubject model mixtures require additional vectors of individual parameters representing the proportions of the M models within each individual i:
$$f\left( t_{ij}; \psi_i, z_i \right) = \sum_{m=1}^M \pi_{m,i} f_m\left( t_{ij}; \psi_i \right)$$
The proportions are now individual parameters in the model and the problem is transformed into a standard mixed effects model. These proportions are assumed to be positive and to sum to 1 for each patient.
Between subject mixture models
Supervised learning
 bsmm1_project (data = ‘pdmixt1_data.txt’, model = ‘bsmm1_model.txt’)
We consider a very simple example here with two subpopulations of individuals who receive a given treatment. The outcome of interest is the measured effect of the treatment (a viral load for instance). The two populations are non responders and responders. We assume here that the status of the patient is known. Then, the data file contains an additional column GROUP. This column is duplicated because Monolix
uses it
 i) as a regression variable (REGRESSOR): it is used in the model to distinguish responders and non responders,
 ii) as a categorical covariate (CATEGORICAL COVARIATE): it is used to stratify the diagnosis plots.
We can then display the data
and use the categorical covariate GROUP_CAT to split the plot into responders and non responders:
We use different structural models for non responders and responders. The predicted effect for non responders is constant f(t) = A1 while the predicted effect for responders decreases exponentially f(t) = A2 exp(kt).
The model is implemented in the model file bsmm1_model.txt
(note that the names of the regression variable in the data file and in the model script do not need to match):
[LONGITUDINAL] input = {A1, A2, k, g} g = {use=regressor} EQUATION: if g==1 f = A1 else f = A2*exp(k*max(t,0)) end OUTPUT: output = f
The plot of individual fits exhibit the two different structural models:
VPCs should then be splitted according to the GROUP_CAT
as well as the prediction distribution for non responders and responders:
Unsupervised learning
 bsmm2_project (data = ‘pdmixt2_data.txt’, model = ‘bsmm2_model.txt’)
The status of the patient is unknown in this project (which means that the column GROUP is not available anymore). Let p be the proportion of non responders in the population. Then, the structural model for a given subject is f1 with probability p and f2 with probability 1p. The structural model is therefore a BSMM:
[LONGITUDINAL] input = {A1, A2, k, p} EQUATION: f1 = A1 f2 = A2*exp(k*max(t,0)) f = bsmm(f1, p, f2, 1p) OUTPUT: output = f
Important:
 The bsmm function must be used on the last line of the structural model, just before “OUTPUT:”. It is not possible to reuse the variable returned by the bsmm function (here f) in another equation.
 p is a population parameter of the model to estimate. There is no interpatient variability on p: all the subjects have the same probability of being a non responder in this example. We use a logitnormal distribution for p in order to constrain it to be between 0 and 1, but without variability:
p is estimated with the other population parameters:
Then, the group to which a patient belongs is also estimated as the group of highest conditional probability:
$$\begin{aligned}\hat{z}_i &= 1~~~~\textrm{if}~~~~ \mathbb{P}(z_i=1  (y_{ij}), \hat{\psi}_i, \hat{\theta})> \mathbb{P}(z_i=2  (y_{ij}),\hat{\psi}_i, \hat{\theta}),\\ &=0~~~~\textrm{otherwise}\end{aligned}$$
The estimated groups can be used as a stratifying variable to split some plots such as VPCs
Bsmm function with ODEs
The bsmm function can also be used with models defined via ODE systems. The syntax in that case follows this example, with model M defined as a mixture of M1 and M2:
M1_0 = ... ; initial condition for M1 ddt_M1 = ... ; ODE for M1 M2_0 = ... ; initial condition for M2 ddt_M2 = ... ; ODE for M2 M = bsmm(M1,p1,M2,1p1)
Unsupervised learning with latent covariates
If the models composing the mixture have a similar structure, it is sometimes possible and easier to implement the mixture with a latent categorical covariate instead of the bsmm function. It also has the advantage of allowing more than two mixture groups, while the bsmm function can only define two mixture groups.
Within subject mixture models
 wsmm_project (data = ‘pdmixt2_data.txt’, model = ‘wsmm_model.txt’)
It may be too simplistic to assume that each individual is represented by only one welldefined model from the mixture. We consider here that the mixture of models happens within each individual and use a WSMM: f = p*f1 + (1p)*f2
[LONGITUDINAL] input = {A1, A2, k, p} EQUATION: f1 = A1 f2 = A2*exp(k*max(t,0)) f = wsmm(f1, p, f2, 1p) OUTPUT: output = f
Remark: Here, writing f = wsmm(f1, p, f2, 1p) is equivalent to writing f = p*f1 + (1p)*f2
Important: Here, p is an individual parameter: the subjects have different proportions of non responder cells. We use a probitnormal distribution for p in order to constrain it to be between 0 and 1, with variability:
There is no latent covariate when using WSMM: mixtures are continuous mixtures. We therefore cannot split anymore the VPC and the prediction distribution anymore.
2.4.Models for non continuous outcomes
2.4.1.Timetoevent data models
 Introduction
 Formatting of timetoevent data in the MonolixSuite
 Single event
 Repeated events
 User defined likelihood function for timetoevent data
Objectives: learn how to implement a model for (repeated) timetoevent data with different censoring processes.
Projects: tte1_project, tte2_project, tte3_project, tte4_project, rtteWeibull_project, rtteWeibullCount_project
Introduction
Here, observations are the “times at which events occur”. An event may be oneoff (e.g., death, hardware failure) or repeated (e.g., epileptic seizures, mechanical incidents, strikes). Several functions play key roles in timetoevent analysis: the survival, hazard and cumulative hazard functions. We are still working under a population approach here so these functions, detailed below, are thus individual functions, i.e., each subject has its own. As we are using parametric models, this means that these functions depend on individual parameters \((\psi_i)\).
 The survival function \(S(t, \psi_i)\) gives the probability that the event happens to individual i after time \(t>t_{\text{start}}\):
$$S(t,\psi_i) = \mathbb{P}(T_i>t; \psi_i) $$
 The hazard function \(h(t,psi_i)\) is defined for individual i as the instantaneous rate of the event at time t, given that the event has not already occurred:
$$h(t, \psi_i) = \lim_{dt \to 0} \frac{S(t, \psi_i) – S(t + dt, \psi_i)}{ S(t, \psi_i) dt} $$
This is equivalent to
$$h(t, \psi_i) = \frac{d}{dt} \left(\log{S(t, \psi_i)}\right)$$
 Another useful quantity is the cumulative hazard function \(H(a,b; \psi_i)\), defined for individual i as
$$H(a,b; \psi_i) = \int_a^b h(t,\psi_i) dt $$
Note that \(S(t, \psi_i) = e^{H(t_{\text{start}},t; \psi_i)}\). Then, the hazard function \(h(t,\psi_i)\) characterizes the problem, because knowing it is the same as knowing the survival function \(S(t, \psi_i)\). The probability distribution of survival data is therefore completely defined by the hazard function.
Timetoevent (TTE) models are thus defined in Monolix via the hazard function. Monolix also holds a TTE library that contains typical hazard functions for timetoevent data. More details and modeling guidelines can be found on the TTE dedicated webpage, along with case studies.
Formatting of timetoevent data in the MonolixSuite
In the data set, exactly observed events, interval censored events and right censoring are recorded for each individual. Contrary to other softwares for survival analysis, the MonolixSuite requires to specify the time at which the observation period starts. This allows to define the data set using absolute times, in addition to durations (if the start time is zero, the records represent durations between the start time and the event).
The column TIME also contains the end of the observation period or the time intervals for intervalcensoring. The column OBSERVATION contains an integer that indicates how to interpret the associated time. The different values for each type of event and observation are summarized in the table below:
The figure below summarizes the different situations with examples:
For instance for single events, exactly observed (with or without right censoring), one must indicate the start time of the observation period (Y=0), and the time of event (Y=1) or the time of the end of the observation period if no event has occurred (Y=0). In the following example:
ID TIME Y 1 0 0 1 34 1 2 0 0 2 80 0
the observation period lasts from starting time t=0 to the final time t=80. For individual 1, the event is observed at t=34, and for individual 2, no event is observed during the period. Thus it is noticed that at the final time (t=80), no event had occurred. Using absolute times instead of duration, we could equivalently write:
ID TIME Y 1 20 0 1 54 1 2 33 0 2 113 0
The duration between start time and event (or end of the observation period) are the same as before, but this time we record the day at which the patients enter the study and the days at which they have events or leave the study. Different patients may enter the study at different times.
Single event
To begin with, we will consider a oneoff event. Depending on the application, the length of time to this event may be called the survival time (until death, for instance), failure time (until hardware fails), and so on. In general, we simply say “timetoevent”. The random variable representing the timetoevent for subject i is typically written Ti.
Single event exactly observed or right censored
 tte1_project (data = tte1_data.txt , model=lib:exponential_model_singleEvent.txt)
The event time may be exactly observed at time \(t_i\), but if we assume that the trial ends at time \(t_{\text{stop}}\), the event may happen after the end. This is “right censoring”. Here, Y=0 at time t means that the event happened after t and Y=1 means that the event happened at time t. The rows with t=0 are included to show the trial start time \(t_{\text{start}}=0\):
By clicking on the button Observed data
, it is possible to display the Kaplan Meier plot (i.e. the empirical survival function) before fitting any model:
A very basic model with constant hazard is used for this data:
[LONGITUDINAL] input = Te EQUATION: h = 1/Te DEFINITION: Event = {type=event, maxEventNumber=1, hazard=h} OUTPUT: output = {Event}
Here, Te is the expected time to event. Specification of the maximum number of events is required both for the estimation procedure and for the diagnosis plots based on simulation, such as the predicted interval for the Kaplan Meier plot which is obtained by Monte Carlo simulation:
Single event interval censored or right censored
 tte2_project (data = tte2_data.txt , model=exponentialIntervalCensored_model.txt)
We may know the event has happened in an interval \(I_i\) but not know the exact time \(t_i\). This is interval censoring. Here, Y=0 at time t means that the event happened after t and Y=1 means that the event happened before time t.
Event for individual 1 happened between t=10 and t=15. No event was observed until the end of the experiment (t=100) for individual 5. We use the same basic model, but we now need to specify that the events are interval censored:
[LONGITUDINAL] input = Te EQUATION: h = 1/Te DEFINITION: Event = {type=event, maxEventNumber=1, eventType=intervalCensored, hazard = h intervalLength=5 ; used for the plots (not mandatory) } OUTPUT: output = Event
Repeated events
Sometimes, an event can potentially happen again and again, e.g., epileptic seizures, heart attacks. For any given hazard function h, the survival function S for individual i now represents the survival since the previous event at \(t_{i,j1}\), given here in terms of the cumulative hazard from \(t_{i,j1}\) to \(t_{i,j}\):
$$S(t_{i,j}  t_{i,j1}; \psi_i) = \mathbb{P}(T_{i,j} > t_{i,j}  T_{i,j1} = t_{i,j1}; \psi_i) = \exp(\int_{t_{i,j1}}^{t_{i,j}}h(t,\psi_i) dt)$$
Repeated events exactly observed or right censored
 tte3_project (data = tte3_data.txt , model=lib:exponential_model_repeatedEvents.txt)
A sequence of \(n_i\) event times is precisely observed before \(t_{\text{stop}} = 200\): We can then display the Kaplan Meier plot for the first event and the mean number of events per individual:
After fitting the model, prediction intervals for these two curves can also be displayed on the same graph as on the following
Repeated events interval censored or right censored
 tte4_project (data = tte4_data.txt , model=exponentialIntervalCensored_repeated_model.txt)
We do not know the exact event times, but the number of events that occurred for each individual in each interval of time.
User defined likelihood function for timetoevent data
 weibullRTTE (data = weibull_data.txt , model=weibullRTTE_model.txt)
A Weibull model is used in this example:
[LONGITUDINAL] input = {lambda, beta} EQUATION: h = (beta/lambda)*(t/lambda)^(beta1) DEFINITION: Event = {type=event, hazard=h, eventType=intervalCensored, intervalLength=5} OUTPUT: output = Event
 weibullCount (data = weibull_data.txt , model=weibullCount_model.txt)
Instead of defining the data as events, it is possible to consider the data as count data: indeed, we count the number of events per interval. An additional column with the start of the interval is added in the data file and defined as a regression variable. We then use a model for count data (see rtteWeibullCount_model.txt).
2.4.2.Count data model
 Introduction
 Formatting of count data in the MonolixSuite
 Count data with constant distribution over time
 Count data with time varying distribution
Objectives: learn how to implement a model for count data.
Projects: count1a_project, count1a_project, count1a_project, count2_project
Introduction
Longitudinal count data is a special type of longitudinal data that can take only nonnegative integer values {0, 1, 2, …} that come from counting something, e.g., the number of seizures, hemorrhages or lesions in each given time period . In this context, data from individual j is the sequence \(y_i=(y_{ij},1\leq j \leq n_i)\) where \(y_{ij}\) is the number of events observed in the jth time interval \(I_{ij}\).
Count data models can also be used for modeling other types of data such as the number of trials required for completing a given task or the number of successes (or failures) during some exercise. Here, \(y_{ij}\) is either the number of trials or successes (or failures) for subject i at time \(t_{ij}\). For any of these data types we will then model \(y_i=(y_{ij},1 \leq j \leq n_i)\) as a sequence of random variables that take their values in {0, 1, 2, …}. If we assume that they are independent, then the model is completely defined by the probability mass functions \(\mathbb{P}(y_{ij}=k)\) for \(k \geq 0\) and \(1 \leq j \leq n_i\). Here, we will only consider parametric distributions for count data.
Formatting of count data in the MonolixSuite
Count data can only take nonnegative integer values that come from counting something, e.g., the number of trials required for completing a given task. The task can for instance be repeated several times and the individual’s performance followed. In the following data set:
ID TIME Y 1 0 10 1 24 6 1 48 5 1 72 2
10 trials are necessary the first day (t=0), 6 the second day (t=24), etc. Count data can also represent the number of events happening in regularly spaced intervals, e.g the number of seizures every week. If the time intervals are not regular, the data may be considered as repeated timetoevent interval censored, or the interval length can be given as regressor to be used to define the probability distribution in the model.
One can see the epilepsy attacks data set for a more practical example.
Modling count data in the MonolixSuite
Link to the detailed description of the library of count models integrated within Monolix.
Count data with constant distribution over time
 count1a_project (data = ‘count1_data.txt’, model = ‘count_library/poisson_mlxt.txt’)
A Poisson model is used for fitting the data:
[LONGITUDINAL] input = lambda DEFINITION: Y = {type = count, log(P(Y=k)) = lambda + k*log(lambda)  factln(k) } OUTPUT: output = Y
Residuals for noncontinuous data reduce to NPDEs. We can compare the empirical distribution of the NPDEs with the distribution of a standardized normal distribution either with the pdf (top) or the cdf (bottom):
VPCs for count data compare the observed and predicted frequencies of the categorized data over time:
 count1b_project (data = ‘count1_data.txt’, model = ‘count_library/poissonMixture_mlxt.txt’)
A mixture of two Poisson distributions is used to fit the same data. For that, we define the probability of k occurrences as the weigthed sum of two Poisson distributions with two expected numbers of occurrences lambda1 and lambda2. The structural model file writes
[LONGITUDINAL] input = {lambda1, alpha, mp} EQUATION: lambda2 = (1+alpha)*lambda1 DEFINITION: Y = { type = count, P(Y=k) = mp*exp(lambda1 + k*log(lambda1)  factln(k)) + (1mp)*exp(lambda2 + k*log(lambda2)  factln(k)) } OUTPUT: output = Y
Thus, the parameter alpha has to be strictly positive to ensure different expected number of occurrences in the two poisson distributions and mp has to be in [0, 1] to ensure the probability is correctly defined. Thus those parameters should be defined with lognormal and probitnormal distribution respectively as shown on the following figure.
We see on the VPC below that the data set is well modeled using this mixture of Poisson distributions.
In addition, we can compute the prediction distribution of the modalities as on the following figure
Count data with time varying distribution
 count2_project (data = ‘count2_data.txt’, model = ‘count_library/poissonTimeVarying_mlxt.txt’)
The distribution of the data changes with time in this example:
We then use a Poisson distribution with a time varying intensity:
[LONGITUDINAL] input = {a,b} EQUATION: lambda= a*exp(b*t) DEFINITION: y = {type=count, P(y=k)=exp(lambda)*(lambda^k)/factorial(k)} OUTPUT: output = y
This model seems to fit the data very well:
2.4.3.Categorical data model
 Introduction
 Formatting of categorical data in the MonolixSuite
 Ordered categorical data
 Ordered categorical data with regression variables
 Discretetime Markov chain
 Continuoustime Markov chain
Objectives: learn how to implement a model for categorical data, assuming either independence or a Markovian dependence between observations.
Projects: categorical1_project, categorical2_project, markov0_project, markov1a_project, markov1b_project, markov1c_project, markov2_project, markov3a_project, markov3b_project
Introduction
Assume now that the observed data takes its values in a fixed and finite set of nominal categories \(\{c_1, c_2,\ldots , c_K\}\). Considering the observations \((y_{ij},\, 1 \leq j \leq n_i)\) for any individual \(i\) as a sequence of conditionally independent random variables, the model is completely defined by the probability mass functions \(\mathbb{P}(y_{ij}=c_k  \psi_i)\) for \(k=1,\ldots, K\) and \(1 \leq j \leq n_i\). For a given (i,j), the sum of the K probabilities is 1, so in fact only K1 of them need to be defined. In the most general way possible, any model can be considered so long as it defines a probability distribution, i.e., for each k, \(\mathbb{P}(y_{ij}=c_k  \psi_i) \in [0,1]\), and \(\sum_{k=1}^{K} \mathbb{P}(y_{ij}=c_k  \psi_i) =1\). Ordinal data further assumed that the categories are ordered, i.e., there exists an order \(\prec\) such that
$$c_1 \prec c_2,\prec \ldots \prec c_K $$
We can think, for instance, of levels of pain (low \(\prec\) moderate \(\prec\) severe) or scores on a discrete scale, e.g., from 1 to 10. Instead of defining the probabilities of each category, it may be convenient to define the cumulative probabilities \(\mathbb{P}(y_{ij} \preceq c_k  \psi_i)\) for \(k=1,\ldots ,K1\), or in the other direction: \(\mathbb{P}(y_{ij} \succeq c_k  \psi_i)\) for \(k=2,\ldots, K\). Any model is possible as long as it defines a probability distribution, i.e., it satisfies
$$0 \leq \mathbb{P}(y_{ij} \preceq c_1  \psi_i) \leq \mathbb{P}(y_{ij} \preceq c_2  \psi_i)\leq \ldots \leq \mathbb{P}(y_{ij} \preceq c_K  \psi_i) =1 .$$
It is possible to introduce dependence between observations from the same individual by assuming that \((y_{ij},\,j=1,2,\ldots,n_i)\) forms a Markov chain. For instance, a Markov chain with memory 1 assumes that all that is required from the past to determine the distribution of \(y_{ij}\) is the value of the previous observation \(y_{i,j1}\)., i.e., for all \(k=1,2,\ldots ,K\),
$$\mathbb{P}(y_{ij} = c_k\,\,y_{i,j1}, y_{i,j2}, y_{i,j3},\ldots,\psi_i) = \mathbb{P}(y_{ij} = c_k  y_{i,j1},\psi_i)$$
Formatting of categorical data in the MonolixSuite
In case of categorical data, the observations at each time point can only take values in a fixed and finite set of nominal categories. In the data set, the output categories must be coded as integers, as in the following example:
ID TIME Y 1 0.5 3 1 1 0 1 1.5 2 1 2 2 1 2.5 3
One can see the respiratory status data set and the warfarin data set for example for more practical examples on a categorical and a joint continuous and categorical data set respectively.
Ordered categorical data
 categorical1_project (data = ‘categorical1_data.txt’, model = ‘categorical1_model.txt’)
In this example, observations are ordinal data that take their values in {0, 1, 2, 3}:
 Cumulative odds ratio are used in this example to define the model
$$\textrm{logit}(\mathbb{P}(y_{ij} \leq k))= \log \left( \frac{\mathbb{P}(y_{ij} \leq k)}{1 – \mathbb{P}(y_{ij} \leq k )} \right)$$
where
$$\begin{array}{ccl} \text{logit}(\mathbb{P}(y_{ij} \leq 0)) &=& \theta_{i,1}\\ \text{logit}(\mathbb{P}(y_{ij} \leq 1)) &=& \theta_{i,1}+\theta_{i,2}\\ \text{logit}(\mathbb{P}(y_{ij} \leq 2)) &=& \theta_{i,1}+\theta_{i,2}+\theta_{i,3}\end{array}$$
This model is implemented in categorical1_model.txt
:
[LONGITUDINAL] input = {th1, th2, th3} DEFINITION: level = { type = categorical, categories = {0, 1, 2, 3}, logit(P(level<=0)) = th1 logit(P(level<=1)) = th1 + th2 logit(P(level<=2)) = th1 + th2 + th3 }
A normal distribution is used for \(\theta_{1}\), while lognormal distributions for \(\theta_{2}\) and \(\theta_{3}\) ensure that these parameters are positive (even without variability). Residuals for noncontinuous data reduce to NPDE’s. We can compare the empirical distribution of the NPDE’s with the distribution of a standardized normal distribution:
VPC’s for categorical data compare the observed and predicted frequencies of each category over time:
The prediction distribution can also be computed by MonteCarlo:
Ordered categorical data with regression variables
 categorical2_project (data = ‘categorical2_data.txt’, model = ‘categorical2_model.txt’)
A proportional odds model is used in this example, where PERIOD and DOSE are used as regression variables (i.e. timevarying covariates)
Discretetime Markov chain
If observation times are regularly spaced (constant length of time between successive observations), we can consider the observations \((y_{ij},j=1,2,\ldots,n_i)\) to be a discretetime Markov chain.
 markov0_project (data = ‘markov1a_data.txt’, model = ‘markov0_model.txt’)
In this project, states are assumed to be independent and identically distributed:
\( \mathbb{P}(y_{ij} = 1) = 1 – \mathbb{P}(y_{ij} = 2) = p_{i,1} \)
Observations in markov1a_data.txt
take their values in {1, 2}.
 markov1a_project (data = ‘markov1a_data.txt’, model = ‘markov1a_model.txt’)
Here,
\(\begin{aligned}\mathbb{P}(y_{i,j} = 1  y_{i,j1} = 1) = 1 – \mathbb{P}(y_{i,j} = 2  y_{i,j1} = 1) = p_{i,11}\\ \mathbb{P}(y_{i,j} = 1  y_{i,j1} = 2) = 1 – \mathbb{P}(y_{i,j} = 2  y_{i,j1} = 2) = p_{i,12} \end{aligned}\)
[LONGITUDINAL] input = {p11, p21} DEFINITION: State = {type = categorical, categories = {1,2}, dependence = Markov P(State=1State_p=1) = p11 P(State=1State_p=2) = p21 }
The distribution of the initial state is not defined in the model, which means that, by default,
\( \mathbb{P}(y_{i,1} = 1) = \mathbb{P}(y_{i,1} = 2) = 0.5 \)
 markov1b_project (data = ‘markov1b_data.txt’, model = ‘markov1b_model.txt’)
The distribution of the initial state, \(p = \mathbb{P}(y_{i,1} = 1)\), is estimated in this example
DEFINITION: State = {type = categorical, categories = {1,2}, dependence = Markov P(State_1=1)= p P(State=1State_p=1) = p11 P(State=1State_p=2) = p21 }
 markov3a_project (data = ‘markov3a_data.txt’, model = ‘markov3a_model.txt’)
Transition probabilities change with time in this example. We then define time varying transition probabilities in the model:
[LONGITUDINAL] input = {a1, b1, a2, b2} EQUATION: lp11 = a1 + b1*t/100 lp21 = a2 + b2*t/100 DEFINITION: State = {type = categorical, categories = {1,2}, dependence = Markov logit(P(State=1State_p=1)) = lp11 logit(P(State=1State_p=2)) = lp21 }
 markov2_project (data = ‘markov2_data.txt’, model = ‘markov2_model.txt’)
Observations in markov2_data.txt
take their values in {1, 2, 3}. Then, 6 transition probabilities need to be defined in the model.
Continuoustime Markov chain
The previous situation can be extended to the case where time intervals between observations are irregular by modeling the sequence of states as a continuoustime Markov process. The difference is that rather than transitioning to a new (possibly the same) state at each time step, the system remains in the current state for some random amount of time before transitioning. This process is now characterized by transition rates instead of transition probabilities:
\( \mathbb{P}(y_{i}(t+h) = k,,y_{i}(t)=\ell , \psi_i) = h \rho_{\ell k}(t,\psi_i) + o(h),\qquad k \neq \ell .\)
The probability that no transition happens between \(t\) and \(t+h\) is
\( \mathbb{P}(y_{i}(s) = \ell, \forall s\in(t, t+h)  y_{i}(t)=\ell , \psi_i) = e^{h , \rho_{\ell \ell}(t,\psi_i)} .\)
Furthermore, for any individual i and time t, the transition rates \((\rho_{\ell,k}(t, \psi_i))\) satisfy for any \(1\leq \ell \leq K\),
\( \sum_{k=1}^K \rho_{\ell k}(t, \psi_i) = 0\)
Constructing a model therefore means defining parametric functions of time \((\rho_{\ell,k})\) that satisfy this condition.
 markov1c_project (data = ‘markov1c_data.txt’, model = ‘markov1c_model.txt’)
Observation times are irregular in this example. Then, a continuous time Markov chain should be used in order to take into account the Markovian dependence of the data:
DEFINITION: State = { type = categorical, categories = {1,2}, dependence = Markov transitionRate(1,2) = q12 transitionRate(2,1) = q21 }
 markov3b_project (data = ‘markov3b_data.txt’, model = ‘markov3b_model.txt’)
Time varying transition rates are used in this example.
2.5.Joint models for multivariate outcomes
2.5.1.Joint models for continuous outcomes
 Introduction
 Fitting first a PK model to the PK data
 Simultaneous PKPD modeling
 Sequential PKPD modelling
 Fitting a PKPD model to the PD data only
 Case studies
Objectives: learn how to implement a joint model for continuous PKPD data.
Projects: warfarinPK_project, warfarin_PKPDimmediate_project, warfarin_PKPDeffect_project, warfarin_PKPDturnover_project, warfarin_PKPDseq1_project, warfarin_PKPDseq2_project, warfarinPD_project
Introduction
A “joint model” describes two or more types of observation that typically depend on each other. A PKPD model is a “joint model” because the PD depends on the PK. Here we demonstrate how several observations can be modeled simultaneously. We also discuss the special case of sequential PK and PD modelling, using either the population PK parameters or the individual PK parameters as an input for the PD model.
Fitting first a PK model to the PK data
 warfarinPK_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
The column DV of the data file contains both the PK and the PD measurements: in Monolix
this column is tagged as an OBSERVATION column. The column DVID is a flag defining the type of observation: DVID=1 for PK data and DVID=2 for PD data: the keyword OBSERVATION ID is then used for this column.
We will use the model oral1_1cpt_TlagkaVCl from the Monolix
PK library
[LONGITUDINAL] input = {Tlag, ka, V, Cl} EQUATION: Cc = pkmodel(Tlag, ka, V, Cl) OUTPUT: output = {Cc}
Only the predicted concentration Cc is defined as an output of this model. Then, this prediction will be automatically associated to the outcome of type 1 (DVID=1) while the other observations (DVID=2) will be ignored.
Remark: any other ordered values could be used for OBSERVATION ID column: the smallest one will always be associated to the first prediction defined in the model.
Simultaneous PKPD modeling
 warfarin_PKPDimmediate_project (data = ‘warfarin_data.txt’, model = ‘immediateResponse_model.txt’)
It is also possible for the user to write his own PKPD model. The same PK model used previously and an immediate response model are defined in the model file immediateResponse_model.txt
[LONGITUDINAL] input = {Tlag, ka, V, Cl, Imax, IC50, S0} EQUATION: Cc = pkmodel(Tlag, ka, V, Cl) E = S0 * (1  Imax*Cc/(Cc+IC50)) OUTPUT: output = {Cc, E}
Two predictions are now defined in the model: Cc for the PK (DVID=1) and E for the PD (DVID=2).
 warfarin_PKPDeffect_project (data = ‘warfarin_data.txt’, model = ‘effectCompartment_model.txt’)
An effect compartment is defined in the model file effectCompartment_model.txt
[LONGITUDINAL] input = {Tlag, ka, V, Cl, ke0, Imax, IC50, S0} EQUATION: {Cc, Ce} = pkmodel(Tlag, ka, V, Cl, ke0) E = S0 * (1  Imax*Ce/(Ce+IC50)) OUTPUT: output = {Cc, E}
Ce is the concentration in the effect compartment
 warfarin_PKPDturnover_project (data = ‘warfarin_data.txt’, model = ‘turnover1_model.txt’)
An indirect response (turnover) model is defined in the model file turnover1_model.txt
[LONGITUDINAL] input = {Tlag, ka, V, Cl, Imax, IC50, Rin, kout} EQUATION: Cc = pkmodel(Tlag, ka, V, Cl) E_0 = Rin/kout ddt_E = Rin*(1Imax*Cc/(Cc+IC50))  kout*E OUTPUT: output = {Cc, E}
Sequential PKPD modelling
In the sequential approach, a PK model is developed and parameters are estimated in the first step. For a given PD model, different strategies are then possible for the second step, i.e., for estimating the population PD parameters:
Using estimated population PK parameters
 warfarin_PKPDseq1_project (data = ‘warfarin_data.txt’, model = ‘turnover1_model.txt’)
Population PK parameters are set to their estimated values but individual PK parameters are not assumed to be known and sampled from their conditional distributions at each SAEM iteration. In Monolix
, this simply means changing the status of the population PK parameter values so that they are no longer used as initial estimates for SAEM but considered fixed as on the figure below.
To fix parameters, click on the green option button (framed in green) and choose the Fixed method as on the figure below
The joint PKPD model defined in turnover1_model.txt is again used with this project.
Using estimated individual PK parameters
 warfarin_PKPDseq2_project (data = ‘warfarinSeq_data.txt’, model = ‘turnoverSeq_model.txt’)
In htis case, individual PK parameters are set to their estimated values and used as constants in the PKPD model to fit the PD data. To do so, the individual PK parameters need to be added to the PD dataset (or PK/PD dataset) and tagged as regressors.
The PK project (that was executed before and through which the estimated PK parameters were obtained) contains in the result folder the individual parameter values “..\warfarin_PKPDseq1_project\IndividualParameters\estimatedIndividualParameters.txt”. These estimated PK parameters can be added to the PD dataset by using the data formatting tool integrated in Monolix version 2023R1. Depending which tasks have been run in the PK project, the individual parameters corresponding to the conditional mode (EBEs, with “_mode”), the conditional mean (mean of the samples from the conditional distributions, with “_mean”) and an approximation of the conditional mean obtained at the end of the SAEM step (with “_SAEM”) are available in the file. All columns are added to the PD dataset.
At the data tagging step, the user can choose which individual parameters to use. The most common is to use the EBEs so to tag the columns with “_mode” as regressor and leave the others as IGNORE (purple frame below). By activating the toggle button (green frame), the ignored columns flagged with the keyword IGNORE can be hidden.
We use the same turnover model for the PD data. Here, the PK parameters are defined as regression variables (i.e. regressors).
[LONGITUDINAL] input = {Imax, IC50, Rin, kout, Tlag, ka, V, Cl} Tlag = {use = regressor} ka = {use = regressor} V = {use = regressor} Cl = {use = regressor} EQUATION: Cc = pkmodel(Tlag,ka,V,Cl) E_0 = Rin/kout ddt_E= Rin*(1Imax*Cc/(Cc+IC50))  kout*E OUTPUT: output = {E}
As you can see, the names of the regressors do not match the parameter names. The regressors are matched by order (not by name) between the data set and the model input statement.
If there are multiple observation types in the data as well as different response vectors in the output statement of the structural model, then these must be mapped accordingly in the mapping panel.
Fitting a PKPD model to the PD data only
 warfarinPD_project (data = ‘warfarinPD_data.txt’, model = ‘turnoverPD_model.txt’)
In this example, only PD data is available. Nevertheless, a PKPD model – where only the effect is defined as a prediction – can be used for fitting this data and thus defined in the OUTPUT section.
[LONGITUDINAL] input = {Tlag, ka, V, Cl, Imax, IC50, Rin, kout} EQUATION: Cc = pkmodel(Tlag, ka, V, Cl) E_0 = Rin/kout ddt_E = Rin*(1Imax*Cc/(Cc+IC50))  kout*E OUTPUT: output = {E}
Case studies
 8.case_studies/PKVK_project (data = ‘PKVK_data.txt’, model = ‘PKVK_model.txt’)
 8.case_studies/hiv_project (data = ‘hiv_data.txt’, model = ‘hivLatent_model.txt’)
2.5.2.Joint models for non continuous outcomes
 Joint model for continuous PK and categorical PD data
 Joint model for continuous PK and count PD data
 Joint model for continuous PK and timetoevent data
Objectives: learn how to implement a joint model for continuous and non continuous data.
Projects: warfarin_cat_project, PKcount_project, PKrtte_project
Joint model for continuous PK and categorical PD data
 warfarin_cat_project (data = ‘warfarin_cat_data.txt’, model = ‘PKcategorical1_model.txt’)
In this example, the original PD data has been recorded as 1 (Low), 2 (Medium) and 3 (High).
More details about the data
International Normalized Ratio (INR) values are commonly used in clinical practice to target optimal warfarin therapy. Low INR values (<2) are associated with high blood clot risk and high ones (>3) with high risk of bleeding, so the targeted value of INR, corresponding to optimal therapy, is between 2 and 3.
Prothrombin complex activity is inversely proportional to the INR. We can therefore associate the three ordered categories for the INR to three ordered categories for PCA: Low PCA values if PCA is less than 33% (corresponding to INR>3), medium if PCA is between 33% and 50% (INR between 2 and 3) and high if PCA is more than 50% (INR<2).
The column dv contains both the PK and the new categorized PD measurements. Instead of modeling the original PD data, we can model the probabilities of each of these categories, which have direct clinical interpretations. The model is still a joint PKPD model since this probability distribution is expected to depend on exposure, i.e., the plasmatic concentration predicted by the PK model. We introduce an effect compartment to mimic the effect delay. Let \(y_{ij}^{(2)}\) be the PCA level for patient i at time \(t_{ij}^{(2)}\). We can then use a proportional odds model for modeling this categorical data:
$$\begin{array}{ccl}\text{logit} \left(\mathbb{P}(y_{ij}^{(2)} \leq 1  \psi_i)\right) &= &\alpha_{i} + \beta_{i} Ce(t_{ij}^{(2)},\phi_i^{(1)}) \\ \text{logit} \left(\mathbb{P}(y_{ij}^{(2)} \leq 2  \psi_i)\right) &=& \alpha_{i} + \gamma_{i} + \beta_{i}Ce(t_{ij}^{(2)},\phi_i^{(1)}) \\ \text{logit} \left(\mathbb{P}(y_{ij}^{(2)} \leq 3  \psi_i)\right) &= & 1,\end{array}$$
where \(C_e(t,\phi_i^{(1)})\) is the predicted concentration of warfarin in the effect compartment at time t for patient i with PK parameters \(\phi_i^{(1)}\). This model defines a probability distribution for \(y_{ij}\) if \(\gamma_i\geq 0\).
If \(\beta_i>0\), the probability of low PCA at time \(t_{ij}^{(2)}\) (\(y_{ij}^{(2)}=1\)) increases along with the predicted concentration \(Ce(t_{ij}^{(2)},\phi_i^{(1)})\). The joint model is implemented in the model file PKcategorical1_model.txt
[LONGITUDINAL] input = {Tlag, ka, V, Cl, ke0, alpha, beta, gamma} EQUATION: {Cc,Ce} = pkmodel(Tlag,ka,V,Cl,ke0) lp1 = alpha + beta*Ce lp2 = lp1+ gamma ; gamma >= 0 DEFINITION: Level = {type=categorical, categories={1,2,3} logit(P(Level<=1)) = lp1 logit(P(Level<=2)) = lp2 } OUTPUT: output = {Cc, Level}
See Categorical data model for more details about categorical data models.
Joint model for continuous PK and count PD data
 PKcount_project (data = ‘PKcount_data.txt’, model = ‘PKcount1_model.txt’)
The data file used for this project is PKcount_data.txt where the PK and the count PD measurements are simulated. We use a Poisson distribution for the count data, assuming that the Poisson parameter is function of the predicted concentration. For any individual i, we have
$$\lambda_i(t) = \lambda_{0,i} \left( 1 – \frac{Cc_i(t)}{Cc_i(t) + IC50_i} \right)$$
where \(Cc_i(t)\) is the predicted concentration for individual i at time t and
$$ \log\left(P(y_{ij}^{(2)} = k)\right) = \lambda_i(t_{ij}) + k\,\log(\lambda_i(t_{ij})) – \log(k!)$$
The joint model is implemented in the model file PKcount1_model.txt
[LONGITUDINAL] input = {ka, V, Cl, lambda0, IC50} EQUATION: Cc = pkmodel(ka,V,Cl) lambda=lambda0*(1  Cc/(IC50+Cc)) DEFINITION: Seizure = {type = count, log(P(Seizure=k)) = lambda + k*log(lambda)  factln(k) } OUTPUT: output = {Cc,Seizure}
See Count data model for more details about count data models.
Joint model for continuous PK and timetoevent data
 PKrtte_project (data = ‘PKrtte_data.txt’, model = ‘PKrtteWeibull1_model.txt’)
The data file used for this project is PKrtte_data.txt where the PK and the timetoevent data are simulated. We use a Weibull model for the events count data, assuming that the baseline is function of the predicted concentration. For any individual i, we define the hazard function as
$$h_i(t) = \gamma_{i} Cc_i(t) t^{\beta1}$$
where \(Cc_i(t)\) is the predicted concentration for individual i at time t. The joint model is implemented in the model file PKrtteWeibull1_model.txt
[LONGITUDINAL] input = {ka, V, Cl, gamma, beta} EQUATION: Cc = pkmodel(ka, V, Cl) if t<0.1 haz = 0 else haz = gamma*Cc*(t^(beta1)) end DEFINITION: Hemorrhaging = {type=event, hazard=haz} OUTPUT: output = {Cc, Hemorrhaging}
See Timetoevent data model for more details about timetoevent data models.
2.6.Models for the individual parameters
2.6.1.Model for the individual parameters: introduction
What is the individual model and where is it defined in Monolix?
The population approach considers that parameters of the structural model can have a different value for each individual, and the way these values are distributed over individuals and impacted by covariate values is defined in the individual model. The individual model is defined in the lower part of the statistical model tab. This model includes
 distributions for the individual parameters
 which parameters have interindividual variability (random effects)
 correlation structure of the random effects
 covariate effects on the individual parameters
Theory for the individual model
A model for observations depends on a vector of individual parameters \(\psi_i\). As we want to work with a population approach, we now suppose that \(\psi_i\) comes from some probability distribution \(p_{{\psi_i}}\).
In this section, we are interested in the implementation of individual parameter distributions \((p_{{\psi_i}}, 1\leq i \leq N)\). Generally speaking, we assume that individuals are independent. This means that in the following analysis, it is sufficient to take a closer look at the distribution \(p_{{\psi_i}}\) of a unique individual i. The distribution \(p_{{\psi_i}}\) plays a fundamental role since it describes the interindividual variability of the individual parameter \(\psi_i\). In Monolix, we consider that some transformation of the individual parameters is normally distributed and is a linear function of the covariates:
\(h(\psi_i) = h(\psi_{\rm pop})+ \beta \cdot ({c}_i – {c}_{\rm pop}) + \eta_i \,, \quad \eta_i \sim {\cal N}(0,\Omega).\)
This model gives a clear and easily interpreted decomposition of the variability of \(h(\psi_i)\) around \(h(\psi_{\rm pop})\), i.e., of \(\psi_i\) around \(\psi_{\rm pop}\):
The component \(\beta \cdot ({c}_i – {c}_{\rm pop})\) describes part of this variability by way of covariates \({c}_i\) that fluctuate around a typical value \({c}_{\rm pop}\).
The random component \(\eta_i\) describes the remaining variability, i.e., variability between subjects that have the same covariate values. By definition, a mixed effects model combines these two components: fixed and random effects. In linear covariate models, these two effects combine. In the present context, the vector of population parameters to estimate is \(\theta = (\psi_{\rm pop},\beta,\Omega)\). Several extensions of this basic model are possible:
We can suppose for instance that the individual parameters of a given individual can fluctuate over time. Assuming that the parameter values remain constant over some periods of time called occasions, the model needs to be able to describe the interoccasion variability (IOV) of the individual parameters.
If we assume that the population consists of several homogeneous subpopulations, a straightforward extension of mixed effects models is a finite mixture of mixed effects models, assuming for instance that the distribution \(p_{{\psi_i}}\) is a mixture of distributions.
2.6.2.Probability distribution of the individual parameters
 Introduction
 Marginal distributions of the individual parameters
 Correlation structure of the random effects
 Parameters without random effects
 Custom parameter distribution
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
\(h(\psi_i) \sim {\cal N}(h(\psi_{pop}), \omega^2)\)
The transformation \(h\) defines the distribution of \(\psi_i\). Some predefined distributions/transformations are available in Monolix
:
 Normal distribution in ]inf,+inf[:
In that case, \(h(\psi_i) = \psi_i\).
Note: the two mathematical representations for normal distributions are equivalent:
\( \psi_i \sim {\cal N}(\bar{\psi}_{i}, \omega^2) ~~\Leftrightarrow~~ \psi_i = \bar{\psi}_i + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2).\)
 Lognormal distribution in ]0,+inf[:
In that case, \(h(\psi_i) = log(\psi_i)\). A lognormally random variable takes positive values only. A lognormal 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.
Note: the two mathematical representations for lognormal distributions are equivalent:
\(\log(\psi_i) \sim {\cal N}(\log(\bar{\psi}_{i}), \omega^2) ~~\Leftrightarrow~~ \log(\psi_i)=\log(\bar{\psi}_{i})+\eta_i~~\Leftrightarrow~~ \psi_i = \bar{\psi}_i e^{\eta_i}, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2).\)
\(\bar{\psi}_i\) represents the typical value (fixed effect) and \(\omega\) the standard deviation of the random effects, which is interpreted as the interindividual variability. Note that \(\bar{\psi}_i\) is the median of the distribution (neither the mean, nor the mode).
 Logitnormal distribution in ]0,1[:
In that case, \(h(\psi_i) = log\left(\frac{\psi_i}{1\psi_i}\right)\). A random variable \(\psi_i\) with a logitnormal distribution takes its values in ]0,1[. The logit of \(\psi_i\) is normally distributed, i.e.,
\(\text{logit}(\psi_i) = \log \left(\frac{\psi_i}{1\psi_i}\right) \ \sim \ \ {\cal N}( \text{logit}(\bar{\psi}_i), \omega^2) ~~\Leftrightarrow~~ \text{logit}(\psi_i) = \text{logit}(\bar{\psi}_i) + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2)\)
Note that:
\( m = \text{logit}(\psi_i) = \log \left(\frac{\psi_i}{1\psi_i}\right) ~~\Leftrightarrow~~ \psi_i = \frac{\exp(m)}{1+\exp(m)} \)
 Generalized logitnormal distribution in ]a,b[:
In that case, \(h(\psi_i) = log\left(\frac{\psi_i – a}{b\psi_i}\right)\). A random variable \(\psi_i\) with a logitnormal distribution takes its values in ]a,b[. The logit of \(\psi_i\) is normally distributed, i.e.,
\(\text{logit}_{(a,b)}(\psi_i) = \log \left(\frac{\psi_i – a}{b\psi_i}\right) \ \sim \ \ {\cal N}( \text{logit}_{(a,b)}(\bar{\psi}_i), \omega^2) ~~\Leftrightarrow~~ \text{logit}_{(a,b)}(\psi_i) = \text{logit}_{(a,b)}(\bar{\psi}_i) + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2)\)
Note that:
\( m = \text{logit}_{(a,b)}(\psi_i) = \log \left(\frac{\psi_i – a}{b\psi_i}\right) ~~\Leftrightarrow~~ \psi_i = \frac{b \exp(m)+a}{1+\exp(m)} \)
 Probitnormal 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 probitnormal 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:
 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 log normal distribution, make sure the associated population parameter is strictly positive.
 When creating a project, the default proposed distribution is lognormal.
 Logit transformations can be generalized to any interval (a,b) by setting \( \psi_{(a,b)} = a + (ba)\psi_{(0,1)}\) where \(\psi_{(0,1)}\) is a random variable that takes values in (0,1) with a logitnormal 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. Examples are shown on this page.
 “Adapted” Logitnormal distribution:
Another interesting possibility is to “extend” the logit distribution to be bounded in [a, b] rather than in [0, 1]. It is possible starting from the 2019 version. For that, set your parameter in a logit normal distribution. The setting button appear next to the distribution.
Clicking on it will allow to define your bounds as in the following figure.
Notice that if your parameter initial value is not in [0, 1], the bounds are automatically adapted and the following warning message is proposed “The initial value of XX is greater than 1: the logit limit is adjusted”
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 lognormally distributed. LOGNORMAL
distribution is then used for these four lognormal distributions in the main Monolix
graphical user interface:
The distribution of the 4 PK parameters defined in the Monolix
GUI 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 lognormal 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 lognormally 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
Dependency can be introduced between individual parameters by supposing that the random effects \(\eta_i\) are not independent. This means considering them to be linearly correlated.
 warfarin_distribution3_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
Defining correlation between random effects in the interface
To introduce correlations between random effects in Monolix, one can define correlation groups. For example, two correlation groups are defined on the interface below, 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 (#2 in that case):
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. When three of more parameters are included in a correlation groups, all pairwise correlations will be estimated. It is not instance not possible to estimate the correlation between \(\eta_{ka,i}\) and \(\eta_{V,i}\) and between \(\eta_{Cl,i}\) and \(\eta_{V,i}\) but not between \(\eta_{Cl,i}\) and \(\eta_{ka,i}\).
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).
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 interoccasion variability, you can define the correlation group for each level of variability independently.
 The initial value for the correlations is zero and cannot be changed.
 The correlation value cannot be fixed.
Estimated population parameters now include these 2 correlations:
Notice that the high uncertainty on \(\text{corr_ka_Tlag}\) suggests that the correlation between \(\eta_{Tlag,i}\) and \(\eta_{ka,i}\) is not reliable.
How to decide to include correlations between random effects?
The scatterplots of the random effects can hint at correlations to include in the model. This plot represents the joint empirical distributions of each pair of random effects. The regression line (in pink below) and the correlation coefficient (“information” toggle in the settings) permits to visually detect tendencies. If “conditional distribution” (default) is chosen in the display settings, the displayed random effects are calculated using individual parameters sampled from the conditional distribution, which permits to avoid spurious correlations (see the page on shrinkage for more details). If a large correlation is present between a pair of random effects, this correlation can be added to the model in order to be estimated as a population parameter.
Depending on a number of random effects values used to calculate the correlation coefficient, a same correlation value can be more or less significant. To help the user identify significant correlations, Pearson’s correlation tests are performed in the “Result” tab, “Tests” section. If no significant correlation is found, like for the pair \(\eta_{Tlag}\) and \(\eta_{Cl}\) below, the distributions can be assumed to be independent. However, if a significant correlation appears, like for the pair \(\eta_V\) and \(\eta_{Cl}\) below, it can be hypothesized that the distributions are not independent and that the correlation must be included in the model and estimated. Once the correlation is included in the model, the random effects for \(V\) and \(Cl\) are drawn from the joint distribution rather than from two independent distributions.
How do the correlations between random effects affect the individual model?
In this example the model has four parameters Tlag, ka, V and Cl. Without correlation, the individual model is:
\(log(Tlag) = log(Tlag_{pop}) + \eta_{Tlag}\)
\(log(ka) = log(ka_{pop}) + \eta_{ka}\)
\(log(V) = log(V_{pop}) + \eta_V\)
\(log(Cl) = log(Cl_{pop}) + \eta_{Cl}\)
The random effects follow normal distributions: \((\eta_{Tlag,i},\eta_{ka,i},\eta_{V,i},\eta_{Cl,i}) \sim \mathcal{N}(0,\Omega)\)
\(\Omega\) is the variancecovariance matrix defining the distributions of the vectors of random effects, here:
\(\Omega = \begin{pmatrix} \omega_{Tlag}^2 & 0 & 0 & 0 \\ 0 & \omega_{ka}^2 & 0 & 0 \\ 0 & 0 & \omega_V^2 & 0 \\ 0 & 0 & 0 & \omega_{Cl}^2 \end{pmatrix}\)
In this example, two correlations between \(\eta_{Tlag}\) and \(\eta_{ka}\) and between \(\eta_{V}\) and \(\eta_{Cl}\) are added to the model. They are defined with two population parameters called \(\text{corr_Tlag_ka}\) and \(\text{corr_V_Cl}\) that appear in the variancecovariance matrix. So the only difference in the individual model is in \(\Omega\), that is now:
\(\Omega = \begin{pmatrix} \omega_{Tlag}^2 & \omega_{Tlag} \omega_{ka} \text{corr_Tlag_ka} & 0 & 0 \\ \omega_{Tlag} \omega_{ka} \text{corr_Tlag_ka} & \omega_{ka}^2 & 0 & 0 \\ 0 & 0 & \omega_V^2 & \omega_{V} \omega_{Cl} \text{corr_V_Cl} \\ 0 & 0 & \omega_{V} \omega_{Cl} \text{corr_V_Cl} & \omega_{Cl}^2 \end{pmatrix}\)
So the correlation matrix is related to the variancecovariance matrix \(\Omega\) as:
$$\text{corr}(\theta_i,\theta_j)=\frac{\text{covar}(\theta_i,\theta_j)}{\sqrt{\text{var}(\theta_i)}\sqrt{\text{var}(\theta_j)}}$$
Why should the correlation be estimated as part of the population parameters?
The effect of correlations is especially important when simulating parameters from the model. This is the case in the VPC or when simulating new individuals in Simulx to assess the outcome of a different dosing scenario for instance. If in reality individuals with a large distribution volume also have a large clearance (i.e there is a positive correlation between the random effects of the volume and the clearance), but this correlation has not been included in the model, then the concentrations predicted by the model for a new cohort of individuals will display a larger variability than they would in reality.
How do the EBEs change after having included correlation in the model?
Before adding correlation in the model, the EBEs or the individual parameters sampled from the conditional distribution may already be correlated, as can be seen in the “correlation between random effects” plot. This is because the individual parameters (EBEs or sampled) are based on the individual conditional distributions, which takes into account the information given by the data. Especially when the data is rich, the data can indicate that individuals with a large volume of distribution also have a large clearance, even if this correlation is not yet included in the model.
Including the correlation in the model as a population parameter to estimate allows to precisely estimate its value. Usually, one can see a stronger correlation for the corresponding pair of random effects when the correlation is included in the model compared to when it is not. In this example, after including the correlations in the individual model, the joint distribution of \(\eta_{V}\) and \(\eta_{Cl}\) displays a higher correlation coefficient (0.439 compared to 0.375 previously):
 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 \(\eta_{Cl,i}\) :
Parameters without random effects
Adding/removing interindividual variability
By default, all parameters have interindividual variability. To remove it, click on the checkbox of the random effect column:
How the parameters with no variability are estimated is explained here.
Custom parameter distributions
Some datasets may require more complex parameter distributions that those preimplemented in the Monolix GUI. This video shows how to implement a lognormal distribution with BoxCox transformation and how to bound a parameter between two values using a transformed logit distribution (this latter case can be handled automatically from Monolix2019R1).
2.6.3.Model for individual covariates
 Model with continuous covariates
 Model with categorical covariates
 Transforming categorical covariates
 Complex parameter covariate relationships (such as MichaelisMenten or Hill dependencies, timedependent covariates, or covariatedependent standard deviations of random effects)
Objectives: learn how to implement a model for continuous and/or categorical covariates.
Projects: warfarin_covariate1_project, warfarin_covariate2_project, warfarin_covariate3_project, phenobarbital_project
Model with continuous covariates
 warfarin_covariate1_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
The warfarin data contains 2 individual covariates: weight which is a continuous covariate and sex which is a categorical covariate with 2 categories (1=Male, 0=Female). We can ignore these columns if are sure not to use them, or declare them using respectively the reserved keywords CONTINUOUS COVARIATE
and CATEGORICAL COVARIATE
to define continuous and categorical covariate.
Even if these 2 covariates are now available, we can choose to define a model without any covariate by not clicking on any check box in the covariate model.
Here, an unchecked box in the line of the parameter V and the column of the covariate wt means that there is no relationship between weight and volume in the model. A diagnosis plot Individual parameters vs covariates
is generated which displays possible relationships between covariates and individual parameters (even if these covariates are not used in the model):
On the figure, we can see a strong correlation between the volume V and both the weight wt and the sex. One can also see a correlation between the clearance and the weight wt. Therefore, the next step is to add some covariate to our model.
 warfarin_covariate2_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
We decide to use the weight in this project in order to explain part of the variability of \(V_i\) and \(Cl_i\). We will implement the following model for these two parameters:
$$\log(V_i) = \log(V_{\rm pop}) + \beta_V \log(w_i/70) + \eta_{V,i} ~~\text{and}~~\log(Cl_i) = \log(Cl_{\rm pop}) + \beta_{Cl} \log(w_i/70) + \eta_{Cl,i}$$
which means that population parameters of the PK parameters are defined for a typical individual of the population with weight = 70kg.
More details about the model
The model for \(V_{i}\) and \(Cl_{i}\) can be equivalently written as follows:
$$ V_i = V_{\rm pop} ( w_i/70 )^{\beta_V} e^{ \eta_{V,i} } ~~\text{and}~~ Cl_i = Cl_{\rm pop} ( w_i/70 )^{\beta_{Cl}} e^{ \eta_{Cl,i} }$$
The individual predicted values for \(V_i\) and \(Cl_i\) are therefore
$$\bar{V}_i = V_{\rm pop} \left( w_i/70 \right)^{\beta_V} ~~\text{and}~~ \bar{Cl}_i = Cl_{\rm pop} \left( w_i/70 \right)^{\beta_{Cl}} $$
and the statistical model describes how \(V_i\) and \(Cl_i\) are distributed around these predicted values:
$$ \log(V_i) \sim {\cal N}( \log(\bar{V}_i) , \omega^2_V) ~~\text{and}~~\log(Cl_i) \sim {\cal N}( \log(\bar{Cl}_i) , \omega^2_{Cl}) $$
Here, \(\log(V_i)\) and \(\log(Cl_i)\) are linear functions of \(\log(w_i/70)\): we then need to transform first the original covariate \(w_i\) into \(\log(w_i/70)\) by clicking on the button CONTINUOUS
next to ADD COVARIATE (blue button). Then, the following pop up arises
You have to
 define the name of the covariate you want to add (the blue frame).
 define the associated equation (the green frame).
 click on the ACCEPT button
Remarks
 You can define any formula for your covariate as long as you use mathematical functions available in the Mlxtran language.
 You can use any covariate available in the list of covariates proposed in the window. Thus, if you have a Height and Weight as covariates, you can directly compute the Body Mass Index.
 If you go over a covariate with your mouse, all the information (min, mean, median, max and weighted mean) are displayed as a tooltip. The weighted mean is defined as \[ \text{weighted mean} (cov) = \exp \Big( \sum_i \frac{\text{nbObs}_i }{\text{nbObs}} \log(cov_i) \Big)\] where \(\text{nbObs}_i\) is the number of observation for the \(i^{th}\) individual and \(\text{nbObs}\) is the total number of observations.
 If you click on the covariate name, it will be written in the formula.
 You can use this formula box to replace missing continuous covariate values by an imputed value. This is explained in the feature of the week #141 below. For example, if your continuous covariate takes only positive values, you can use a negative value for the missing values in your dataset, for example 99, and enter the following formula: max(COV,0) + min(COV,0)/COV*ImputedValue with the desired ImputedValue (COV is thAse name of your covariate).
We then define a new covariate model, where \(\log(V_i)\) and \(\log(Cl_i)\) are linear functions of the transformed weight \(lw70_i\) as shown on the following figure:
Notice that by clicking on the button FORMULA, you have the display of all the individual model equations. Coefficients \(\beta_{V}\) and \(\beta_{Cl}\) are now estimated with their s.e. and the pvalues of the Wald tests are derived to test if these coefficients are different from 0.
Again, a diagnosis plot Individual parameters vs covariates
is generated which displays possible relationships between covariates and individual parameters (even if these covariates are not used in the model) as one can see on the figure below on the left. However, as there are covariates on the model, what is interesting is to see if there still are correlation between the random effects and the covariates as one can see on the figure below on the right.
Note: To make it automatically, starting from the 2019 version, there is an arrow next (in purple in the next figure) to the continuous covariate from the data set and propose to add a log transformed covariate centered by the weighted mean.
Model with categorical covariates
 warfarin_covariate3_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
We use sex instead of weight in this project, assuming different population values of volume and clearance for males and females. More precisely, we consider the following model for \(V_i\) and \(Cl_i\):
$$\log(V_i) = \log(V_{\rm pop}) + \beta_V 1_{sex_i=F} + \eta_{V,i}~~\text{and}~~\log(Cl_i) = \log(Cl_{\rm pop}) + \beta_{Cl} 1_{sex_i=F} + \eta_{Cl,i}$$
where \(1_{sex_i=F} =1\) if individual i is a female and 0 otherwise. Then, \(V_{\rm pop}\) and \(Cl_{\rm pop}\) are the population volume and clearance for males while \(V_{\rm pop}, e^{\beta_V}\) and \(Cl_{\rm pop} e^{\beta_{Cl}}\) are the population volume and clearance for females. By clicking on the purple button DISCRETE
, the following window pops up
You have to
 define the name of the covariate you want to add (the blue frame).
 define the associated categories (the green frame).
 click on the ALLOCATE button to define all the categories.
Then, you can
 define the name of the categories (the blue frame).
 define the reference category (the green frame).
 click on ACCEPT
Then, define the covariate model in the main GUI:
Estimated population parameters, including the coefficients \(\beta_V\) and \(\beta_{Cl}\) are displayed with the results:
We can display the probability distribution functions of the 4 PK parameters using the Individual parameter
graphic:
Notice that for the volume and the clearance, the theoretical curve is is not the PDF of a lognormal distribution, due to the impact of the covariate sex.
Calculating the typical value for each category
Cl_pop represents the typical value for the reference category (in the example above SEX=0). The typical value for the other categories can be calculated based on the estimated beta parameters:
 normal distribution: \( Cl_{SEX=1} = Cl_{pop} + beta\_Cl\_SEX\_1 \)
 lognormal distribution: \( Cl_{SEX=1} = Cl_{pop} \times e^{beta\_Cl\_SEX\_1 } \)
 logit distribution: \( F_{SEX=1} = \frac{1}{1+ e^{\left( \log \left(\frac{F_{pop}}{1F_{pop}} \right) + beta\_F\_SEX\_1 \right) }} \)
Transforming categorical covariates
 phenobarbital_project (data = ‘phenobarbital_data.txt’, model = ‘lib:bolus_1cpt_Vk.txt’)
The phenobarbital data contains 2 covariates: the weight and the APGAR score which is considered as a categorical covariate. Instead of using the 10 original levels of the APGAR score, we will transform this categorical covariate and create 3 categories: Low = {1,2,3}, Medium = {4, 5, 6, 7} and High={8,9,10}.
If we assume, for instance that the volume is related to the APGAR score, then \(\beta_{V,Low}\) and \(\beta_{V,High}\) are estimated (assuming that Medium is the reference level).
In that case, one can see that both pvalues concerning the transformed APGAR covariate are over .05.
2.6.4.Inter occasion variability (IOV)
 Introduction
 Occasion definition in a data set
 Cross over study
 Occasions with washout
 Occasions without washout
 Multiple levels of occasions
Objectives: learn how to take into account inter occasion variability (IOV).
Projects: iov1_project, iov1_Evid_project, iov2_project, iov3_project, iov4_project
Introduction
A simple model consists of splitting the study into K time periods or occasions and assuming that individual parameters can vary from occasion to occasion but remain constant within occasions. Then, we can try to explain part of the intraindividual variability of the individual parameters by piecewiseconstant covariates, i.e., occasiondependent or occasionvarying (varying from occasion to occasion and constant within an occasion) ones. The remaining part must then be described by random effects. We will need some additional notation to describe this new statistical model. Let
 \(\psi_{ik}\) be the vector of individual parameters of individual i for occasion k, where \(1\leq i \leq N\) and \(1\leq k \leq K\).
 \({c}_{ik}\) be the vector of covariates of individual i for occasion k. Some of these covariates remain constant (gender, group treatment, ethnicity, etc.) and others can vary (weight, treatment, etc.).
Let \(\psi_i = (\psi_{i1}, \psi_{i2}, \ldots , \psi_{iK})\) be the sequence of K individual parameters for individual i. We also need to define:
 \(\eta_i^{(0)}\), the vector of random effects which describes the random interindividual variability of the individual parameters,
 \(\eta_{ik}^{(1)}\), the vector of random effects which describes the random intraindividual variability of the individual parameters in occasion k, for each \(1\leq k \leq K\).
Here and in the following, the superscript (0) is used to represent interindividual variability, i.e., variability at the individual level, while superscript (1) represents interoccasion variability, i.e., variability at the occasion level for each individual. The model now combines these two sequences of random effects:
\(h(\psi_{ik}) = h(\psi_{\rm pop})+ \beta(c_{ik} – c_{\rm pop}) + \eta_i^{(0)} + \eta_{ik}^{(1)} \)
Remark: Individuals do not need to share the same sequence of occasions: the number of occasions and the times defining the occasions can differ from one individual to another.
Occasion definition in a data set
There are two ways to define occasions in a data set:
 Explicitly using an OCCASION column. It is possible to have, in a data set, one or several columns with the columntype OCCASION. It corresponds to the same subject (ID should remain the same) but under different circumstances, occasions. For example, if the same subject has two successive different treatments, it should be considered as the same subject with two occasions. The OCC columns can contain only integers.
 Implicitly using EVID column. If there is an EVID column with a value 4 then Monolix defines a washout and creates an occasion. Thus, if there are several times where EVID equals 4 for a subject, it will create the same number of occasions. Notice that if EVID equals 4 happens only once at the beginning, only one occasion will be defined and no inter occasion variability would be possible.
There are three kinds of occasions
 Cross over study: In that case, data is collected for each patient during two independent treatment periods of time, there is an overlap on the time definition of the periods. A column OCCASION can be used to identify the period. An alternative way is to define an EVID column starting for all occasions with EVID equals 4. Both types of definition will be presented in the iov1 example.
 Occasions with washout: In that case, data is collected for each patient during one period and there is no overlap between the periods. The time is increasing but the dynamical system (i.e. the compartments) is reset when the second period starts. In particular, EVID=4 indicates that the system is reset (washout) for example, when a new dose is administrated.
 Occasions without washout: In that case, data is collected for each patient during one period and there is no overlap between the periods. The time is increasing and we want to differentiate periods in terms of occasions without any reset of the dynamical system. Multiple doses are administrated to each patient. each period of time between successive doses is defined as a statistical occasion. A column OCCASION is therefore necessary in the data file to define it.
Cross over study
 iov1_project (data = ‘iov1_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
In this example, PK data is collected for each patient during two independent treatment periods of time (each one starting at time 0). A column OCCASION is used to identify the study:
This column is defined using the reserved keyword OCCASION. Then, the model associated to the individual parameter is as presented below
First, to define the variability of each parameter on each level, you just have to go on the good level, and you’ll see the associated random effects on each level. On the figure above, we see that all parameters have variability on the ID level, which means that all parameters have interindividual variability. On the figure below, we see the OCC level. In the presented case, only the volume V has interstudy variability and thus inter occasion variability. Thus, this is the only one having variability on the occasion level.
In terms of covariates, we then see two parts as displayed below. We see the covariates
 associated to the level ID (in green). It corresponds to all the covariates that are constant for each subject.
 associated to the level OCC (in blue). It corresponds to all the covariates that are constant for each occasion but not on each subject.
In the presented case, the treatment TRT varies for each individual. It contains interoccasion information and is thus displayed with the occasion level. On the other hand, the SEX is constant for each subject. It contains then interindividual information but no interoccasion information. It is then displayed with the ID level.
What is the impact?
Covariates can be associated to the parameter if and only if their level of variability is coherent with the level of variability of the parameter. In the presented case,
 TRT has interoccasion variability. It can only be used with the parameter V that has interoccasion variability. The two other parameters have only interindividual variability and can therefore not use this TRT information. The interface is greyed and the user can not add this covariate to the parameters ka and Cl.
 SEX has only interindividual variability. It can therefore be associated to any parameter that has interindividual variability.
The population parameters now include the standard deviations of the random effects for the 2 levels of variability (omega is used fo IIV and gamma for IOV):
Two important features are proposed in the plots. Firstly, in the individual fits, you can split or merge the occasions. When split is done, the name of the subjectoccasion is the name of the subject, #, and the name of the occasion.
Secondly, you can use the occasion to split the plots
 iov1_Evid_project (data = ‘iov1_Evid_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
Another way to describe this cross over study is to use EVID=4 as explained in the data set definition. In that example, the EVID creates a washout and another occasion.
Occasions with washout
 iov2_project (data = ‘iov2_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
The time is increasing in this example, but the dynamical system (i.e. the compartments) is reset when the second period starts. Column EVID provides some information about events concerning dose administration. In particular, EVID=4 indicates that the system is reset (washout) when a new dose is administrated
Monolix
automatically proposes to define the treatment periods (between successive resetting) as statistical occasions and introduce IOV, as we did in the previous example. We can display the individual fit by splitting each occasion for each individual
Or by merging the different occasions in a unique plot for each individual:
Remark: If you are modeling a PK as in this example, the washout implies that the occasions are independent. Thus, the cpu time is much faster as we do not have to compute predictions between occasions.
Occasions without washout
 iov3_project (data = ‘iov3_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
Multiple doses are administrated to each patient. We consider each period of time between successive doses as a statistical occasion. A column OCCASION is therefore necessary in the data file.
We can color the observed data by their occasion to have a better representation
The model for IIV and IOV can then be defined as usual. The plot of individual fits allows us to check that the predicted concentration is now continuous over the different occasions for each individual:
Multiple levels of occasions
 iov4_project (data = ‘iov4_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)
We can easily extend such an approach to multiple levels of variability. In this example, columns P1 and P2 define embedded occasions. They are both defined as occasions:
We then define a statistical model for each level of variability.
2.6.5.Mixture of distributions
 Introduction
 Mixture of distributions based on a categorical covariate
 Mixture of distributions based on unsupervised classification with a latent covariate
Objectives: learn how to implement a mixture of distributions for the individual parameters.
Projects: PKgroup_project, PKmixt_project
Introduction
Mixed effects models allow us to take into account betweensubject variability.
One complicating factor arises when data is obtained from a population with some underlying heterogeneity. If we assume that the population consists of several homogeneous subpopulations, a straightforward extension of mixed effects models is a finite mixture of mixed effects models.
There are two approaches to define a mixture of models:
 defining a mixture of structural models (via a regressor or via the bsmm function) –> click here to go to the page dedicated to this approach,
 introducing a categorical covariate (known or latent). This approach is detailed here.
The second approach assumes that the probability distribution of some individual parameters vary from one subpopulation to another one. The introduction of a categorical covariate (e.g., sex, phenotype, treatment, status, etc.) into such a model already supposes that the whole population can be decomposed into subpopulations. The covariate then serves as a label for assigning each individual to a subpopulation.
In practice, the covariate can either be known or not. If it is unknown, the covariate is called a latent covariate and is defined as a random variable with a userdefined number of modalities in the statistical model. Differences in estimation and diagnosis methods appear to deal with this additional random variable: this difference represents a task of unsupervised classification.
Mixture models usually refer to models for which the categorical covariate is unknown and unsupervised classification is needed.
For the sake of simplicity, we will consider a basic model that involves individual parameters \((\psi_i,1\leq i \leq N)\) and observations \((y_{ij}, i \leq N, 1\leq j \leq n_i)\). Then, the easiest way to model a finite mixture model is to introduce a label sequence \((z_i , 1\leq i \leq N)\) that takes its values in \(\{1,2,\ldots,M\}\) such that \(z_i=m\) if subject i belongs to subpopulation m.
In some situations, the label sequence \((z_i , 1\leq i \leq N)\) is known and can be used as a categorical covariate in the model. If \((z_i)\) is unknown, it can be modeled as a set of independent random variables taking their values in \(\{1,2,\ldots,M\}\) where for \(i=1,2,\ldots, N\), \(P(z_i = m)\) is the probability that individual i belongs to group m. We will assume furthermore that the \((z_i)\) are identically distributed, i.e., \(P(z_i = m)\) does not depend on i for \(m=1, \ldots, M\).
Mixture of distributions based on a categorical covariate
 PKgroup_project (data = ‘PKmixt_data.txt’, model = ‘lib:oral1_1cpt_kaVCl.txt’)
The sequence of labels is known as GROUP in this project and comes from the dataset. It is therefore defined as a categorical covariate that classifies We can then assume, for instance different population values for the volume in the two groups and estimate the population parameters using this covariate model.
Then, this covariate GROUP can be used as a stratification variable and is very important in the modeling.
Mixture of distributions based on unsupervised classification with a latent covariate
A latent covariate is defined as a random variable, and the probability of each modality is part of the statistical model and is estimated as well. Methods for estimation and diagnosis are different. After the estimation, for each individual the categorical covariate is not perfectly known, only the probabilities of each modality are estimated.
Note also that latent covariates can be useful to model statistical mixtures of populations, but they provide no biological interpretation for the cause of the heterogeneity in the population since they do not come from the dataset.
Latent covariates can not be handled with IOV.
 PKmixt_project (data = ‘PKmixt_data.txt’, model = ‘lib:oral1_1cpt_kaVCl.txt’)
We will use the same data with this project but ignoring the column GROUP (which is equivalent to assuming that the label is unknown). If we suspect some heterogeneity in the population, we can introduce a “latent covariate” by clicking on the grey button MIXTURE.
It is possible to change the name and the number of modalities of this latent covariate.
Remark: several latent covariates can be introduced in the model, with different number of categories.
We can then use this latent covariate lcat as any observed categorical covariate. Again, we can assume again different population values for the volume in the two groups by applying it on the volume random effect and estimating the population parameters using this covariate model. Proportions of each group are also estimated, plcat_1 which is the probability to have modality 1:
Once the population parameters are estimated, the sequence of latent covariates, i.e. the group to which belongs each subject, can be estimated together with the individual parameters, as the modes of the conditional distributions.
The sequence of estimated latent covariates lcat
can be used as a stratification variable. We can for example display the VPC in the 2 groups:
By plotting the distribution of the individual parameters, we see that V has a bimodal distribution
2.7.Pharmacokinetic models
2.7.1.PK model: single route of administration
 Introduction
 Intravenous bolus injection
 Intravenous infusion
 Oral administration
 Using different parametrizations
Objectives: learn how to define and use a PK model for single route of administration.
Projects: bolusLinear_project, bolusMM_project, bolusMixed_project, infusion_project, oral1_project, oral0_project, sequentialOral0Oral1_project, simultaneousOral0Oral1_project, oralAlpha_project, oralTransitComp_project
Introduction
Once a drug is administered, we usually describe subsequent processes within the organism by the pharmacokinetics (PK) process known as ADME: absorption, distribution, metabolism, excretion. A PK model is a dynamical system mathematically represented by a system of ordinary differential equations (ODEs) which describes transfers between compartments and elimination from the central compartment.
See this web animation for more details.
Mlxtran
is remarkably efficient for implementing simple and complex PK models:
 The function
pkmodel
can be used for standard PK models. The model is defined according to the provided set of named arguments. Thepkmodel
function enables different parametrizations, different models of absorption, distribution and elimination, defined here and summarized in the following..  PK macros define the different components of a compartmental model. Combining such PK components provide a high degree of flexibility for complex PK models. They can also extend a custom ODE system.
 A system of ordinary differential equations (ODEs) can be implemented very easily.
It is also important to highlight the fact that the data file used by Monolix
for PK modelling only contains information about dosing, i.e. how and when the drug is administrated. There is no need to integrate in the data file any information related to the PK model. This is an important remark since it means that any (complex) PK model can be used with the same data file. In particular, we make a clear distinction between administration (related to the data) and absorption (related to the model).
The pkmodel function
The PK model is defined by the names of the input parameters of the pkmodel
function. These names are reserved keywords.
Absorption
 p: Fraction of dose which is absorbed
 ka: absorption constant rate (first order absorption)
 or, Tk0: absorption duration (zero order absorption)
 Tlag: lag time before absorption
 or, Mtt, Ktr: mean transit time & transit rate constant
Distribution
 V: Volume of distribution of the central compartment
 k12, k21: Transfer rate constants between compartments 1 (central) & 2 (peripheral)
 or V2, Q2: Volume of compartment 2 (peripheral) & inter compartment clearance, between compartments 1 and 2,
 k13, k31: Transfer rate constants between compartments 1 (central) & 3 (peripheral)
 or V3, Q3: Volume of compartment 3 (peripheral) & inter compartment clearance, between compartments 1 and 3.
Elimination
 k: Elimination rate constant
 or Cl: Clearance
 Vm, Km: Michaelis Menten elimination parameters
Effect compartment
 ke0: Effect compartment transfer rate constant
Intravenous bolus injection
Linear elimination
 bolusLinear_project
A single iv bolus is administered at time 0 to each patient. The data file bolus1_data.txt contains 4 columns: id, time, amt (the amount of drug in mg) and y (the measured concentration). The names of these columns are recognized as keywords by Monolix
:
It is important to note that, in this data file, a row contains either some information about the dose (in which case y = ".") or a measurement (in which case amt = "."). We could equivalently use the data file bolus2_data.txt which contains 2 additional columns: EVID (in the green frame) and IGNORED OBSERVATION (in the blue frame):
Here, the EVENT ID column allows the identification of an event. It is an integer between 0 and 4. It helps to define the type of line. EVID=1 means that this record describes a dose while EVID=0 means that this record contains an observed value.
On the other hand, the IGNORED OBSERVATION column enables to tag lines for which the information in the OBSERVATION columntype is missing. MDV=1 means that the observed value of this record should be ignored while MDV=0 means that this record contains an observed value. The two data files bolus1_data.txt and bolus2_data.txt contain exactly the same information and provide exactly the same results. A one compartment model with linear elimination is used with this project:
$$\begin{array}{ccl} \frac{dA_c}{dt} &=& – k A_c(t) \\ A_c(t) &= &0 ~~\text{for}~~ t<0 \end{array} $$
Here, \(A_c(t)\) and \(C_c(t)=A_c(t)/V\) are, respectively, the amount and the concentration of drug in the central compartment at time t. When a dose D arrives in the central compartment at time \(\tau\), an iv bolus administration assumes that
$$A_c(\tau^+) = A_c(\tau^) + D$$
where \(A_c(\tau^)\) (resp. \(A_c(\tau^+)\)) is the amount of drug in the central compartment just before (resp. after) \(\tau\) Parameters of this model are V and k. We therefore use the model bolus_1cpt_Vk from the Monolix
PK library:
[LONGITUDINAL] input = {V, k} EQUATION: Cc = pkmodel(V, k) OUTPUT: output = Cc
We could equivalently use the model bolusLinearMacro.txt (click on the button Model
and select the new PK model in the library 6.PK_models/model)
[LONGITUDINAL] input = {V, k} PK: compartment(cmt=1, amount=Ac) iv(cmt=1) elimination(cmt=1, k) Cc = Ac/V OUTPUT: output = Cc
These two implementations generate exactly the same C++ code and then provide exactly the same results. Here, the ODE system is linear and Monolix
uses its analytical solution. Of course, it is also possible (but not recommended with this model) to use the ODE based PK model bolusLinearODE.txt :
[LONGITUDINAL] input = {V, k} PK: depot(target = Ac) EQUATION: ddt_Ac =  k*Ac Cc = Ac/V OUTPUT: output = Cc
Results obtained with this model are slightly different from the ones obtained with the previous implementations since a numeric scheme is used here for solving the ODE. Moreover, the computation time is longer (between 3 and 4 time longer in that case) when using the ODE compared to the analytical solution.
Individual fits obtained with this model look nice
but the VPC show some misspecification in the elimination process:
Michaelis Menten elimination
 bolusMM_project
A non linear elimination is used with this project:
$$\frac{dA_c}{dt} = – \frac{ V_m \, A_c(t)}{V\, K_m + A_c(t) }$$
This model is available in the Monolix
PK library as bolus_1cpt_VVmKm:
[LONGITUDINAL] input = {V, Vm, Km} PK: Cc = pkmodel(V, Vm, Km) OUTPUT: output = Cc
Instead of this model, we could equivalently use PK macros with bolusNonLinearMacro.txt from the library 6.PK_models/model:
[LONGITUDINAL] input = {V, Vm, Km} PK: compartment(cmt=1, amount=Ac, volume=V) iv(cmt=1) elimination(cmt=1, Vm, Km) Cc = Ac/V OUTPUT: output = Cc
or an ODE with bolusNonLinearODE:
[LONGITUDINAL] input = {V, Vm, Km} PK: depot(target = Ac) EQUATION: ddt_Ac = Vm*Ac/(V*Km+Ac) Cc=Ac/V OUTPUT: output = Cc
Results obtained with these three implementations are identical since no analytical solution is available for this non linear ODE. We can then check that this PK model seems to describe much better the elimination process of the data:
Mixed elimination
 bolusMixed_project
THe Monolix
PK library contains “standard” PK models. More complex models should be implemented by the user in a model file. For instance, we assume in this project that the elimination process is a combination of linear and nonlinear elimination processes:
$$ \frac{dA_c}{dt} = \frac{ V_m A_c(t)}{V K_m + A_c(t) } – k A_c(t) $$
This model is not available in the Monolix
PK library. It is implemented in bolusMixed.txt:
[LONGITUDINAL] input = {V, k, Vm, Km} PK: depot(target = Ac) EQUATION: ddt_Ac = Vm*Ac/(V*Km+Ac)  k*Ac Cc=Ac/V OUTPUT: output = Cc
This model, with a combined error model, seems to describe very well the data:
Intravenous infusion
 infusion_project
Intravenous infusion assumes that the drug is administrated intravenously with a constant rate (infusion rate), during a given time (infusion time). Since the amount is the product of infusion rate and infusion time, an additional column INFUSION RATE or INFUSION DURATION is required in the data file: Monolix
can use both indifferently. Data file infusion_rate_data.txt has an additional column rate:
It can be replaced by infusion_tinf_data.txt which contains exactly the same information:
We use with this project a 2 compartment model with non linear elimination and parameters , , , , :
$$\begin{aligned} k_{12} &= Q/V_1 \\ k_{21} &= Q/V_2 \\\frac{dA_c}{dt} & = k_{21} \, Ap(t) – k_{12} \, Ac(t) \frac{ V_m \, A_c(t)}{V_1\, K_m + A_c(t) } \\ \frac{dA_p}{dt} & = – k_{21} \, Ap(t) + k_{12} \, Ac(t) \\ Cc(t) &= \frac{Ac(t)}{V_1} \end{aligned}$$
This model is available in the Monolix
PK library as infusion_2cpt_V1QV2VmKm:
[LONGITUDINAL] input = {V1, Q, V2, Vm, Km} PK: V = V1 k12 = Q/V1 k21 = Q/V2 Cc = pkmodel(V, k12, k21, Vm, Km) OUTPUT: output = Cc
Oral administration
firstorder absorption
 oral1_project
This project uses the data file oral_data.txt. For each patient, information about dosing is the time of administration and the amount. A one compartment model with first order absorption and linear elimination is used with this project. Parameters of the model are ka, V and Cl. we will then use model oral1_kaVCl.txt from the Monolix
PK library
[LONGITUDINAL] input = {ka, V, Cl} EQUATION: Cc = pkmodel(ka, V, Cl) OUTPUT: output = Cc
Both the individual fits and the VPCs show that this model doesn’t describe the absorption process properly.
Many options for implementing this PK model with Mlxtran
exists:
– using PK macros: oralMacro.txt:
[LONGITUDINAL] input = {ka, V, Cl} PK: compartment(cmt=1, amount=Ac) oral(cmt=1, ka) elimination(cmt=1, k=Cl/V) Cc=Ac/V OUTPUT: output = Cc
– using a system of two ODEs as in oralODEb.txt:
[LONGITUDINAL] input = {ka, V, Cl} PK: depot(target=Ad) EQUATION: k = Cl/V ddt_Ad = ka*Ad ddt_Ac = ka*Ad  k*Ac Cc = Ac/V OUTPUT: output = Cc
– combining PK macros and ODE as in oralMacroODE.txt (macros are used for the absorption and ODE for the elimination):
[LONGITUDINAL] input = {ka, V, Cl} PK: compartment(cmt=1, amount=Ac) oral(cmt=1, ka) EQUATION: k = Cl/V ddt_Ac =  k*Ac Cc = Ac/V OUTPUT: output = Cc
– or equivalently, as in oralODEa.txt:
[LONGITUDINAL] input = {ka, V, Cl} PK: depot(target=Ac, ka) EQUATION: k = Cl/V ddt_Ac =  k*Ac Cc = Ac/V< OUTPUT: output = Cc
Remark: Models using the pkmodel function or PK macros only use an analytical solution of the ODE system.
zeroorder absorption
 oral0_project
A one compartment model with zero order absorption and linear elimination is used to fit the same PK data with this project. Parameters of the model are Tk0, V and Cl. We will then use model oral0_1cpt_Tk0Vk.txt from the Monolix
PK library
[LONGITUDINAL] input = {Tk0, V, Cl} EQUATION: Cc = pkmodel(Tk0, V, Cl) OUTPUT: output = Cc
Remark 1: implementing a zeroorder absorption process using ODEs is not easy… on the other hand, it becomes extremely easy to implement using either the pkmodel function or the PK macro oral(Tk0).
Remark 2: The duration of a zeroorder absorption has nothing to do with an infusion time: it is a parameter of the PK model (exactly as the absorption rate constant ka for instance), it is not part of the data.
sequential zeroorder firstorder absorption
 sequentialOral0Oral1_project
More complex PK models can be implemented using Mlxtran
. A sequential zeroorder firstorder absorption process assumes that a fraction Fr of the dose is first absorbed during a time Tk0 with a zeroorder process, then, the remaining fraction is absorbed with a firstorder process. This model is implemented in sequentialOral0Oral1.txt using PK macros:
[LONGITUDINAL] input = {Fr, Tk0, ka, V, Cl} PK: compartment(amount=Ac) absorption(Tk0, p=Fr) absorption(ka, Tlag=Tk0, p=1Fr) elimination(k=Cl/V) Cc=Ac/V OUTPUT: output = Cc
Both the individual fits and the VPCs show that this PK model describes very well the whole ADME process for the same PK data:
simultaneous zeroorder firstorder absorption
 simultaneousOral0Oral1_project
A simultaneous zeroorder firstorder absorption process assumes that a fraction Fr of the dose is absorbed with a zeroorder process while the remaining fraction is absorbed simultaneously with a firstorder process. This model is implemented in simultaneousOral0Oral1.txt using PK macros:
[LONGITUDINAL] input = {Fr, Tk0, ka, V, Cl} PK: compartment(amount=Ac) absorption(Tk0, p=Fr) absorption(ka, p=1Fr) elimination(k=Cl/V) Cc=Ac/V OUTPUT: output = Cc
alphaorder absorption
 oralAlpha_project
An order absorption process assumes that the rate of absorption is proportional to some power of the amount of drug in the depot compartment:
This model is implemented in oralAlpha.txt using ODEs:
[LONGITUDINAL] input = {r, alpha, V, Cl} PK: depot(target = Ad) EQUATION: dAd = Ad^alpha ddt_Ad = r*dAd ddt_Ac = r*Ad  (Cl/V)*Ac Cc = Ac/V OUTPUT: output = Cc
transit compartment model
 oralTransitComp_project
A PK model with a transit compartment of transit rate Ktr and mean transit time Mtt can be implemented using the PK macro oral(ka, Mtt, Ktr), or using the pkmodel function, as in oralTransitComp.txt:
[LONGITUDINAL] input = {Mtt, Ktr, ka, V, Cl} EQUATION: Cc = pkmodel(Mtt, Ktr, ka, V, Cl) OUTPUT: output = Cc
Using different parametrizations
The PK macros and the function pkmodel use some preferred parametrizations and some reserved names as input arguments: Tlag, ka, Tk0, V, Cl, k12, k21. It is however possible to use another parametrization and/or other parameter names. As an example, consider a 2compartment model for oral administration with a lag, a first order absorption and a linear elimination. We can use the pkmodel function with, for instance, parameters ka, V, k, k12 and k21:
[LONGITUDINAL] input = {ka, V, k, k12, k21} PK: Cc = pkmodel(ka, V, k, k12, k21) OUTPUT: output = Cc
Imagine now that we want i) to use the clearance instead of the elimination rate constant , ii) to use capital letters for the parameter names. We can still use the pkmodel function as follows:
[LONGITUDINAL] input = {KA, V, CL, K12, K21} PK: Cc = pkmodel(ka=KA, V, k=CL/V, k12=K12, k21=K21) OUTPUT: output = Cc
2.7.2.PK model: multiple routes of administration
Objectives: learn how to define and use a PK model for multiple routes of administration..
Projects: ivOral1_project, ivOral2_project
Some drugs can display complex absorption kinetics. Common examples are mixed firstorder and zeroorder absorptions, either sequentially or simultaneously, and fast and slow parallel firstorder absorptions. A few examples of those kinds of absorption kinetics are proposed below. Various absorption models are proposed here as examples.
Combining iv and oral administrations – Example 1
 ivOral1_project (data = ‘ivOral1_data.txt’ , model = ‘ivOral1Macro_model.txt’)
In this example, we combine oral and iv administrations of the same drug. The data file ivOral1_data.txt contains an additional column ADMINISTRATION ID which indicates the route of administration (1=iv, 2=oral)
We assume here a one compartment model with firstorder absorption process from the depot compartment (oral administration) and a linear elimination process from the central compartment. We further assume that only a fraction F (bioavailability) of the drug orally administered is absorbed. This model is implemented in ivOral1Macro_model.txt
using PK macros:
[LONGITUDINAL] input = {F, ka, V, k} PK: compartment(cmt=1, amount=Ac) iv(adm=1, cmt=1) oral(adm=2, cmt=1, ka, p=F) elimination(cmt=1, k) Cc = Ac/V OUTPUT: output = Cc
A logitnormal distribution is used for bioavability F that takes it values in (0,1). The model properly fits the data as can be seen on the individual fits of the 6 first individuals
Remark: the same PK model could be implemented using ODEs instead of PK macros.
Let \(A_d\) and \(A_c\) be, respectively, the amounts in the depot compartment (gut) and the central compartment (bloodtsream). Kinetics of \(A_d\) and \(A_c\) are described by the following system of ODEs
$$\dot{A}_d(t) = – k_a A_d(t)~~\text{and}~~ \dot{A}_c(t) = k_a A_d(t) – k A_c(t)$$
The target compartment is the depot compartment (\(A_d\)) for oral administrations and the central compartment (\(A_c\)) for iv administrations. This model is implemented in ivOral1ODE_model.txt
using a system of ODEs:
[LONGITUDINAL] input = {F, ka, V, k} PK: depot(type=1, target=Ad, p=F) depot(type=2, target=Ac) EQUATION: ddt_Ad = ka*Ad ddt_Ac = ka*Ad  k*Ac Cc = Ac/V OUTPUT: output = Cc
Solving this ODEs system is less efficient than using the PK macros which uses the analytical solution of the linear system.
Combining iv and oral administrations – Example 2
 ivOral2_project (data = ‘ivOral2_data.txt’ , model = ‘ivOral2Macro_model.txt’)
In this example (based on simulated PK data), we combine intraveinous injection with 3 different types of oral administrations of the same drug. The datafile ivOral2_data.txt contains column ADM which indicates the route of administration (1,2,3=oral, 4=iv). We assume that one type of oral dose (adm=1) is absorbed into a latent compartment following a zeroorder absorption process. The 2 oral doses (adm=2,3) are absorbed into the central compartment following firstorder absorption processes with different rates. Bioavailabilities are supposed to be different for the 3 oral doses. There is linear transfer from the latent to the central compartment. A peripheral compartment is linked to the central compartment. The drug is eliminated by a linear process from the central compartment:
This model is implemented in ivOral2Macro_model.txt
using PK macros:
[LONGITUDINAL] input = {F1, F2, F3, Tk01, ka2, ka3, kl, k23, k32, V, Cl} PK: compartment(cmt=1, amount=Al) compartment(cmt=2, amount=Ac) peripheral(k23,k32) oral(type=1, cmt=1, Tk0=Tk01, p=F1) oral(type=2, cmt=2, ka=ka2, p=F2) oral(type=3, cmt=2, ka=ka3, p=F3) iv(type=4, cmt=2) transfer(from=1, to=2, kt=kl) elimination(cmt=2, k=Cl/V) Cc = Ac/V OUTPUT: output = Cc
Here, logitnormal distributions are used for bioavabilities \(F_1\), \(F_2\) and \(F_3\). The model fits the data properly :
Remark: the number and type of doses vary from one patient to another in this example.
2.7.3.From multiple doses to steadystate
Objectives: learn how to define and use a PK model with multiple doses or assuming steadystate.
Projects: multidose_project, addl_project, ss1_project, ss2_project, ss3_project
Multiple doses
 multidose_project (data = ‘multidose_data.txt’ , model = ‘lib:bolus_1cpt_Vk.txt’)
In this project, each patient receives several iv bolus injections. Each dose is represented by a row in the data file multidose_data.txt:
The PK model and the statistical model used in this project properly fit the observed data of each individual. Even if there is no observation between 12h and 72h, predicted concentrations computed on this time interval exhibit the multiple doses received by each patient:
VPCs, which is a diagnosis tool, are based on the design of the observations and therefore “ignore” what may happen between 12h and 72h:
On the other hand, the prediction distribution, which is not a diagnosis tool, computes the distribution of the predicted concentration at any time point:
Additional doses (ADDL)
 addl_project (data = ‘addl_data.txt’ , model = ‘lib:bolus_1cpt_Vk.txt’)
We can note in the previous project, that, for each patient, the interval time between two successive doses is the same (12 hours for each patient) and the amount of drug which is administrated is always the same as well (40mg for each patient). We can take advantage of this design in order to simplify the data file by defining, for each patient, a unique amount (AMT), the number of additional doses which are administrated after the first one (ADDITIONAL DOSES) and the time interval between successive doses (INTERDOSE INTERVAL):
The keywords ADDL and II are automatically recognized by Monolix.
Remarks:
 Results obtained with this project, i.e. with this data file, are identical to the ones obtained with the previous project.
 It is possible to combine single doses (using ADDL=0) and repeated doses in a same data file.
Steadystate
 ss1_project (data = ‘ss1_data.txt’ , model = ‘lib:oral0_1cpt_Tk0VCl.txt’)
The dose, orally administrated at time 0 to each patient, is assumed to be a “steadystate dose” which means that a “large” number of doses before time 0 have been administrated, with a constant amount and a constant interval dosing, such that steadystate, i.e. equilibrium, is reached at time 0. The data file ss1_data contains a column STEADY STATE which indicates if the dose is a steadystate dose or not and a column INTERDOSE INTERVAL for the interdose interval:
Click on Check the initial fixed effects
to display the predicted concentration between the last dose administrated at time 0. One can see that the initial concentration is not 0 but the result of the steady state calculation.
We see on this plot that Monolix adds 5 doses before the last dose to reach steadystate. Individual fits display the predicted concentrations computed with these additional doses:
If the dynamics is slow, adding 5 doses before the last dose might not be sufficient. You can adapt the number of doses in the frame data and thus define it for all individuals as on the following figure.
leading to the following check initial fixed effects
 ss2_project (data = ‘ss2_data.txt’ , model = ‘lib:oral0_1cpt_Tk0VCl.txt’)
Steadystate and non steadysates doses are combined in this project:
Individual fits display the predicted concentrations computed with this combination of doses:
2.8.Extensions
2.8.1.Using regression variables
 Introduction
 Regressor definition in a data set
 Continuous regression variables
 Categorical regression variables
Objectives: learn how to define and use regression variables (time varying covariates).
Projects: reg1_project, reg2_project
Introduction
A regression variable is a variable x which is a given function of time, which is not defined in the model but which is used in the model. x is only defined at some time points \(t_1, t_2, \ldots, t_m\) (possibly different from the observation time points), but x is a function of time that should be defined for any t (if is used in an ODE for instance, or if a prediction is computed on a fine grid). Then, Mlxtran defines the function x by interpolating the given values \((x_1, x_2, \ldots, x_m)\). In the current version of Mlxtran, interpolation is performed by using the last given value:
\( x(t) = x_j \quad~~\text{for}~~t_j \leq t < t_{j+1} \)
The way to introduce it in the Mlxtran longitudinal model is defined here.
Regressor definition in a data set
It is possible to have in a data set one or several columns with columntype REGRESSOR. Within a given subjectoccasion, string “.” will be interpolated (last value carried forward interpolation is used) for observation and doselines. Lines with no observation and no dose but with regressor values are also taken into account by Monolix for regressor interpolation.
Several points have to be noticed:
 The name of the regressor in the data set and the name of the regressor used in the longitudinal model do not need to be identical.
 If there are several regressors, the mapping will be done by order of definition.
 Regressors can only be used in the longitudinal model.
Continuous regression variables
 reg1_project (data = reg1_data.txt , model=reg1_model.txt)
We consider a basic PD model in this example, where some concentration values are used as a regression variable. The data set is defined as follows
[LONGITUDINAL] input = {Emax, EC50, Cc} Cc = {use=regressor} EQUATION: E = Emax*Cc/(EC50 + Cc) OUTPUT: output = E
As explained in the previous subsection, there is no name correspondance between the regressor in the data set and the regressor in the model file. Thus, in that case, the values of Cc with respect to time will be taken from the y1 column.
In addition, in that case, the predicted effect is therefore piece wise constant because
 the regressor interpolation is performed by using the last given value, and then Cc is piece wise constant.
 The effect model is direct with respect to the concentration.
Thus, it changes at the time points where concentration values are provided:
Categorical regression variables
 reg2_project (data = reg2_data.txt , model=reg2_model.txt)
The variable \(z_{ij}\) takes its values in {1, 2} in this example and represents the state of individual i at time \(t_{ij}\). We then assume that the observed data \(y_{ij}\) has a Poisson distribution with parameter lambda1 if \(z_{ij}=1\) and parameter lambda2 if \(z_{ij}=2\). z is known in this example: it is then defined as a regression variable in the model:
[LONGITUDINAL] input = {lambda1, lambda2, z} z = {use=regressor} EQUATION: if z==0 lambda=lambda1 else lambda=lambda2 end DEFINITION: y = {type=count, log(P(y=k)) = lambda + k*log(lambda)  factln(k) } OUTPUT: output = y
2.8.2.Bayesian estimation
In the tab “Initial estimates”, clicking on the wheel icon next to a population parameters opens a window to choose among three estimation methods (see image below). “Maximum Likelihood Estimation” corresponds to the default method using SAEM, detailed on this page. “Maximum A Posteriori Estimation” corresponds to Bayesian estimation, and “Fixed” to a fixed parameter.
Bayesian estimation
Purpose
Bayesian estimation allows to take into account prior information in the estimation of parameters. It is called in Monolix Maximum A Posteriori estimation, and it corresponds to a penalized maximum likelihood estimation, based on a prior distribution defined for a parameter. The weight of the prior in the estimation is given by the standard deviation of the prior distribution.
Objectives: learn how to combine maximum likelihood estimation and Bayesian estimation of the population parameters.
Projects: theobayes1_project, theobayes2_project,
Introduction
The Bayesian approach considers the vector of population parameters \(\theta\) as a random vector with a prior distribution \(\pi_\theta\). We can then define the *posterior distribution* of \(\theta\):
\(\begin{aligned} p(\theta  y ) &= \frac{\pi_\theta( \theta )p(y  \theta )}{p(y)} \\ &= \frac{\pi_\theta( \theta ) \int p(y,\psi \theta) \, d \psi}{p(y)} . \end{aligned} \)
We can estimate this conditional distribution and derive statistics (posterior mean, standard deviation, quantiles, etc.) and the socalled maximum a posteriori (MAP) estimate of \(\theta\):
\(\begin{aligned} \hat{\theta}^{\rm MAP} &=\text{arg~max}_{\theta} p(\theta  y ) \\ &=\text{arg~max}_{\theta} \left\{ {\cal LL}_y(\theta) + \log( \pi_\theta( \theta ) ) \right\} . \end{aligned} \)
The MAP estimate maximizes a penalized version of the observed likelihood. In other words, MAP estimation is the same as penalized maximum likelihood estimation. Suppose for instance that \(\theta\) is a scalar parameter and the prior is a normal distribution with mean \(\theta_0\) and variance \(\gamma^2\). Then, the MAP estimate is the solution of the following minimization problem:
\(\hat{\theta}^{\rm MAP} =\text{arg~min}_{\theta} \left\{ 2{\cal LL}_y(\theta) + \frac{1}{\gamma^2}(\theta – \theta_0)^2 \right\} .\)
This is a tradeoff between the MLE which minimizes the deviance, \(2{\cal LL}_y(\theta)\), and \(\theta_0\) which minimizes \((\theta – \theta_0)^2\). The weight given to the prior directly depends on the variance of the prior distribution: the smaller \(\gamma^2\) is, the closer to \(\theta_0\) the MAP is. In the limiting case, \(\gamma^2=0\); this means that \(\theta\) is fixed at \(\theta_0\) and no longer needs to be estimated. Both the Bayesian and frequentist approaches have their supporters and detractors. But rather than being dogmatic and following the same rulebook every time, we need to be pragmatic and ask the right methodological questions when confronted with a new problem.
All things considered, the problem comes down to knowing whether the data contains sufficient information to answer a given question, and whether some other information may be available to help answer it. This is the essence of the art of modeling: find the right compromise between the confidence we have in the data and our prior knowledge of the problem. Each problem is different and requires a specific approach. For instance, if all the patients in a clinical trial have essentially the same weight, it is pointless to estimate a relationship between weight and the model’s PK parameters using the trial data. A modeler would be better served trying to use prior information based on physiological knowledge rather than just some statistical criterion.
Generally speaking, if prior information is available it should be used, on the condition of course that it is relevant. For continuous data for example, what does putting a prior on the residual error model’s parameters mean in reality? A reasoned statistical approach consists of including prior information only for certain parameters (those for which we have real prior information) and having confidence in the data for the others. Monolix allows this hybrid approach which reconciles the Bayesian and frequentist approaches. A given parameter can be
 a fixed constant if we have absolute confidence in its value or the data does not allow it to be estimated, essentially due to lack of identifiability.
 estimated by maximum likelihood, either because we have great confidence in the data or no information on the parameter.
 estimated by introducing a prior and calculating the MAP estimate or estimating the posterior distribution.
Computing the Maximum a posteriori (MAP) estimate
 demo project: theobayes1_project (data = ‘theophylline_data.txt’ , model = ‘lib:oral1_1cpt_kaVCl.txt’)
We want to introduce a prior distribution for \(ka_{\rm pop}\) in this example. Click on the option button
and select Maximum A Poteriori Estimation
We propose a typical value, here 2 and standard deviation 0.1 for \(ka_{\rm pop}\) and to compute the MAP estimate for \(ka_{\rm pop}\). The parameter is then colored in purple.
Starting from the 2021 version, it is possible to select maximum a posteriori estimation also for the omega parameters (standard deviations of the random effects). In this case, an inverse Wishart is set as a prior distribution for the omega matrix.
The following distributions for the priors are used:
 typical value (*_pop): the distribution of the prior is the same as the distribution of the parameter. For instance, if ka has been set with a lognormal distribution in the “Statistical model & Tasks” tab, a lognormal distribution is also used for the prior on ka_pop. When a lognormal distribution is used, setting sd=0.1 roughtly corresponds to 10% uncertainty in the provided prior value for ka_pop.
 covariate effects (beta_*): a normal distribution is used, to allow betas to be either positive or negative.
 standard deviations (omega_*) [starting version 2021]: an inverse Wishart distribution is used. Inverse Wisharts are common prior distributions for variancecovariance matrices as they allow to fulfill the positivedefinite matrix requirement. The weight of the prior in the estimation is based on the number of degrees of freedom (df) of the inverse Wishart, instead of a standard deviation. More degrees of freedom correspond to a higher constrain of the prior in the omega estimation. In Monolix, each omega parameter is handled as a 1×1 matrix with its own degree of freedom, independently from the other omegas. The univariate inverse Wishart \( W^{1}(df, (df+2)\omega_{typ}) \) simplifies to an inverse gamma distribution with shape parameter \(\alpha=df/2 \) and scale parameter \(\beta=\omega_{typ}*(df+2)/2\). The coefficient of variation is thus \(CV=\frac{1}{\sqrt{\frac{df}{2}2}}\). To obtain a 20% uncertainty on omega (CV=0.2), the user can set df=50.
 correlations: it is currently not possible to set a prior on the correlation parameters
It is common to set the initial value of the parameter to be the same as the typical value of the prior. Note that the default value for the typical value of the prior is set to the initial value of the parameter. However, if the initial value of the parameter is modified afterwards, the typical value of the prior is not updated automatically.
Fixing the value of a parameter
Population parameters can be fixed to their initial values, in this case they are not estimated. It is possible to fix one, several or all population parameters, among the fixed effects, standard deviations of random effects, and error model parameters. In Monolix2021, it is also possible to fix correlation parameters.
To fix a population parameter, click on the wheel next to the parameter in the tab “Initial estimates” and select “Fixed”, like on the image below:
Fixed parameters appear on this tab in red. In Monolix2021, they are also colored in red in the subtab “Check initial estimates”.
 theobayes2_project (data = ‘theophylline_data.txt’ , model = ‘lib:oral1_1cpt_kaVCl.txt’)
We can combine different strategies for the population parameters: Bayesian estimation for \(ka_{\rm pop}\), fixed value for \(V_{\rm pop}\) and maximum likelihood estimation for \(Cl_{\rm pop}\), for instance.
Remark:
 The parameter \(V_{\rm pop}\) is fixed and then colored in red.
 \(V_{\rm pop}\) is not estimated (it’s s.e. is not computed) but the standard deviation \(\omega_{V}\) is estimated as usual.
2.8.3.Delayed differential equations
 Ordinary differential equations based model
 Don’t forget the initial conditions!
 Delayed differential equations based model
Objectives: learn how to implement a model with ordinary differential equations (ODE) and delayed differential equations (DDE).
Projects: tgi_project, seir_project
Ordinary differential equations based model
 tgi_project (data = tgi_data.txt , model = tgi_model.txt)
Here, we consider the tumor growth inhibition (TGI) model proposed by Ribba et al. (Ribba, B., Kaloshi, G., Peyre, M., Ricard, D., Calvez, V., Tod, M., . & Ducray, F., *A tumor growth inhibition model for lowgrade glioma treated with chemotherapy or radiotherapy*. Clinical Cancer Research, 18(18), 50715080, 2012.). This model is defined by a set of ordinary differential equations
where is the total tumor size. This set of ODEs is valid for t greater than 0, while
This model (derivatives and initial conditions) can easily be implemented with Mlxtran:
DESCRIPTION: Tumor Growth Inhibition (TGI) model proposed by Ribba et al A tumor growth inhibition model for lowgrade glioma treated with chemotherapy or radiotherapy. Clinical Cancer Research, 18(18), 50715080, 2012. Variables  PT: proliferative equiescent tissue  QT: nonproliferative equiescent tissue  QP: damaged quiescent cells  C: concentration of a virtual drug encompassing the 3 chemotherapeutic components of the PCV regimen Parameters  K : maximal tumor size (should be fixed a priori)  KDE : the rate constant for the decay of the PCV concentration in plasma  kPQ : the rate constant for transition from proliferation to quiescence  kQpP : the rate constant for transfer from damaged quiescent tissue to proliferative tissue  lambdaP: the rate constant of growth for the proliferative tissue  gamma : the rate of damages in proliferative and quiescent tissue  deltaQP: the rate constant for elimination of the damaged quiescent tissue  PT0 : initial proliferative equiescent tissue  QT0 : initial nonproliferative equiescent tissue [LONGITUDINAL] input = {K, KDE, kPQ, kQpP, lambdaP, gamma, deltaQP, PT0, QT0} PK: depot(target=C) EQUATION: ; Initial conditions t0 = 0 C_0 = 0 PT_0 = PT0 QT_0 = QT0 QP_0 = 0 ; Dynamical model PSTAR = PT + QT + QP ddt_C = KDE*C ddt_PT = lambdaP*PT*(1PSTAR/K) + kQpP*QP  kPQ*PT  gamma*KDE*PT*C ddt_QT = kPQ*PT  gamma*KDE*QT*C ddt_QP = gamma*KDE*QT*C  kQpP*QP  deltaQP*QP OUTPUT: output = PSTAR
Remark: t0, PT_0 and QT_0 are reserved keywords that define the initial conditions.
Then, the graphic of individual fits clearly shows that the tumor size is constant until and starts changing according to the model at t=0.
Don’t forget the initial conditions!
 tgiNoT0_project (data = tgi_data.txt , model = tgiNoT0_model.txt)
The initial time t0 is not specified in this example. Since t0 is missing, Monolix uses the first time value encountered for each individual. If, for instance, the tumor size has not been computed before 5 for the individual fits, then t0=5 will be used for defining the initial conditions for this individual, which introduces a shift in the plot:
As defined here, the following rule applies
 When no starting time t0 is defined in the Mlxtran model for Monolix then by default t0 is selected to be equal to the first dose or the first observation, whatever comes first.
 If t0 is defined, a differential equation needs to be defined.
Conclusion: don’t forget to properly specify the initial conditions of a system of ODEs!
Delayed differential equations based model
A system of delay differential equations (DDEs) can be implemented in a block EQUATION
of the section [LONGITUDINAL]
of a script Mlxtran. Mlxtran provides the command delay(x,T) where x is a onedimensional component and T is the explicit delay. Therefore, DDEs with a nonconstant past of the form
$$ \begin{array}{ccl} \frac{dx}{dt} &=& f(x(t),x(tT_1), x(tT_2), …), ~~\text{for}~~t \geq 0\ x(t) &=& x_0(t) ~~~~\text{for}~~\text{min}(T_k) \leq t \leq 0 \end{array} $$
can be solved. The syntax and rules are explained here.
 seir_project (data = seir_data.txt , model = seir_model.txt)
The model is a system of 4 DDEs and defined with the following mode:
DESCRIPTION: SEIR model, using delayed differential equations. "An Epidemic Model with RecruitmentDeath Demographics and Discrete Delays", Genik & van den Driessche (1999) Decomposition of the total population into four epidemiological classes S (succeptibles), E (exposed), I (infectious), and R (recovered) The parameters corresponds to  birthRate: the birth rate,  deathRate: the natural death rate,  infectionRate: the contact rate of infective individuals,  recoveryRate: the rate of recovery,  excessDeathRate: the excess death rate for infective individuals There is a time delay in the model:  tauImmunity: a temporary immunity delay [LONGITUDINAL] input = {birthRate, deathRate, infectionRate, recoveryRate, excessDeathRate, tauImmunity, tauLatency} EQUATION: ; Initial conditions t0 = 0 S_0 = 15 E_0 = 0 I_0 = 2 R_0 = 3 ; Dynamical model N = S + E + I + R ddt_S = birthRate  deathRate*S  infectionRate*S*I/N + recoveryRate*delay(I,tauImmunity)*exp(deathRate*tauImmunity) ddt_E = infectionRate*S*I/N  deathRate*E  infectionRate*delay(S,tauLatency)*delay(I,tauLatency)*exp(deathRate*tauLatency)/(delay(I,tauLatency)+delay(S,tauLatency)+delay(E,tauLatency)+delay(R,tauLatency)) ddt_I = (recoveryRate+excessDeathRate+deathRate)*I + infectionRate*delay(S,tauLatency)*delay(I,tauLatency)*exp(deathRate*tauLatency)/(delay(I,tauLatency)+delay(S,tauLatency)+delay(E,tauLatency)+delay(R,tauLatency)) ddt_R = recoveryRate*I  deathRate*R  recoveryRate*delay(I,tauImmunity)*exp(deathRate*tauImmunity) OUTPUT: output = {S, E, I, R}
Introducing these delays allows to obtain nice fits for the 4 outcomes, including (corresponding to the output y4):
Case studies
 8.case_studies/arthritis_project
3.Tasks
Monolix tasks
Monolix allows a workflow with several tasks.
On the interface, one can see six different tasks
 POP. PARAM.: it corresponds to the estimation of the population parameters,
 EBEs: it corresponds to the estimation of the individual parameters using the conditional mode, i.e. the most probable individual parameters.
 CONDITIONAL DISTRIBUTION: It corresponds to the draws individual parameters based on the conditional distribution. It allows to compute the mean value of the conditional distribution.
 STD. ERRORS.: it corresponds to the calculation of the Fisher information matrix and standard errors. Two methods are proposed for it. Either using the linearization method or using the stochastic approximation. The choice between those methods is done with the “Use linearization method” toggle under the tasks.
 LIKELIHOOD: it corresponds to the explicit calculation of the loglikelihood. A specificity of the SAEM algorithm is that it does not explicitly compute the objective function. Thus, a dedicated task is proposed. Two methods are proposed for it. Either using the linearization method or using the importance sampling. The choice between those methods is done with the “Use linearization method” toggle under the tasks. This toogle is for both STD ERRORS and LIKELIHOOD tasks to be more relevant.
 PLOTS: it corresponds to the generation of the plots.
Also, different types of results are available in the form of plots and tables. The tasks can be run individually by clicking on the associated button, or you can define a workflow by clicking on the tasks to run (on the small light blue checks) and click on the play button (in green) as proposed on the figure below.
Notice that you can initialize all the parameters and the associated methods in the “Initial Estimates” frame as described here.
Moreover, Monolix includes a convergence assessment tool. It is possible to execute a workflow as defined above but for different, randomly generated, initial values of fixed effects.
Monolix results
All the output files are detailed here.
MonolixR functions
Monolix is now proposed with an API leading to the possibility to have access to the project exactly by the same way as you would do with the interface. All the functions are described here.
3.1.Initialization
Parameter initial estimates and associated methods
 Initialization of the estimates
 What method can I use for the parameters estimations?
 How to initialize your parameters?
Initial values are specified for the fixed effects, for the standard deviations of the random effects and for the residual error parameters. These initial values are available through the frame “Initial estimates” of the interface as can be seen on the following figure. It is recommended to initialize the estimation to have faster convergence.
Initialization of the estimates
Initialization of the “Fixed effects”
The user can modify all the initial values of the fixed effects. When initializing the project, the values are set by default to 1. To change it, the user can click on the parameter and change the value
Notice that when you click on the parameter, an information is provided to tell what value is possible. The constraint depends on the distribution chosen for the parameter. For exemple, if the volume parameter V is defined as lognormal, its initial value should be strictly greater than 0. In that case, if you set a negative value, an error will be thrown and the previous parameter will be displayed.
When a parameter depends on a covariate, initial values for the dependency (named with \(\beta\) prefix, for instance beta_V_SEX_M to add the dependency of SEX, on parameter V) are displayed. The default initial value is 0. In case of a continuous covariate, the covariate is added linearly to the transformed parameter, with a coefficient \(\beta\). For categorical covariates, the initial value for the reference category will be the one of the fixed effect, while for all other categories it will be the initial value for the fixed effect plus the initial value of the \(\beta\), in the transformed parameter space. It is possible to define different initial values for the nonreference categories. The equations for the parameters can be visualized by clicking on button formula in the “Statistical model & Tasks” frame
Initialization of the “Standard deviation of the random effects”
The user can modify all the initial values of the standard deviations of the random effects. The default value is set to 1. We recommend to keep these values high in order for SAEM to have the possibility to explore the domain.
Initialization of the “Residual error parameters”
The user can modify all the initial values of the residual error parameters. There are as many lines as continuous outputs of the model. The default value depends on the parameter (1 for “a”, 0.3 for “b” and 1 for “c”).
What method can I use for the parameters estimations?
For all the parameters, there are several methods for the estimation
 “Fixed”: the parameter is kept to its initial value and so, it will not be estimated. In that case, the parameter name is set to orange.
 “Maximum Likelihood Estimation”: The parameter is estimated using maximum likelihood. In that case, the parameter name remains grey. This is the default option
 “Maximum A Posteriori Estimation”: The parameter is estimated using maximum a posteriori estimation. In that case, the user has to define both a typical value and a standard deviation. For more about this, see here. In that case, the parameter name is colored in purple.
To change the method, click on the right of the parameter as on the following.
A window pops up to choose the method as on the following figure
Notice that you have buttons to fix all the parameters or estimate all on the top right of the window as can be seen on the following figure
How to initialize your parameters?
On the use of last estimates
If you have already estimated the population parameters for this project, then you can use the “Use the last estimates” buttons to use the previous estimates as initial values. The user has the possibility to use all the last estimates or only the fixed effects. The interest of using only the fixed effects is not to have too low initial standard effects and thus let SAEM explore a larger domain for the next run.
Check initial fixed effects
When clicking on the “Check the initial fixed effects”, the simulations obtained with the initial population fixed effects values are displayed for each individual together with the data points, in case of continuous observations. It allows also an automatic initialization in case of a model of the PK library as described here.
3.2.Population parameter estimation using SAEM
 Purpose
 Calculations: the SAEM algorithm
 Running the population parameter estimation task
 Outputs
 In the graphical user interface
 In the output folder
 Settings
 Good practices, troubleshooting and tips
Purpose
The estimation of the population parameters is the key task in nonlinear mixed effect modeling. In Monolix, it is performed using the Stochastic Approximation ExpectationMaximization (SAEM) algorithm [1]. SAEM has been shown to be extremely efficient for both simple and a wide variety of complex models: categorical data [2], count data [3], timetoevent data [4], mixture models [5–6], differential equation based models, censored data [7], … The convergence of SAEM has been rigorously proven [1] and its implementation in Monolix is particularly efficient. No other algorithms are available in Monolix.
Calculations: the SAEM algorithm
Running the population parameter estimation task
Overview
The popup window which permits to follow the progress of the task is shown below. The algorithm starts with a small number (5 by default) of burnin iterations for initialization which are displayed in the following way: (note that this step can be so fast that it is not visible by the user)
Afterwards, the evolution of the value for each population parameter over the iterations of the algorithm is displayed. The red line marks the switch from the exploratory phase to the smoothing phase. The exact value at each iterations can be followed by hovering over the curve (as for Cl_pop below). The convergence indicator (in purple) helps to detect that convergence has been reached (see below for more details).
Dependencies between tasks
The “Population parameter” estimation task must be launched before running any other task. To skip this task, the user can fix all population parameters. If all population parameters have been set to “fixed”, the estimation will stop after a single iteration and allow the user to continue with the other tasks.
The convergence indicator
The convergence indicator (also sometimes called complete likelihood) is defined as the joint probability distribution of the data and the individual parameters and can be decomposed using Bayes law:
\(\sum_{i=1}^{N_{\text{ind}}}\log\left(p(y_i, \phi_i; \theta)\right)=p(y_i \psi_i; \theta)p(\psi_i; \theta)\)
Those two terms have an analytical expression and are easy to calculate, using as \(\phi_i\) the individual parameters drawn by MCMC for the current iteration of SAEM. This quantity is calculated at each SAEM step and is useful to assess the convergence of the SAEM algorithm.
The convergence indicator aggregates the information from all parameters and can serve to detect if the SAEM algorithm has already converged or not. When the indicator is stable, that is it oscillates around the same value without drifting, then we can be pretty confident that the maximum likelihood has been achieved. The convergence indicator is used, among other measures, in the autostop criteria to switch from the exploratory phase to the smoothing phase.
Note that the likelihood (i.e the objective function) \(\sum_{i=1}^{N_{\text{ind}}}\log\left(p(y_i; \theta)\right)\) cannot be computed in closed form because the individual parameters \(\phi_i\) are unobserved. It requires to integrate over all possible values of the individual parameters. Thus, to estimate the loglikelihood an importance sampling Monte Carlo method is used in a separate task (or an approximation is calculated via linearization of the model).
The simulated annealing
The simulated annealing option (setting enabled by default) permits to keep the explored parameter space large for a longer time (compared to without simulated annealing). This allows to escape local maximums and improve the convergence towards the global maximum.
In practice, the simulated annealing option constrains the variance of the random effects and the residual error parameters to decrease by maximum 5% (by default – the setting “Decreasing rate” can be changed) from one iteration to the next one. As a consequence, the variances decrease more slowly:
The size of the parameter space explored by the SAEM algorithm depends on individual parameters sampled from their conditional distribution via Markov Chain Monte Carlo. If the standard deviation of the conditional distributions is large, the individual parameters sampled at iteration k can be quite far away from those at iteration (k1), meaning a large exploration of the parameter space. The standard deviation of the conditional distribution depends on the standard deviation of the random effects (population parameters ‘omega’). Indeed, the conditional distribution is \(p(\psi_iy_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual \(i\), \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual \(i\). The conditional distribution thus depends on the population parameters, and the larger the population parameters ‘omega’, the larger the standard deviation of the conditional distribution. That’s why we want to keep large ‘omega’ values during the first iterations.
Methods for the parameters without variability
Parameters without variability are not estimated in the same way as parameters with variability. Indeed, the SAEM algorithm requires to draw parameter values from their marginal distribution, which exists only for parameters with variability.
Several methods can be used to estimate the parameters without variability. By default, these parameters are optimized using the NelderMead simplex algorithm (Matlab’s fminsearch method). Other options are also available in the SAEM settings:
 No variability (default): optimization via NelderMead simplex algorithm
 Add decreasing variability: an artificial variability (i.e random effects) is added for these parameters, allowing estimation via SAEM. The variability starts at omega=1 and is progressively decreased such that at the end of the estimation process, the parameter has a variability of 1e5. The decrease in variability is exponential with a rate based on the maximum number of iterations for both the exploratory and smoothing phases. Note that if the autostop is triggered, the resulting variability might me higher.
 Variability at the first stage: during the exploratory phase of SAEM, an artificial variability is added and progressively forced to 1e5 (same as above). In the smoothing phase, the NelderMead simplex algorithm is used.
Depending on the specific project, one or the other method may lead to a better convergence. If the default method does not provide satisfying results, it is worth trying the other methods. In terms of computing time, if all parameters are without variability, the first option will be faster because only the NelderMead simplex algorithm will be used to estimate all the fixed effects. If some parameters have random effects, the first option will be slower because the NelderMead and the SAEM algorithm are computed at each step. In that case the second or third option will be faster because only the SAEM algorithm will be required when the artificial variability is added.
Alternatively, the standard deviation of the random effects can be fixed to a small value, for instance 5% for lognormally distributed parameters. (See next section on how to enforce a fixed value). With this method, the SAEM algorithm can be used, and the variability is kept small.
Other estimation methods: fixing population parameters or Bayesian estimation
Instead of the default estimation method with SAEM, it is possible to fix a population parameter, or set a prior on the estimate and use Bayesian estimation. This page gives details on these methods and how to use them.
Outputs
In the graphical user interface
The estimated population parameters are displayed in the POP.PARAM section of the RESULTS tab. Fixed effects names are “*_pop”, the standard deviation of the random effects “omega_*”, parameters of the error model “a”, “b”, “c”, the correlation between random effects “corr_*_*” and parameters associated to covariates “beta_*_*”. The standard deviation of the random effects is also expressed as coefficient of variation (CV) – a feature present in versions Monolix 2023 or above. The CV calculation depends on the parameter distribution:
 lognormal: \(CV=100*\sqrt{exp(\omega_p^2) 1}\)
 normal: \(CV=100*\frac{\omega_p}{p_{pop}}\)
 logitnormal and probitnormal: the CV is computed by MonteCarlo. 100000 samples X are drawn from the distribution (defined by \(\omega_p\) and \(p_{pop}\)) and the CV is calculated as the ratio of the sample standard deviation over the sample mean: \(CV=100*\frac{sd(X)}{mean(X)}\)
When you run the “Standard errors” task, then the population parameter table contains also the standard error (s.e) and relative standard error (r.s.e).
Information about the SAEM task performance are below the table:
 The total elapsed time for this task
 The number of iterations in the exploratory and smoothing phases, along with a message that indicates whether the convergence has been reached (“autostop”), or if the algorithm arrived at the maximum number of iterations (“stopped at the maximum number of iterations/autostop criteria have not been reached”) or if it was stopped by the user (“manual stop”). (for Monolix versions 2021 or above)
A “Copy table” icon on the top of the table allows to copy it in Excel, Word, etc. The table format and display is kept.
In the output folder
After having run the estimation of the population parameters, the following files are available:
 summary.txt: contains the estimated population parameters (and the number of iterations in Monolix2021R1), in a format easily readable by a human (but not easy to parse for a computer)
 populationParameters.txt: contains the estimated population parameters (by default in csv format), including the CV.
 predictions.txt: contains for each individual and each observation time, the observed data (y), the prediction using the population parameters and population median covariates value from the data set (popPred_medianCOV), the prediction using the population parameters and individual covariates value (popPred), the prediction using the individual approximate conditional mean calculated from the last iterations of SAEM (indivPred_SAEM) and the corresponding weighted residual (indWRes_SAEM).
 IndividualParameters/estimatedIndividualParameters.txt: individual parameters corresponding to the approximate conditional mean, calculated as the average of the individual parameters sampled by MCMC during all iterations of the smoothing phase. When several chains are used (see project settings), the average is also done over all chains. Values are indicated as *_SAEM in the file.
Parameters without variability: method “no variability” or “variability at the first stage”: *_SAEM represents the value at the last SAEM iteration, so the estimated population parameter plus the covariate effects. In absence of covariate, all individuals have the same value.
 method “add decreasing variability”: *_SAEM represents the average of all iterations of the smoothing phase. This value can be slightly different from individual to individual, even in te absence of covariates.
 IndividualParameters/estimatedRandomEffects.txt: individual random effects corresponding to the approximate conditional mean, calculated using the last estimations of SAEM (*_SAEM).
For parameters without variability, see above.
More details about the content of the output files can be found here.
Settings
The settings are accessible through the interface via the button next to the parameter estimation task.
Burnin phase:
The burnin phase corresponds to an initialization of SAEM: individual parameters are sampled from their conditional distribution using MCMC using the initial values for the population parameters (no update of the population parameter estimates).
Note: the meaning of the burnin phase in Monolix is different to what is called burnin in Nonmem algorithms.
 Number of iterations (default: 5): number of iterations of the burnin phase
Exploratory phase:
 Autostop criteria (default: yes): if ticked, autostop criteria are used to automatically detect convergence during the exploratory phase. If convergence is detected, the algorithm switches to the smoothing phase before the maximum number of iterations. The criteria take into account the stability of the convergence indicator, omega parameters and error model parameters.
 Maximum number of iterations (default: 500, if autostop ticked): maximum number of iterations for the exploratory phase. Even if the autostop criteria are not fulfilled, the algorithm switches to the smoothing phase after this maximum number of iterations. A warning message will be displayed in the GUI if the maximum number of iterations is reached while the autostop criteria are not fulfilled.
 Minimum number of iterations (default: 150, if autostop ticked): minimum number of iterations for the exploratory phase. This value also corresponds to the interval length over which the autostop criteria are tested. A larger minimum number of iterations means that the autostop criteria are harder to reach.
 Number of iterations (default: 500, if autostop unticked): fixed number of iterations for the exploratory phase.
 Stepsize exponent (default: 0): The value, comprised between 0 and 1, represents memory of the stochastic process, i.e how much weight is given at iteration k to the value of the previous iteration compared to the new information collected. A value 0 means no memory, i.e the parameter value at iteration k is built based on the information collected at that iteration only, and does not take into account the value of the parameter at the previous iteration.
 Simulated annealing (default: enabled): the Simulated Annealing version of SAEM permits to better explore the parameter space by constraining the standard deviation of the random effects to decrease slowly.
 Decreasing rate for the variance of the residual errors (default: 0.95, if simulated annealing enabled): the residual error variance (parameter “a” for a constant error model for instance) at iteration k is constrained to be larger than the decreasing rate times the variance at the previous iteration.
 Decreasing rate for the variance of the individual parameters (default: 0.95, if simulated annealing enabled): the variance of the random effects at iteration k is constrained to be larger than the decreasing rate times the variance at the previous iteration.
Smoothing phase:
 Autostop criteria (default: yes): if ticked, autostop criteria are used to automatically detect convergence during the smoothing phase. If convergence is detected, the algorithm stops before the maximum number of iterations.
 Maximum number of iterations (default: 200, if autostop ticked): maximum number of iterations for the smoothing phase. Even if the autostop criteria are not fulfilled, the algorithm stops after this maximum number of iterations.
 Minimum number of iterations (default: 50, if autostop ticked): minimum number of iterations for the smoothing phase. This value also corresponds to the interval length over which the autostop criteria is tested. A larger minimum number of iterations means that the autostop criteria is harder to reach.
 Number of iterations (default: 200, if autostop unticked): fixed number of iterations for the smoothing phase.
 Stepsize exponent (default: 0.7): The value, comprised between 0 and 1, represents memory of the stochastic process, i.e how much weight is given at iteration k to the value of the previous iteration compared to the new information collected. The value must be strictly larger than 0.5 for the smoothing phase to converge. Large values (close to 1) will result in a smoother parameter trajectory during the smoothing phase, but may take longer to converge to the maximum likelihood estimate.
Methodology for parameters without variability (if parameters without variability are present in the model):
The SAEM algorithm requires to draw parameter values from their marginal distribution, which does not exist for parameters without variability. These parameters are thus estimated via another method, which can be chosen among:
 No variability (default choice): After each SAEM iteration, the parameter without variability are optimized using the NelderMead simplex algorithm. The absolute tolerance (stopping criteria) is 1e4 and the maximum number of iterations 20 times the number of parameters to calculate via this algorithm.
 Add decreasing variability: an artificial variability is added for these parameters, allowing estimation via SAEM. The variability is progressively decreased such that at the end of the estimation process, the parameter has a variability of 1e5.
 Variability in the first stage: during the exploratory phase, an artificial variability is added and progressively forced to 1e5 (same as above). In the smoothing phase, the NelderMead simplex optimization algorithm is used.
Handling parameters without variability is also discussed here.
Set all SAEM iterations to zero in one click
The icon on the bottom left provides a shortcut to set all SAEM iterations to zero (in all three phases). This is convenient if the user wish to skip the estimation of the population parameters and keep the initial estimates as population estimates to run the other tasks, since running SAEM first is mandatory. It is not equivalent to fixing all population parameters, since standard errors are not estimated for fixed parameters, while they will be estimated for the initial estimates in this case. The action can be easily reversed with the second shortcut to reset default SAEM iterations.
Good practice, troubleshooting and tips
Choosing to enable or disable the simulated annealing
As the simulated annealing option permits to more surely find the global maximum, it should be used during the first runs, when the initial values may be quite far from the final estimates.
On the other side, the simulated annealing option may keep the omega values artificially high, even after a large number of SAEM iterations. This may prevent the identification of parameters for which the variability is in fact zero and lead to NaN in the standard errors. So once good initial values have been found and there is no risk to fall in a local maximum, the simulated annealing option can be disabled. Below we show an example where removing the simulated annealing permits to identify parameters for which the interindividual variability can be removed.
Example: The dataset used in the tobramycin case study is quite sparse. In these conditions, we expect that estimating the interindividual variability for all parameters will be difficult. In this case, the estimation can be done in two steps, as shown below for a twocompartments model on this dataset:
 First, we run SAEM with the simulated annealing option (default setting), which facilitates the convergence towards the global maximum. All four parameters V, k, k12 and k21 have random effects. The estimated parameters are shown below:
The parameters omega_k12 and omega_k21 have high standard errors, suggesting that the variability is difficult to estimate. The omega_k12 and omega_k21 values themselves are also high (100% interindividual variability), suggesting that they may have been kept too high due to the simulated annealing.
 As a second step, we use the last estimates as new initial values (as shown here), and run SAEM again after disabling the simulated annealing option. On the plot showing the convergence of SAEM, we can see omega_V, omega_k12 and omega_k21 decreasing to very low values. The data is too sparse to correctly identify the interindividual variability for V, k12 and k21. Thus, their random effects can be removed, but the random effect of k can be kept.
Note that because the omega_V, omega_k12 and omega_k21 parameters decrease without stabilizing, the convergence indicator does the same.
3.3.Conditional distribution
 Purpose
 Calculation of the conditional distribution
 Conditional distribution
 MCMC algorithm
 Conditional mean
 Samples from the conditional distribution
 Stopping criteria
 Running the conditional distribution estimation task
 Outputs
 In the graphical user interface
 In the output folder
 Settings
Purpose
The conditional distribution represents the uncertainty of the individual parameter values. The conditional distribution estimation task permits to sample from this distribution. The samples are used to calculate the condition mean, or directly as estimators of the individual parameters in the plots to improve their informativeness [1]. They are also used to compute the statistical tests.
Calculation of the conditional distribution
Conditional distribution
The conditional distribution is \(p(\psi_iy_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual i, \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual i. The conditional distribution represents the uncertainty of the individual’s parameter value, taking into account the information at hand for this individual:
 the observed data for that individual,
 the covariate values for that individual,
 and the fact that the individual belongs to the population for which we have already estimated the typical parameter value (fixed effects) and the variability (standard deviation of the random effects).
It is not possible to directly calculate the probability for a given \(\psi_i\) (no closed form), but is possible to obtain samples from the distribution using a MarkovChain MonteCarlo procedure (MCMC).
MCMC algorithm
MCMC methods are a class of algorithms for sampling from probability distributions for which direct sampling is difficult. They consist of constructing a stochastic procedure which, in its stationary state, yields draws from the probability distribution of interest. Among the MCMC class, we use the MetropolisHastings (MH) algorithm, which has the property of being able to sample probability distributions which can be computed up to a constant. This is the case for our conditional distribution, which can be rewritten as:
$$p(\psi_iy_i)=\frac{p(y_i\psi_i)p(\psi_i)}{p(y_i)}$$
\(p(y_i\psi_i)\) is the conditional density function of the data when knowing the individual parameter values and can be computed (closed form solution). \(p(\psi_i)\) is the density function for the individual parameters and can also be computed. The likelihood \(p(y_i)\) has no closed form solution but it is constant.
In brief, the MH algorithm works in the following way: at each iteration k, a new individual parameter value is drawn from a proposal distribution for each individual. The new value is accepted with a probability that depends on \(p(\psi_i)\) and \(p(y_i\psi_i)\). After a transition period, the algorithm reaches a stationary state where the accepted values follow the conditional distribution probability \(p(\psi_iy_i)\). For the proposal distribution, three different distributions are used in turn with a (2,2,2) pattern (setting “Number of iterations of kernel 1/2/3” in Settings > Project Settings): the population distribution, a unidimensional Gaussian random walk, or a multidimensional Gaussian random walk. For the random walks, the variance of the Gaussian is automatically adapted to reach an optimal acceptance ratio (“target acceptance ratio” setting in Settings > Project Settings).
Conditional mean
The draws from the conditional distribution generated by the MCMC algorithm can be used to estimate any summary statistics of the distribution (mean, standard deviation, quantiles, etc). In particular we calculate the conditional mean by averaging over all draws:
$$ \hat{\psi}_i^{mean} = \frac{1}{K}\sum_{k=1}^{K}\psi_i^{k}$$
The standard deviation of the conditional distribution is also calculated. The number of samples used to calculate the mean and standard deviation corresponds to the number of chains times the total number of iterations of the conditional distribution task (not only during the convergence interval length). The mean is calculated over the transformed individual parameters (in the gaussian space), and backtransformed to the nongaussian space.
Samples from the conditional distribution
Among all samples from the conditional distribution, a small number (between 1 and 10, see “Simulated parameters per individual” setting) is kept to be used in the plots. These samples are unbiased estimators and they present the advantage of not being affected by shrinkage, as shown for example on the documentation of the plot “distribution of the individual parameters“.
Stopping criteria
At iteration k, the conditional mean is calculated for each individual by averaging over all k previous iterations. The average conditional means over all individuals (noted E(Xy)), and the standard deviation of the conditional means over all individuals (noted sd(Xy)) are calculated and displayed in the popup window. The algorithm stops when, for all parameters, the average conditional means and standard deviations of the last 50 iterations (“Interval length” setting) do not deviate by more than 5% (2.5% in each direction, “relative interval” setting) from the average and standard deviation values at iteration k.
In some very specific cases (for example with a parameter with a normal distribution and a value very close to 1), it can take many iterations to reach the convergence criteria because the criteria is defined as a percentage. In that case, the toggle “enable maximum number of iterations” can be used to limit the number iterations of this task. If the limit is reached, a warning message will be displayed in the interface.
Running the conditional distribution estimation task
During the evaluation of the conditional distribution, the following plot popups, displaying the average conditional means over all individuals (noted E(Xy)), and the standard deviation of the conditional means over all individuals (noted sd(Xy)) for each iteration of the MCMC algorithm.
The convergence criteria described above means that the blue line, which represents the average over all individuals of the conditional mean, must be within the tube. The tube is centered around the last value of the blue line and spans over 5% of that last value. The algorithm stops when all blue lines are in their tube.
Dependencies between tasks:
 The “Population parameters” task must be run before launching the conditional distribution task.
 The conditional distribution task is recommended before calculating the loglikelihood task without the linearization method (i.e loglikelihood via importance sampling).
 The conditional distribution task is necessary for the statistical tests.
 The samples generated during the conditional distribution task will be reused for the Standard errors task (without linearization).
Outputs
In the graphical user interface
In the Indiv.Param
section of the Results
tab, a summary of the estimated conditional mean is given (min, max and quartiles), as shown in the figure below. Starting from Monolix2021R1, the number of iterations is also displayed, along with a message indicating whether convergence has been reached (“autostop”) or if the task was stopped by the user or reached the maximum number of iterations.
To see the estimated parameter value for each individual, the user can click on the [INDIV. ESTIM.] section. Notice that the user can also see them in the output files, which can be accessed via the folder icon at the bottom left. Notice that there is a “Copy table” icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.
In the output folder
After having run the conditional distribution task, the following files are available:
 summary.txt: contains the summary statistics (as displayed in the GUI)
 IndividualParameters/estimatedIndividualParameters.txt: the individual parameters for each subjectoccasion are displayed. The conditional mean (*_mean) and the standard deviation (*_sd) of the conditional distribution are added to the file. The number of samples used to calculate the mean and standard deviation corresponds to the number of chains times the total number of iterations of the conditional distribution task (not only during the convergence interval length).
 IndividualParameters/estimatedRandomEffects.txt: the individual random effects for each subjectoccasion are displayed. Those corresponding to the conditional mean (*_mean) are added to the file, together with the standard deviation (*_sd).
 IndividualParameters/simulatedIndividualParameters.txt: several simulated individual parameters (draws from the conditional distribution) are recorded for each individual. The rep column permits to distinguish the several simulated parameters for each individual.
 IndividualParameters/simulatedRandomEffects.txt: the random effects corresponding to the simulated individual parameters are recorded.
More details about the content of the output files can be found here.
Settings
To change the settings, you can click on the settings button next the conditional distribution task.
 Interval length (default: 50): number of iterations over which the convergence criteria is checked.
 Relative interval (default: 0.05): size of the interval (relative to the current average or standard deviation) in which the last “interval length” iterations must be for the stopping criteria to be met. A value at 0.05 means that over the last “interval length” iterations, the value should not vary by more than 5% (2.5% in each direction).
 Simulated parameters per individual (default: via calculation): number of draws from the conditional distribution that will be used in the plots. The number is calculated as min(10, idealNb) with idealNb = max(500 / number of subject , 5000 / number of observations). This means that the maximum number is 10 (which is usually the case for small data sets). For large data sets, the number may be reduced, but the number of individual times the number of simulated parameters should be at least 500, and the number of observations times the number of simulated parameters should be at least 5000. This ensures to have a sufficiently large but not unnecessarily large number of dots in the plots such as Observations versus predictions or Correlation between random effects.
If the user sets the number of simulated parameters to a value larger than the number of chains (project settings) times the total number of iterations of the conditional distribution task (maximum number of iterations or when convergence criteria are reached), the number will be restricted to the number of available samples.
If the user sets the number of simulated parameters to a value smaller than interval length times number of chains, the simulated parameters are picked evenly from the interval length and the chains. If the requested number of simulated parameters is larger, the last n (n=number of requested simulated parameters) samples are picked.  Enable maximum iterations limit (default: toggle off) [from version 2020 on]: When the toggle in “on”, a maximum number of iterations can be defined.
 Maximum number of iterations (default: 500, needs to be larger than the interval length, available if “enable maximum iterations limit’ is on): maximum number of iterations for the conditional distribution task. Even if the convergence criteria are not fulfilled, the algorithm stops after this maximum number of iterations. If the maximum number of iterations is reached, a warning message will be displayed in the interface.
3.4.EBEs
 Purpose
 Calculation of the EBEs (conditional mode)
 Conditional distribution
 Mode of the conditional distribution
 Individual random effects
 Algorithm
 Running the EBEs task
 Outputs
 In the graphical user interface
 In the output folder
 Settings
 Calculate EBEs for a new data set using an existing model
Purpose
EBEs stands for Empirical Bayes Estimates. The EBEs are the most probable value of the individual parameters (parameters for each individual), given the estimated population parameters and the data of each individual. In a more mathematical language, they are the mode of the conditional parameter distribution for each individual.
These values are useful to compute the most probable prediction for each individual, for comparison with the data (for instance in the Individual Fits plot).
Calculation of the EBEs (conditional mode)
When launching the “EBEs” task, the mode of the conditional parameter distribution is calculated.
Conditional distribution
The conditional distribution is \( p(\psi_iy_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual i, \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual i. The conditional distribution represents the uncertainty of the individual’s parameter value, taking into account the information at hand for this individual: the observed data for that individual, the covariate values for that individual and the fact that the individual belongs to the population for which we have already estimated the typical parameter value (fixed effects) and the variability (standard deviation of the random effects). It is not possible to directly calculate the probability for a given \(\psi_i\) (no closed form), but is possible to obtain samples from the distribution using a MarkovChain MonteCarlo procedure (MCMC). This is detailed more on the Conditional Distribution page.
Mode of the conditional distribution
The mode is the parameter value with the highest probability:
$$ \hat{\psi}_i^{mode} = \underset{\psi_i}{\textrm{arg max }}p(\psi_iy_i;\hat{\theta})$$
To find the mode, we thus need to maximize the conditional probability with respect to the individual parameter value \(\psi_i\).
Individual random effects
Once the individual parameters values \(\psi_i\) are known, the corresponding individual random effects can be calculated using the population parameters and covariates. Taking the example of a parameter \(\psi\) having a normal distribution within the population and that depends on the covariate \(c\), we can write for individual \(i\):
$$ \psi_i = \psi_{pop} + \beta \times c_i + \eta_i$$
As \(\psi_i\) (estimated conditional mode), \(\psi_{pop}\) and \(\beta\) (population parameters) and \(c_i\) (individual covariate value) are known, the individual random effect \(\eta_i\) can easily be calculated.
Algorithm
For each individual, to find the \(\psi_i\) values that maximizes the conditional distribution, we use the NelderMead Simplex algorithm [1].
As the conditional distribution does not have a closed form solution (i.e \(p(\psi_iy_i;\hat{\theta})\) cannot be directly or easily calculated for a given \(\psi_i\)), we use the Bayes law to rewrite it in the following way (leaving the population parameters \(\hat{\theta}\) out for clarity):
$$p(\psi_iy_i)=\frac{p(y_i\psi_i)p(\psi_i)}{p(y_i)}$$
The conditional density function of the data when knowing the individual parameter values (i.e \(p(y_i\psi_i)\)) is easy to calculate, as well as the density function for the individual parameters (i.e \(p(\psi_i)\)), because they have closed form solutions. On the opposite, the likelihood \(p(y_i)\) has no closed form solution. But as it does not depend on \(\psi_i\), we can leave it out of the optimization procedure and only optimize \(p(y_i\psi_i)p(\psi_i)\).
The initial value used for the NelderMead simplex algorithm is the conditional mean (estimated during the conditional distribution task) if avaible (typical case of a full scenario), or the approximate conditional mean calculated at the end of SAEM otherwise.
Parameters without variability are not estimated, they are set to \( \psi_i = \psi_{pop} + \beta \times c_i \).
Running the EBEs task
When running the EBEs task, the progress is displayed in the popup window:
Dependencies between tasks:
 The “Population parameters” task must be run before launching the EBEs task.
 The EBEs task is recommended before calculating the Standard errors task and the Loglikelihood task using the linearization method.
Outputs
In the graphical user interface
In the Indiv.Param
section of the Results
tab, a summary of the individual parameters is proposed (min, max, median and quartiles) as shown in the figure below. The elapsed time for this task is also shown.
To see the estimated parameter value for each individual, the user can click on the [INDIV. ESTIM.] section. Notice that the user can also see them in the output files, which can be accessed via the folder icon at the bottom left. Notice that there is a “Copy table” icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.
In the output folder
After having run the EBEs task, the following files are available:
 summary.txt: contains the summary statistics (as displayed in the GUI)
 IndividualParameters/estimatedIndividualParameters.txt: the individual parameters for each subjectoccasion are displayed. In addition to the already present approximation conditional mean from SAEM (*_SAEM), the conditional mode (*_mode) is added to the file.
 IndividualParameters/estimatedRandomEffects.txt: the individual random effects for each subjectoccasion are displayed (*_mode), in addition to the already present value based on the approximate conditional mean from SAEM (*_SAEM).
More details about the content of the output files can be found here.
Settings
The settings are accessible through the interface via the button next to the EBEs task.
 Maximum number of iterations (default: 200): maximum number of iterations for the NelderMead Simplex algorithm, for each individual. Even if the tolerance criteria is not met, the algorithm stops after that number of iterations.
 Tolerance (default: 1e6): absolute tolerance criteria. The algorithm stops when the change of the conditional probability value between two iterations is less than the tolerance.
Calculate EBEs for a new data set using an existing model
3.5.Standard error using the Fisher Information Matrix
 Purpose
 Calculation of the standard errors
 Running the standard errors task
 Outputs
 In the graphical user interface
 In the output folder
 Interpreting the correlation matrix of the estimates
 Settings
 Good practice
Purpose
The standard errors represent the uncertainty of the estimated population parameters. In Monolix, they are calculated via the estimation of the Fisher Information Matrix. They can for instance be used to calculate confidence intervals or detect model overparametrization.
Calculation of the standard errors
Several methods have been proposed to estimate the standard errors, such as bootstrapping or via the Fisher Information Matrix (FIM). In the Monolix GUI, the standard errors are estimated via the FIM. Bootstrapping is available via the Rsmlx R package.
The Fisher Information Matrix (FIM)
The observed Fisher information matrix (FIM) \(I \) is minus the second derivatives of the observed loglikelihood:
$$ I(\hat{\theta}) = \frac{\partial^2}{\partial\theta^2}\log({\cal L}_y(\hat{\theta})) $$
The loglikelihood cannot be calculated in closed form and the same applies to the Fisher Information Matrix. Two different methods are available in Monolix for the calculation of the Fisher Information Matrix: by linearization or by stochastic approximation.
Via stochastic approximation
A stochastic approximation algorithm using a Markov chain Monte Carlo (MCMC) algorithm is implemented in Monolix for estimating the FIM. This method is extremely general
and can be used for many data and model types (continuous, categorical, timetoevent, mixtures, etc.).
Via linearization
This method can be applied for continuous data only. A continuous model can be written as:
$$\begin{array}{cl} y_{ij} &= f(t_{ij},z_i)+g(t_{ij},z_i)\epsilon_{ij} \\ z_i &= z_{pop}+\eta_i \end{array}$$
with \( y_{ij} \) the observations, f the prediction, g the error model, \( z_i\) the individual parameter value for individual i, \( z_{pop}\) the typical parameter value within the population and \(\eta_i\) the random effect.
Linearizing the model means using a Taylor expansion in order to approximate the observations \( y_{ij} \) by a normal distribution. In the formulation above, the appearance of the random variable \(\eta_i\) in the prediction f in a nonlinear way leads to a complex (nonnormal) distribution for the observations \( y_{ij} \).
The Taylor expansion is done around the EBEs value, that we note \( z_i^{\textrm{mode}} \).
Standard errors
Once the Fisher Information Matrix has been obtained, the standard errors can be calculated as the square root of the diagonal elements of the inverse of the Fisher Information Matrix. The inverse of the FIM \(I(\hat{\theta})\) is the variancecovariance matrix \(C(\hat{\theta})\):
$$C(\hat{\theta})=I(\hat{\theta})^{1}$$
The standard error for parameter \( \hat{\theta}_k \) can be calculated as:
$$\textrm{s.e}(\hat{\theta}_k)=\sqrt{\tilde{C}_{kk}(\hat{\theta})}$$
Note that in Monolix, the Fisher Information Matrix and variancecovariance matrix are calculated on the transformed normally distributed parameters. The variancecovariance matrix \( \tilde{C} \) for the untransformed parameters can be obtained using the jacobian \(J\):
$$\tilde{C}=J^TC J$$
Correlation matrix
The correlation matrix is calculated from the variancecovariance matrix as:
$$\text{corr}(\theta_i,\theta_j)=\frac{\tilde{C}_{ij}}{\textrm{s.e}(\theta_i)\textrm{ s.e}(\theta_j)}$$
Wald test
For the beta parameters characterizing the influence of the covariates, the relative standard error can be used to perform a Wald test, testing if the estimated beta value is significantly different from zero.
Running the standard errors task
When running the standard error task, the progress is displayed in the popup window. At the end of the task, the correlation matrix is also shown, along with the elapsed time and number of iterations.
Dependencies between tasks:
The “Population parameters” task must be run before launching the Standard errors task. If the Conditional distribution task has already been run, the first iterations of the Standard errors (without linearization) will be very fast, as they will reuse the same draws as those obtained in the Conditional distribution task.
Output
In the graphical user interface
In the Pop.Param
section of the Results
tab, three additional columns appear in addition to the estimated population parameters:
 S.E: the estimated standard errors
 R.S.E: the relative standard error (standard error divided by the estimated parameter value)
To help the user in the interpretation, a color code is used for the pvalue and the RSE:
 For the pvalue: between .01 and .05, between .001 and .01, and less than .001.
 For the RSE: between 50% and 100%, between 100% and 200%, and more than 200%.
When the standard errors were estimated both with and without linearization, the S.E and R.S.E are displayed for both methods.
In the STD.ERRORS
section of the Results
tab, we display:
 R.S.E: the relative standard errors
 Correlation matrix: the correlation matrix of the population parameters
 Eigen values: the smallest and largest eigen values, as well as the condition number (max/min)
 The elapsed time and, starting from Monolix2021R1, the number of iterations for stochastic approximation, as well as a message indicating whether convergence has been reached (“autostop”) or if the task was stopped by the user or reached the maximum number of iterations.
To help the user in the interpretation, a color code is used:
 For the correlation: between .5 and .8, between .8 and .9, and higher than .9.
 For the RSE: between 50% and 100%, between 100% and 200%, and more than 200%.
When the standard errors were estimated both with and without linearization, both results appear in different subtabs.
If you hover on a specific value with the mouse, both parameters are highlighted to know easily which parameter you are looking at:
In the output folder
After having run the Standard errors task, the following files are available:
 summary.txt: contains the s.e, r.s.e, pvalues, correlation matrix and eigenvalues in an easily readable format, as well as elapsed time and number of iterations for stochastic approximation (starting from Monolix2021R1).
 populationParameters.txt: contains the s.e, r.s.e and pvalues in csv format, for the method with (*_lin) or without (*_sa) linearization
 FisherInformation/correlationEstimatesSA.txt: correlation matrix of the population parameter estimates, method without linearization (stochastic approximation)
 FisherInformation/correlationEstimatesLin.txt: correlation matrix of the population parameter estimates, method with linearization
 FisherInformation/covarianceEstimatesSA.txt: variancecovariance matrix of the transformed normally distributed population parameter, method without linearization (stochastic approximation)
 FisherInformation/covarianceEstimatesLin.txt: variancecovariance matrix of the transformed normally distributed population parameter, method with linearization
Interpreting the correlation matrix of the estimates
The color code of Monolix’s results allows to quickly identify population parameter estimates that are strongly correlated. This often reflects model overparameterization and can be further investigated using Mlxplore and the convergence assessment. This is explained in details in this video:
Settings
The settings are accessible through the interface via the button next to the Standard errors task:
 Minimum number of iterations: minimum number of iterations of the stochastic approximation algorithm to calculate the Fisher Information Matrix.
 Maximum number of iterations: maximum number of iterations of the stochastic approximation algorithm to calculate the Fisher Information Matrix. The algorithm stops even if the stopping criteria are not met.
Good practices and tips
When to use “use linearization method”?
Firstly, it is only possible to use the linearization method for continuous data. For the linearization is available, this method is generally much faster than without linearization (i.e stochastic approximation) but less precise. The Fisher Information Matrix by model linearization will generally be able to identify the main features of the model. More precise– and timeconsuming – estimation procedures such as stochastic approximation will have very limited impact in terms of decisions for these most obvious features. Precise results are required for the final runs where it becomes more important to rigorously defend decisions made to choose the final model and provide precise estimates and diagnosis plots.
I have NANs as results for standard errors for parameter estimates. What should I do? Does it impact the likelihood?
NaNs as standard errors often appear when the model is too complex and some parameters are unidentifiable. They can be seen as an infinitely large standard error.
The likelihood is not affected by NaNs in the standard errors. The estimated population parameters having a NaN as standard error are only very uncertain (infinitely large standard error and thus infinitely large confidence intervals).
3.6.Log Likelihood estimation
Purpose
The loglikelihood is the objective function and a key information. The loglikelihood cannot be computed in closed form for nonlinear mixed effects models. It can however be estimated.
Loglikelihood estimation
Performing likelihood ratio tests and computing information criteria for a given model requires computation of the loglikelihood
$$ {\cal L}{\cal L}_y(\hat{\theta}) = \log({\cal L}_y(\hat{\theta})) \triangleq \log(p(y;\hat{\theta})) $$
where \(\hat{\theta}\) is the vector of population parameter estimates for the model being considered, and \(p(y;\hat{\theta})\) is the probability distribution function of the observed data given the population parameter estimates. The loglikelihood cannot be computed in closed form for nonlinear mixed effects models. It can however be estimated in a general framework for all kinds of data and models using the importance sampling Monte Carlo method. This method has the advantage of providing an unbiased estimate of the loglikelihood – even for nonlinear models – whose variance can be controlled by the Monte Carlo size.
Two different algorithms are proposed to estimate the loglikelihood:
 by linearization,
 by Importance sampling.
Loglikelihood by importance sampling
The observed loglikelihood \({\cal LL}(\theta;y)=\log({\cal L}(\theta;y))\) can be estimated without requiring approximation of the model, using a Monte Carlo approach. Since
$${\cal LL}(\theta;y) = \log(p(y;\theta)) = \sum_{i=1}^{N} \log(p (y_i;\theta))$$
we can estimate \(\log(p(y_i;\theta))\) for each individual and derive an estimate of the loglikelihood as the sum of these individual loglikelihoods. We will now explain how to estimate \(\log(p(y_i;\theta))\) for any individual i. Using the \(\phi\)representation of the model (the individual parameters are transformed to be Gaussian), notice first that \(p(y_i;\theta)\) can be decomposed as follows:
$$p(y_i;\theta) = \int p(y_i,\phi_i;\theta)d\phi_i = \int p(y_i\phi_i;\theta)p(\phi_i;\theta)d\phi_i = \mathbb{E}_{p_{\phi_i}}\left(p(y_i\phi_i;\theta)\right)$$
Thus, \(p(y_i;\theta)\) is expressed as a mean. It can therefore be approximated by an empirical mean using a Monte Carlo procedure:
 Draw M independent values \(\phi_i^{(1)}\), \(\phi_i^{(2)}\), …, \(\phi_i^{(M)}\) from the marginal distribution \(p_{\phi_i}(.;\theta)\).
 Estimate \(p(y_i;\theta)\) with \(\hat{p}_{i,M}=\frac{1}{M}\sum_{m=1}^{M}p(y_i  \phi_i^{(m)};\theta)\)
By construction, this estimator is unbiased, and consistent since its variance decreases as 1/M:
$$\mathbb{E}\left(\hat{p}_{i,M}\right)=\mathbb{E}_{p_{\phi_i}}\left(p(y_i\phi_i^{(m)};\theta)\right) = p(y_i;\theta) ~~~~\mbox{Var}\left(\hat{p}_{i,M}\right) = \frac{1}{M} \mbox{Var}_{p_{\phi_i}}\left(p(y_i\phi_i^{(m)};\theta)\right)$$
We could consider ourselves satisfied with this estimator since we “only” have to select M large enough to get an estimator with a small variance. Nevertheless, it is possible to improve the statistical properties of this estimator.
For any distribution \(\tilde{p_{\phi_i}}\) that is absolutely continuous with respect to the marginal distribution \(p_{\phi_i}\), we can write
$$ p(y_i;\theta) = \int p(y_i\phi_i;\theta) \frac{p(\phi_i;\theta)}{\tilde{p}(\phi_i;\theta)} \tilde{p}(\phi_i;\theta)d\phi_i = \mathbb{E}_{\tilde{p}_{\phi_i}}\left(p(y_i\phi_i;\theta)\frac{p(\phi_i;\theta)}{\tilde{p}(\phi_i;\theta)} \right).$$
We can now approximate \(p(y_i;\theta)\) using an importance sampling integration method with \(\tilde{p}_{\phi_i}\) as the proposal distribution:
 Draw M independent values \(\phi_i^{(1)}\), \(\phi_i^{(2)}\), …, \(\phi_i^{(M)}\) from the proposal distribution \(\tilde{p_{\phi_i}}(.;\theta)\).
 Estimate \(p(y_i;\theta)\) with \(\hat{p}_{i,M}=\frac{1}{M}\sum_{m=1}^{M}p(y_i  \phi_i^{(m)};\theta)\frac{p(\phi_i^{(m)};\theta)}{\tilde{p}(\phi_i^{(m)};\theta)}\)
By construction, this estimator is unbiased, and its variance also decreases as 1/M:
$$\mbox{Var}\left(\hat{p}_{i,M}\right) = \frac{1}{M} \mbox{Var}_{\tilde{p_{\phi_i}}}\left(p(y_i\phi_i^{(m)};\theta)\frac{p(\phi_i^{(m)};\theta)}{\tilde{p}(\phi_i^{(m)};\theta)}\right)$$
There exist an infinite number of possible proposal distributions \(\tilde{p}\) which all provide the same rate of convergence 1/M. The trick is to reduce the variance of the estimator by selecting a proposal distribution so that the numerator is as small as possible.
For this purpose, an optimal proposal distribution would be the conditional distribution \(p_{\phi_iy_i}\). Indeed, for any \(m = 1,2, …, M,\)
$$ p(y_i\phi_i^{(m)};\theta)\frac{p(\phi_i^{(m)};\theta)}{p(\phi_i^{(m)}y_i;\theta)} = p(y_i;\theta) $$
which has a zero variance, so that only one draw from \(p_{\phi_iy_i}\) is required to exactly compute the likelihood \(p(y_i;\theta)\).
The problem is that it is not possible to generate the \(\phi_i^{(m)}\) with this exact conditional distribution, since that would require computing a normalizing constant, which here is precisely \(p(y_i;\theta)\).
Nevertheless, this conditional distribution can be estimated using the MetropolisHastings algorithm and a practical proposal “close” to the optimal proposal \(p_{\phi_iy_i}\) can be derived. We can then expect to get a very accurate estimate with a relatively small Monte Carlo size M.
The mean and variance of the conditional distribution \(p_{\phi_iy_i}\) are estimated by MetropolisHastings for each individual i. Then, the \(\phi_i^{(m)}\) are drawn with a noncentral student tdistribution:
$$ \phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m}$$
where \(\mu_i\) and \(\sigma^2_i\) are estimates of \(\mathbb{E}\left(\phi_iy_i;\theta\right)\) and \(\mbox{Var}\left(\phi_iy_i;\theta\right)\), and \((T_{i,m})\) is a sequence of i.i.d. random variables distributed with a Student’s tdistribution with \(\nu\) degrees of freedom (see section Advanced settings for the loglikelihood for the number of degrees of freedom).
Remark: The standard error of the LL on all the draws is proposed. It represents the impact of the variability of the draws on the LL uncertainty, given the estimated population parameters, but it does not take into account the uncertainty of the model that comes from the uncertainty on the population parameters.
Remark: Even if \(\hat{\cal L}_y(\theta)=\prod_{i=1}^{N}\hat{p}_{i,M}\) is an unbiased estimator of \({\cal L}_y(\theta)\), \(\hat{\cal LL}_y(\theta)\) is a biased estimator of \({\cal LL}_y(\theta)\). Indeed, by Jensen’s inequality, we have :
$$\mathbb{E}\left(\log(\hat{\cal L}_y(\theta))\right) \leq \log \left(\mathbb{E}\left(\hat{\cal L}_y(\theta)\right)\right)=\log\left({\cal L}_y(\theta)\right)$$
Best practice: the bias decreases as M increases and also if \(\hat{\cal L}_y(\theta)\) is close to \({\cal L}_y(\theta)\). It is therefore highly recommended to use a proposal as close as possible to the conditional distribution \(p_{\phi_iy_i}\), which means having to estimate this conditional distribution before estimating the loglikelihood (i.e. run task “Conditional distribution” before).
Loglikelihood by linearization
The likelihood of the nonlinear mixed effects model cannot be computed in a closedform. An alternative is to approximate this likelihood by the likelihood of the Gaussian model deduced from the nonlinear mixed effects model after linearization of the function f (defining the structural model) around the predictions of the individual parameters \((\phi_i; 1 \leq i \leq N)\).
Notice that the loglikelihood can not be computed by linearization for discrete outputs (categorical, count, etc.) nor for mixture models.
Best practice: We strongly recommend to compute the conditional mode before computing the loglikelihood by linearization. Indeed, the linearization should be made around the most probable values as they are the same for both the linear and the nonlinear model.
Best practices: When should I use the linearization and when should I use the importance sampling?
Firstly, it is only possible to use the linearization algorithm for the continuous data. In that case, this method is generally much faster than importance sampling method and also gives good estimates of the LL. The LL calculation by model linearization will generally be able to identify the main features of the model. More precise– and timeconsuming – estimation procedures such as stochastic approximation and importance sampling will have very limited impact in terms of decisions for these most obvious features. Selection of the final model should instead use the unbiased estimator obtained by Monte Carlo.
Display and outputs
In case of estimation using the importance sampling method, a graphical representation is proposed to see the valuation of the mean value over the Monte Carlo iterations as on the following:
The final estimations are displayed in the result frame as below. Notice that there is a “Copy table” icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.
The loglikelihood is given in Monolix together with the Akaike information criterion (AIC) and Bayesian information criterion (BIC):
$$ AIC = 2 {\cal L}{\cal L}_y(\hat{\theta}) +2P $$
$$ BIC = 2 {\cal L}{\cal L}_y(\hat{\theta}) +log(N)P $$
where P is the total number of parameters to be estimated and N the number of subjects.
The new BIC criterion penalizes the size of \(\theta_R\) (which represents random effects and fixed covariate effects involved in a random model for individual parameters) with the log of the number of subjects (\(N\)) and the size of \(\theta_F\) (which represents all other fixed effects, so typical values for parameters in the population, beta parameters involved in a nonrandom model for individual parameters, as well as error parameters) with the log of the total number of observations (\(n_{tot}\)), as follows:
$$ BIC_c = 2 {\cal L}{\cal L}_y(\hat{\theta}) + \dim(\theta_R)\log N+\dim(\theta_F)\log n_{tot}$$
If the loglikelihood has been computed by importance sampling, the number of degrees of freedom used for the proposal tdistribution (5 by default) is also displayed, together with the standard error of the LL on the individual parameters drawn from the tdistribution.
In terms of output, a folder called LogLikelihood is created in the result folder where the following files are created
 logLikelihood.txt: containing for each computed method, the 2 x loglikelihood, the Akaike Information Criteria (AIC), the Bayesian Information Criteria (BIC), and the corrected Bayesian Information Criteria (BICc).
 individualLL.txt: containing the 2 x loglikelihood for each individual for each computed method.
Advanced settings for the loglikelihood
Monolix uses a tdistribution as proposal. By default, the number of degrees of freedom of this distribution is fixed to 5. In the settings of the task, it is also possible to optimize the number of degrees of freedom. In such a case, the default possible values are 1, 2, 5, 10 and 20 degrees of freedom. A distribution with a small number of degree of freedom (i.e. heavy tails) should be avoided in case of stiff ODE’s defined models.
3.7.Algorithms convergence assessment
Monolix includes a convergence assessment tool. It allows to execute a workflow of estimation tasks several times, with different, randomly generated, initial values of fixed effects, as well as different seeds. The goal is to assess the robustness of the convergence.
Running the convergence assessment
For that, click on the shortcut button in the “Tasks” part.
A dedicated panel opens as in the figure below. The first shortcut button next to Run can be used to go back to the estimation.
The user can define

 the number of runs, or replicates
 the type of assessment:
 Estimation of the standard errors and loglikelihood
 Use the linearization method if the previous option is selected.
 the initial parameters. By default, initial values are uniformly drawn from intervals defined around the estimated values if population parameters have been estimated, the initial estimates otherwise. Notice that it is possible to set one initial parameter constant while generating the others. The minimum and maximum of the generated parameters can be modified by the user.
All settings used are saved and reloaded with the run containing the convergence assessment results.
Notice that
 In the case of estimation of the standard errors and loglikelihood by linearization, the individual parameters with the conditional mode method are computed as well to have more relevant linearization.
 In the case of estimation of the standard errors and loglikelihood without the linearization, the conditional distribution method is computed too to have more relevant estimation.
 The workflow is the same between the runs and is not the one defined in the interface.
Click on Run to execute the tool. Thus you are able to estimate the population parameters using several initial seeds and/or several initial conditions.
Display and outputs
Several kinds of plots are given as a summary of the results.
First of all, the SAEM convergence assessment is proposed. The convergence of each parameter on each run is proposed. It allows to see if the convergence for each run is ok.
Then, a plot showing the estimated values for each replicate is proposed. If the estimation of the standard errors was included in the scenario, the estimated standard errors are also displayed as horizontal bars. It allows to see if all parameters converge statistically to the same values.
Starting from the 2019 version, it is possible to export manually all the plots in the Assessment folder in your result folder by clicking on the “export” icon (purple box on the previous figure).
Finally, if loglikelihood without linearization is used, the curves for convergence of importance sampling are proposed.
In addition, a Monolix project and its result folder is generated for each set of initial parameters. They are located in the Assessment subfolder of the main project’s result folder (located by default next to the .mlxtran project file). Along with all the runs, there is a summary of all the runs “assessment.txt” providing all the individual parameter estimates along with the 2LL, as in the following:
Parameters,Run_1,Run_2,Run_3,Run_4,Run_5 Cl_pop,0.03994527,0.04017999,0.04016216,0.04012077,0.0400175 V_pop,0.4575748,0.4556463,0.4560732,0.4557009,0.4569431 a,0.4239969,0.42482,0.4227559,0.4294611,0.435585 b,0.05653124,0.05684357,0.05700663,0.054965,0.05450724 ka_pop,1.527947,1.521184,1.5226,1.519333,1.519678 omega_Cl,0.2653109,0.2643172,0.268475,0.266199,0.2693083 omega_V,0.1293328,0.1274441,0.122951,0.1301242,0.1261098 omega_ka,0.6530206,0.6655251,0.643456,0.6425528,0.6424614 2LL,339.387,339.417,339.429,339.444,339.462
Notice that, starting from the 2019 version, it is possible to reload all the results of a previous convergence if nothing has changed in the project. Starting from version 2021, the settings of the convergence assessment are also reloaded.
Best practices: what is the use the convergence assessment tool?
We cannot claim that SAEM always converges (i.e., with probability 1) to the global maximum of the likelihood. We can only say that it converges under quite general hypotheses to a maximum – global or perhaps local – of the likelihood. A large number of simulation studies have shown that SAEM converges with high probability to a “good” solution – hopefully the global maximum – after a small number of iterations. The purpose of this tool is to evaluate the SAEM algorithm with initial conditions and see if the estimated parameters are the “global” minimum.
The trajectory of the outputs of SAEM depends on the sequence of random numbers used by the algorithm. This sequence is entirely determined by the “seed.” In this way, two runs of SAEM using the same seed will produce exactly the same results. If different seeds are used, the trajectories will be different but convergence occurs to the same solution under quite general hypotheses. However, if the trajectories converge to different solutions, that does not mean that any of these results are “false”. It just means that the model is sensitive to the seed or to the initial conditions. The purpose of this tool is to evaluate the SAEM algorithm with several seeds to see the robustness of the convergence.
3.8.What result files are generated by Monolix?
Monolix generates a lot of different output files depending on the tasks done by the user. Here is a complete listing of the files, along with the condition for their creation and their content.
 Task: Population parameter estimation
 Task: Individual parameter estimation
 Task: Fisher Information Matrix calculation
 Task: Loglikelihood calculation
 Tests
 Tables
 ChartsData
Population parameter estimation
summary.txt
Description: summary file.
Outputs:
 Header: project file name, date and time of run, Monolix version
 Estimation of the population parameters: Estimated population parameters & computation time
populationParameters.txt
Description: estimated population parameters (with SAEM).
Outputs:
 First column (no name): contains the parameter names (e.g ‘V_pop’ and ‘omega_V’).
 value: contains the estimated parameter values.
Individual parameters estimation
All the files are in the IndividualParameters folder of the result folder
estimatedIndividualParameters.txt
Description: Individual parameters (from SAEM, mode, and mean of the conditional distribution)
Outputs:
 ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
 parameterName_SAEM: individual parameter estimated by SAEM, it corresponds to the average of the individual parameters sampled by MCMC during all iterations of the smoothing phase. When several chains are used (see project settings), the average is also done over all chains. This value is an approximation of the conditional mean.
 parameterName_mode (if conditional mode was computed): individual parameter estimated by the conditional mode task, i.e mode of the conditional distribution \(p(\psi_iy_i;\hat{\theta})\).
 parameterName_mean (if conditional distribution was computed) : individual parameter estimated by the conditional distribution task, i.e mean of the conditional distribution \(p(\psi_iy_i;\hat{\theta})\) . The average of samples from all chains and all iterations is computed in the gaussian space (eg mean of the log values in case of a lognormal distribution), and backtransformed.
 parameterName_sd (if conditional distribution was computed): standard deviation of the conditional distribution \(p(\psi_iy_i;\hat{\theta})\) calculated during the conditional distribution task.
 COVname: continuous covariates values corresponding to all data set columns tagged as “Continuous covariate” and all the associated transformed covariates.
 CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.
estimatedRandomEffects.txt
Description: individual random effect, calculated using the population parameters, the covariates and the conditional mode or conditional mean. For instance if we have a parameter defined as \(k_i=k_{pop}+\beta_{k,WT}WT_i+\eta_i\), we calculate \(\eta_i=k_i – k_{pop}\beta_{k,WT}WT_i\) with \(k_i\) the estimated individual parameter (mode or mean of the conditional distribution), \(WT_i\) the individual’s covariate, and \(k_{pop}\) and \(\beta_{k,WT}\) the estimated population parameters.
Outputs:
 ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
 eta_parameterName_SAEM: individual random effect estimated by SAEM, it corresponds to the last iteration of SAEM.
 eta_parameterName_mode (if conditional mode was computed): individual random effect estimated by the conditional mode task, i.e mode of the conditional distribution \(p(\psi_iy_i;\hat{\theta})\).
 eta_parameterName_mean (if conditional distribution was computed) : individual random effect estimated by the conditional distribution task, i.e mean of the conditional distribution \(p(\psi_iy_i;\hat{\theta})\) . The average of random effect samples from all chains and all iterations is computed.
 eta_parameterName_sd (if conditional distribution was computed): standard deviation of the conditional distribution \(p(\psi_iy_i;\hat{\theta})\) calculated during the conditional distribution task.
 COVname: continuous covariates values corresponding to all data set columns tagged as “Continuous covariate” and all the associated transformed covariates.
 CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.
simulatedIndividualParameters.txt
Description: Simulated individual parameter (by the conditional distribution)
Outputs:
 rep: replicate of the simulation
 ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
 parameterName: simulated individual parameter corresponding to the draw rep.
 COVname: continuous covariates values corresponding to all data set columns tagged as “Continuous covariate” and all the associated transformed covariates.
 CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.
simulatedRandomEffects.txt
Description: Simulated individual random effect (by the conditional distribution)
Outputs:
 rep: replicate of the simulation
 ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
 eta_parameterName: simulated individual random effect corresponding to the draw rep.
 COVname: continuous covariates values corresponding to all data set columns tagged as”Continuous covariate” and all the associated transformed covariates.
 CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.
Fisher Information Matrix calculation
summary.txt
Description: summary file.
Outputs:
 Header: project file name, date and time of run, Monolix version (outputted population parameter estimation task)
 Estimation of the population parameters: Estimated population parameters & computation time (outputted population parameter estimation task). Standard errors and relative standard errors are added.
 Correlation matrix of the estimates: correlation matrix by block, eigenvalues and computation time
populationParameters.txt
Description: estimated population parameters, associated standard errors and pvalue.
Outputs:
 First column (no name): contains the parameter names (outputted population parameter estimation task)
 Column ‘parameter’: contains the estimated parameter values (outputted population parameter estimation task)
 se_lin / se_sa: contains the standard errors (s.e.) for the (untransformed) parameter, obtained by linearization of the system (lin) or stochastic approximation (sa).
 rse_lin / rse_sa: contains the parameter relative standard errors (r.s.e.) in % (param_r.s.e. = 100*param_s.e./param), obtained by linearization of the system (lin) or stochastic approximation (sa).
 pvalues_lin / pvalues_sa: for beta parameters associated to covariates, the line contains the pvalue obtained from a Wald test of whether beta=0. If the parameter is not a beta parameter, ‘NaN’ is displayed.
Notice that if the Fisher Information Matrix is difficult to invert, some parameter’s standard error can maybe not be computed leading to NaN in the corresponding columns.
All the more detailed files are in the FisherInformation folder of the result folder.
covarianceEstimatesSA.txt and/or covarianceEstimatesLin.txt
Description: variancecovariance matrix of the estimates for the (untransformed) parameters or transformed normally distributed parameters depending on the MonolixSuite version (see below)
Outputs: matrix with the project parameters as lines and columns. First column contains the parameter names.
The Fisher information matrix (FIM) is calculated for the transformed normally distributed parameters (i.e log(V_pop) if V has a lognormal distribution). By inverting the FIM, we obtain the variancecovariance matrix \(\Gamma\) for the transformed normally distributed parameters \(\zeta\). This matrix is then multiplied by the jacobian J (which elements are defined by \(J_{ij}=\frac{\partial\theta_i}{\partial\zeta_j}\)) to obtain the variancecovariance matrix \(\tilde{\Gamma}\) for the untransformed parameters \(\theta\):
$$\tilde{\Gamma}=J^T\Gamma J$$
The diagonal elements of the variancecovariance matrix \(\tilde{\Gamma}\) for the untransformed parameters are finally used to calculate the standard errors.
correlationEstimatesSA.txt and/or correlationEstimatesLin.txt
Description: correlation matrix for the (untransformed) parameters
Outputs: matrix with the project parameters as lines and columns. First column contains the parameter names.
The correlation matrix is calculated as:
$$\text{corr}(\theta_i,\theta_j)=\frac{\text{covar}(\theta_i,\theta_j)}{\sqrt{\text{var}(\theta_i)}\sqrt{\text{var}(\theta_j)}}$$
This implies that the diagonal is unitary. The variancecovariance matrix for the untransformed parameters \(\theta\) is obtained from the inverse of the Fisher Information Matrix and the jacobian. See above for the formula.
LogLikelihood calculation
summary.txt
Description: summary file.
Outputs:
 Header: project file name, date and time of run, Monolix version (outputted population parameter estimation task)
 Estimation of the population parameters: Estimated population parameters & computation time (outputted population parameter estimation task). Standard errors and relative standard errors are added.
 Correlation matrix of the estimates: correlation matrix by block, eigenvalues and computation time
 Loglikelihood Estimation: 2*loglikelihood, AIC and BIC values, together with the computation time
All the more detailed files are in the LogLikelihood folder of the result folder
logLikelihood.txt
Description: Summary of the loglikelihood calculation with the two methods.
Outputs:
 criteria: OFV (Objective Function Value), AIC (Akaike Information Criteria), and BIC (Bayesian Information Criteria )
 method: ImportanceSampling and/or linearization
individualLL.txt
Description: 2LL for each individual. Notice that we only have one by individual even if there are occasions.
Outputs:
 ID: subject name
 method: ImportanceSampling and/or linearization
Tests
Tables
predictions.txt
Description: predictions at the observation times
Outputs:
 ID: subject name. If there are occasions, additional columns will be added to describe the occasions.
 time: Time from the data set.
 MeasurementName: Measurement from the data set.
 RegressorName: Regressor value.
 popPred_medianCOV: prediction using the population parameters and the median covariates.
 popPred: prediction using the population parameters and the covariates, e.g \(V_i=V_{pop}\left(\frac{WT_i}{70}\right)^{\beta}\) (without random effects).
 indivPred_SAEM: prediction using the mean of the conditional distribution, calculated using the last iterations of the SAEM algorithm.
 indPred_mean (if conditional distribution was computed): prediction using the mean of the conditional distribution, calculated in the Conditional distribution task.
 indPred_mode (if conditional mode was computed): prediction using the mode of the conditional distribution, calculated in the EBEs task.
 indWRes_SAEM: weighted residuals \(IWRES_{ij}=\frac{y_{ij}f(t_{ij}, \psi_i)}{g(t_{ij}, \psi_i)}\) with \(\psi_i\) the mean of the conditional distribution, calculated using the last iterations of the SAEM algorithm.
 indWRes_mean (if conditional distribution was computed): weighted residuals \(IWRES_{ij}=\frac{y_{ij}f(t_{ij}, \psi_i)}{g(t_{ij}, \psi_i)}\) with \(\psi_i\) the mean of the conditional distribution, calculated in the Conditional distribution task.
 indWRes_mode (if conditional mode was computed): weighted residuals \(IWRES_{ij}=\frac{y_{ij}f(t_{ij}, \psi_i)}{g(t_{ij}, \psi_i)}\) with \(\psi_i\) the mode of the conditional distribution, calculated in the EBEs task.
Notice that in case of several outputs, Monolix generates predictions1.txt, predictions2.txt, …
Below is a correspondence of the terms used for predictions in Nonmem versus Monolix:
Charts data
All plots generated by Monolix can be exported as a figure or as text files in order to be able to plot it in another way or with other software for more flexibility. The description of all generated text files is described here.
3.9.Tests
Several statistical tests may be automatically performed to test the different components of the model. These tests use individual parameters drawn from the conditional distribution, which means that you need to run the task “Conditional distribution” in order to get these results. In addition, the tests for the residuals require to have first generated the residuals diagnostic plots (scatter plot or distribution).
The tests are all performed using the individual parameters sampled from the conditional distribution (or the random effects and residuals derived thereof). They are thus not subject to bias in case of shrinkage. For each individual, several samples from the conditional distribution may be used. The used tests include a correction to take into account that these samples are correlated among each other.
Results of the tests are available in the tab “Results” and selecting “Tests” in the left menu
The model for the individual parameters
Consider a PK example (warfarin data set from the demos) with the following model for the individual PK parameters (ka, V, Cl):
In this example, the different assumptions we make about the model are:
 The 3 parameters are lognormally distributed
 ka is function of sex
 V is function of sex and weight. More precisely, the logvolume log(V) is a linear function of the logweight \({\rm lw70 }= \log({\rm wt}/70)\).
 Cl is not function of any of the covariates.
 The random effects \(\eta_V\) and \(\eta_{Cl}\) are linearly correlated
 \(\eta_{ka}\) is not correlated with \(\eta_V\) and \(\eta_{Cl}\)
Let’s see how each of these assumptions are tested:
Covariate model
Individual parameters vs covariates – Test whether covariates should be removed from the model
If an individual parameter is function of a continuous covariate, the linear correlation between the transformed parameter and the covariate is not 0 and the associated \(\beta\) coefficient is not 0 either. To detect covariates bringing redundant information, we check if these beta coefficients are different from 0. For this, we perform two different tests: a correlation test based on betas coefficients estimated with a linear regression, and a Wald test relying on the estimated population parameters and their standard error.
In both cases, a small pvalue indicates that the null hypothesis can be rejected and thus that the estimated beta parameter is significantly different from zero. If this is the case, the covariate should be kept in the model. On the opposite, if the pvalue is large, the null hypothesis cannot be rejected and this suggests to remove the covariate from the model. Note that if beta is equal to zero, then the covariate has no impact on the parameter. High pvalues are colored in yellow (pvalue in [0.010.05]), orange (pvalue in [0.050.10]) or red (pvalue in [0.101]) to draw attention on parametercovariate relationships that can be removed from the model from a statistical point of view.
Correlation test
Briefly, we perform a linear regression between the covariates and the transformed parameters and test if the resulting beta coefficient are different from 0.
More precisely: for each individual \(i\), let \(z_i^l\) be the transformed individual parameters (e.g log(V) for lognormally distributed parameters or logit(F) for logitdistributed parameters) sampled from the conditional distribution (called replicates, index l). Here we will call covariates all continuous covariates and all nonreferent categories of categorical covariates \(cov^{(c)}, c = 1..n_C\). For each individual \(i\), \(cov_i^{(c)}\) is the value of the c\(^{th}\) covariate, equal to 0 or 1 if the covariate is a category.
The transformed individual parameters are first averaged over replicates for each individual:
$$z_i^{(L)}=\frac{1}{L} \sum_{l=1}^{L} z_i^l$$
We then perform the following linear regression:
$$z_i^{(L)}=\alpha_0 + \sum_{c=1}^{n_c} \beta_c \text{cov}_i^{(c)} + e_i$$
If two covariates \(cov^{(1)}\) and \(cov^{(2)}\) (for example WT and BMI) are strongly correlated with a parameter \(z^{(L)}\) (for example the volume), only one of them is needed in the model because they are redundant. In the linear regression, only one of the estimated \(\hat{\beta_1}\) and \(\hat{\beta_2}\) will be significantly different from zero.
For each covariate \(c\) we conduct a ttest on the \(\hat{\beta_c}\) with the null hypothesis:
H0: \(\beta_c = 0\).
The test statistic is
$$T_0 = \frac{\hat{\beta_c}}{se(\hat{\beta_c})}$$
where \(se(\hat{\beta_c})\) is the estimated standard error of \(\hat{\beta_c}\) (obtained by least squares estimation during the regression). If the null hypothesis is true, \(T_0\) follows a t distribution with \(Nn_C1\) degrees of freedom (where N is the number of individuals).
In our example, the correlation test suggests to remove sex from ka:
Wald test
The Wald test relies on the standard errors. Thus the task “Standard errors” must have been calculated to see the test results. The test can be performed using the standard errors calculated using either the “linearization method” (indicated as “linearization”) or not (indicated as “stochastic approximation” in the tests).
The Wald test tests the following null hypothesis:
H0: the beta parameter estimated by SAEM is equal to zero.
The math behind: Let’s note \( \hat{\beta} \) the estimated beta value (which is a population parameter) and \(se(\hat{\beta}) \) the associated standard error calculated during the task “Standard errors”. The Wald test statistic is:
$$W=\frac{\hat{\beta}}{se(\hat{\beta})} $$
The test statistic is compared to a tdistribution with 1 degree of freedom.
In our example, the Wald test suggests to remove sex from ka and V:
Remark: the Wald test and the correlation test may suggest different covariates to keep or remove. Note that the null hypothesis tested is not the same.
Random effects vs covariates – Test whether covariates should be added to the model
Pearson’s correlation tests and ANOVA are performed to check if some relationships between random effects and covariates not yet included in the model should be added to the model.
For continuous covariates, the Pearson’s correlation test tests the following null hypothesis:
H0: the person correlation coefficient between the random effects (calculated from the individual parameters sampled from the conditional distribution) and the covariate values is zero
For categorical covariates, the oneway ANOVA tests the following nullhypothesis:
H0: the mean of the random effects (calculated from the individual parameters sampled from the conditional distribution) is the same for each category of the categorical covariate
A small pvalue indicates that the null hypothesis can be rejected and thus that the correlation between the random effects and the covariate values is significant. If this is the case, it is probably worth considering to add the covariate in the model. Note that the decision of adding a covariate in the model should not only be driven by statistical considerations but also biological relevance. Note also that for parametercovariate relationships already included in the model, the correlation between the random effects and covariates is not significant (while the correlation between the parameter and the covariate can be – see above). Small pvalues are colored in yellow (pvalue in [0.050.10]), orange (pvalue in [0.010.05]) or red (pvalue in [0.000.01]) to draw attention on parametercovariate relationships that can be considered for addition in the model from a statistical point of view.
In our example, we already have sex on ka and V, and lw70 on V in the model. The only remaining relationship that could possibly be worth investigating is between weight (or the logtransformed weight “lw70”) and clearance.
The math behind:
Continuous covariate: Let \(\eta_i^l\) the random effects corresponding to the \(L\) individual parameters sampled from the conditional distribution (called replicates) for individual \(i\), and \(cov_i\) the covariate value for individual \(i\). The random effects are first averaged over replicates for each individual:
$$ \eta_i^{(L)}=\frac{1}{L} \sum_{l=1}^{L} \eta_i^l $$
We note \(\overline{cov} = \sum_{i=1}^N cov_i \) the average covariate value over the N subjects and \(\overline{\eta}=\sum_{i=1}^N \eta_i^{(L)} \) the average random effect. The Pearson correlation coefficient is calculated as:
$$r=\frac{\sum_{i=1}^N(cov_i – \overline{cov})(\eta_i^{(L)} – \overline{\eta})}{\sqrt{ \sum_{i=1}^N(cov_i – \overline{cov})^2 \sum_{i=1}^N(\eta_i^{(L)} – \overline{\eta})^2}}$$
The test statistic is:
$$t=\frac{r}{\sqrt{1r^2}}\sqrt{N2}$$
and it is compared to a tdistribution with \(N2\) degrees of freedom with \(N\) the number of individuals.
Categorical covariates: The random effects are first averaged over replicates for each individual and a oneway analysis of variance is performed (simplified to a ttest when the covariate has only two categories).
The model for the random effects
Distribution of the random effects – Test if the random effects are normally distributed
In the individual model, the distributions for the parameters assume that the random effects follow a normal distribution. ShapiroWilk tests are performed to test this hypothesis. The null hypothesis is:
H0: the random effects are normally distributed
If the pvalue is small, there is evidence that the random effects are not normally distributed and this calls the choice of the individual model (parameter distribution and covariates) into question. Small pvalues are colored in yellow (pvalue in [0.050.10]), orange (pvalue in [0.010.05]) or red (pvalue in [0.000.01]).
In our example, there is no reason to reject the nullhypothesis and no reason to question the chosen lognormal distributions for the parameters.
The math behind: Let \(\eta_i^l\) the random effects corresponding to the \(L\) individual parameters sampled from the conditional distribution (called replicates) for individual \(i\). The ShapiroWilk test statistic is calculated for each replicate \(l\) (i.e the first sample from all individuals, then the second sample from all individuals, etc):
$$W^l=\frac{\left( \sum_{i=1}^N a_i \eta_i^l \right)^2}{ \sum_{i=1}^N (\eta_i^l – \overline{\eta}^l)^2}$$
with \(a_i\) tabulated coefficient and \(\overline{\eta}^l=\frac{1}{N}\sum_{i=1}^N \eta_i^l \) the average over all individuals, for each replicate.
The statistic displayed in Monolix corresponds to the average statistic over all replicates \(W=\frac{1}{L}\sum_{l=1}^L W^l \). For the pvalues, one pvalue is calculated for each replicate, using the ShapiroWild table with \(N\) (number of individuals) degrees of freedom. The BenjaminiHochberg (BH) procedure is then applied: the pvalues are ranked by ascending order and the BH critical value is calculated for each as \( \frac{\textrm{rank}}{L}Q \) with \(\textrm{rank}\) the individual pvalue’s rank, \(L\) the total number of pvalues (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest pvalue that is smaller than the corresponding critical value is selected.
Joint distribution of the random effects – Test if the random effects are correlated
Correlation tests are performed to test if the random effects (calculated from the individual parameters sampled from the conditional distribution) are correlated. The nullhypothesis is:
H0: the expectation of the product of the random effects of the first and second parameter is zero
The nullhypothesis is assessed using a ttest.
Remark: In the 2018 version, a Pearson correlation test was used.
For correlations not yet included in the model, a small pvalue indicates that there is a significant correlation between the random effects of two parameters and that this correlation should be estimated as part of the model (otherwise simulations from the model will assume that the random effects of the two parameters are not correlated, which is not what is observed for the random effects estimated using the data). Small pvalues are colored in yellow (pvalue in [0.050.10]), orange (pvalue in [0.010.05]) or red (pvalue in [0.000.01]).
For correlations already included in the model, a large pvalue indicates that one cannot reject the hypothesis that the correlation between the random effects is zero. If the correlation is not significantly different from zero, it may not be worth estimating it in the model. High pvalues are colored in yellow (pvalue in [0.010.05]), orange (pvalue in [0.050.10]) or red (pvalue in [0.101])
In our example, we have assumed in the model that \(\eta_V\) and \(\eta_{Cl}\) are correlated. The high pvalue (0.033, above the 0.01 threshold, see above) indicates that the correlation between the random effects of V and Cl is not significantly different from zero and suggests to remove this correlation from the model.
Remark: as correlations can only be estimated by groups (i.e if a correlation is estimated between (ka, V) and between (V, Cl), then one must also estimate the correlation between (ka, Cl)), it may happen that it is not possible to remove a nonsignificant correlation without removing also a significant one.
The math behind: Let \(\eta_{\psi_1,i}^l\) and \(\eta_{\psi_2,i}^l\) the random effects corresponding to the \(L\) individual parameters \(\psi_1\) and \(\psi_2\) sampled from the conditional distribution (called replicates) for individual \(i\). First we calculate the product of the random effects averaged over the replicates:
$$p_i^{(L)} = \frac{1}{L} \sum_{l=1}^{L} \eta_{\psi_1,i}^l \eta_{\psi_2,i}^l $$
We note \( \overline{p}=\sum_{i=1}^{N} p_i^{(L)} \) the average of the product over the individuals and \(s\) their standard deviation. The test statistic is:
$$ T=\frac{\overline{p}}{\frac{s}{\sqrt{N}}}$$
and it is compared to a tdistribution with \(N1\) degrees of freedom with \(N\) the number of individuals.
The distribution of the individual parameters
Distribution of the individual parameters not dependent on covariates – Test if transformed individual parameters are normally distributed
When an individual parameter doesn’t depend on covariates, its distribution (normal, lognormal, logit or probit) can be transformed into the normal distribution. Then, a ShapiroWilk test can be used to test the normality of the transformed parameter. The null hypothesis is:
H0: the transformed individual parameter values (sampled from the conditional distribution) is normally distributed
If the pvalue is small, there is evidence that the transformed individual parameter values are not normally distributed and this calls the choice of the parameter distribution into question. Small pvalues are colored in yellow (pvalue in [0.050.10]), orange (pvalue in [0.010.05]) or red (pvalue in [0.000.01]).
In our example, there is no reason to reject the null hypothesis of lognormality for Cl.
Remark: testing the normality of a transformed individual parameter that does not depend on covariates is equivalent to testing the normality of the associated random effect. We can check in our example that the ShapiroWilk tests for \(\log(Cl)\) and \(\eta_{Cl}\) are equivalent.
The math behind: Let \(z_i^l\) the transformed individual parameters (e.g log(V) for lognormally distributed parameters and logit(F) for logitdistributed parameters) sampled from the conditional distribution (called replicates, index \(l\) ) for individual \(i\). The ShapiroWilk test statistic is calculated for each replicate \(l\) (i.e the first sample from all individuals, then the second sample from all individuals, etc):
$$W^l=\frac{\left( \sum_{i=1}^N a_i z_i^l \right)^2}{ \sum_{i=1}^N (z_i^l – \overline{z}^l)^2}$$
with \(a_i\) tabulated coefficient and \(\overline{z}^l=\frac{1}{N}\sum_{i=1}^N z_i^l \) the average over all individuals, for each replicate.
The statistic displayed in Monolix corresponds to the average statistic over all replicates \(W=\frac{1}{L}\sum_{l=1}^L W^l \). For the pvalues, one pvalue is calculated for each replicate, using the ShapiroWild table with \(N\) (number of individuals) degrees of freedom. The BenjaminiHochberg (BH) procedure is then applied: the pvalues are ranked by ascending order and the BH critical value is calculated for each as \( \frac{\textrm{rank}}{L}Q \) with \(\textrm{rank}\) the individual pvalue’s rank, \(L\) the total number of pvalues (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest pvalue that is smaller than the corresponding critical value is selected.
Distribution of the individual parameters dependent on covariates – test the marginal distribution of each individual parameter
Individual parameters that depend on covariates are not anymore identically distributed. Each transformed individual parameter is normally distributed, with its own mean that depends on the value of the individual covariate. In other words, the distribution of an individual parameter is a mixture of (transformed) normal distributions. A KolmogorovSmirnov test is used for testing the distributional adequacy of these individual parameters. The nullhypothesis is:
H0: the individual parameters are samples from the mixture of transformed normal distributions (defined by the population parameters and the covariate values)
A small pvalue indicates that the null hypothesis can be rejected. Small pvalues are colored in yellow (pvalue in [0.050.10]), orange (pvalue in [0.010.05]) or red (pvalue in [0.000.01]).
With our example, we obtain:
The model for the observations
A combined1 error model with a normal distribution is assumed in our example:
Distribution of the residuals
Several tests are performed for the individual residuals (IWRES), the NPDE and for the population residuals (PWRES).
Test if the distribution of the residuals is symmetrical around 0
A Miao, Gel and Gastwirth (2006) test (or Van Der Waerden test in the 2018 release) is used to test the symmetry of the residuals. Indeed, symmetry of the residuals around 0 is an important property that deserves to be tested, in order to decide, for instance, if some transformation of the observations should be done. The null hypothesis tested is:
H0: the median of the residuals is equal to its mean
A small pvalue indicates that the null hypothesis can be rejected. Small pvalues are colored in yellow (pvalue in [0.050.10]), orange (pvalue in [0.010.05]) or red (pvalue in [0.000.01]).
With our example, we obtain:
The math behind: Let \(R_i\) the residuals (NPDE, PWRES or IWRES) for each individual \(i\), \(\overline{R}\) the mean of the residuals, and \(M_R\) their median. The MGG test statistic is:
$$T=\frac{\sqrt{n}}{0.9468922}\frac{\overline{R}M_R}{ \sum_{i=1}^{n}R_iM}$$
with \(n\) the number of residuals. The test statistic is compared to a standard normal distribution.
The formula above is valid for i.i.d (independent and identically distributed) residuals. For the IWRES, the residuals corresponding to a given time and given id are not independent (they ressemble each other). To solve the problem, we estimate an effective number of residuals. The number of residuals \(n\) can be split into the number of replicates \(L\) times the number of observations \(m\). We look for the effective number of replicates \(\tilde{L}\) such that:
$$ \frac{\tilde{L}}{L} \sum_{l=1}^L (R_i^l)^2 \approx \chi^2(\tilde{L})$$
using a maximum likelihood estimation. The number of residuals is then calculated as \(n=\tilde{L} \times m \).
Test if the residuals are normally distributed
A Shapiro Wilk test is used for testing the normality of the residuals. The null hypothesis is:
H0: the residuals are normally distributed
If the pvalue is small, there is evidence that the residuals are not normally distributed. The Shapiro Wilk test is known to be very powerful. Then, a small deviation of the empirical distribution from the normal distribution may lead to a very significant test (i.e. a very small pvalue), which does not necessarily means that the model should be rejected. Thus, no color highlight is made for this test.
In our example, we obtain:
The math behind: Let \(R_i^l\) the residuals (NPDE, PWRES or IWRES) for individual \(i\). NPDE and PWRES have one values per time points and per individual. IWRES have one value per time point, per individual and per replicate (corresponding to the \(L\) individual parameters sampled from the conditional distribution). The ShapiroWilk test statistic is calculated for each replicate \(l\) (i.e the first sample from all individuals, then the second sample from all individuals, etc):
$$W^l=\frac{\left( \sum_{i=1}^N a_i R_i^l \right)^2}{ \sum_{i=1}^N (R_i^l – \overline{R}^l)^2}$$
with \(a_i\) tabulated coefficient and \(\overline{R}^l=\frac{1}{N}\sum_{i=1}^N R_i^l \) the average over all individuals, for each replicate.
The statistic displayed in Monolix corresponds to the average statistic over all replicates \(W=\frac{1}{L}\sum_{l=1}^L W^l \). For the pvalues, one pvalue is calculated for each replicate, using the ShapiroWild table with \(N\) (number of individuals) degrees of freedom. The BenjaminiHochberg (BH) procedure is then applied: the pvalues are ranked by ascending order and the BH critical value is calculated for each as \( \frac{\textrm{rank}}{L}Q \) with \(\textrm{rank}\) the individual pvalue’s rank, \(L\) the total number of pvalues (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest pvalue that is smaller than the corresponding critical value is selected.
4.Plots
What kind of plots can be generated by Monolix?
The list of plots below corresponds to all the plots that Monolix can generate. They are computed with the task “Plots”, and the list of plots to compute can be selected by clicking on the button next to the task as shown below, prior to running the task.
By default, only the subset of plots are selected, as one can see on this figure. Plots can be selected or unselected onebyone, by groups or all at once.
In addition to selecting plots, this menu can be used to directly generate one particular plot, by clicking on the green arrow next to it, as can be seen below. The green arrow is not visible if the required information for the chosen plot has not been computed yet. For example, generating the plot “likelihood contribution” requires first to run the “Likelihood” task.
Data
 Observed data: This plot displays the original data w.r.t. time as a spaghetti plot, along with some additional information.
Model for the observations
 Individual fits: This plot displays the individual fits: individual predictions using the individual parameters and the individual covariates w.r.t. time on a continuous grid, with the observed data overlaid.
 Observations vs predictions: This plot displays observations w.r.t. the predictions computed using the population parameters or the individual parameters.
 Scatter plot of the residuals: This plot displays the PWRES (population weighted residuals), the IWRES (individual weighted residuals), and the NPDE (Normalized Prediction Distribution Errors) w.r.t. the time and the prediction.
 Distribution of the residuals: This plot displays the distributions of PWRES, IWRES and NPDE as histograms for the probability density function (PDF) or as cumulative distribution functions (CDF).
Diagnosis plots based on individual parameters
 Distribution of the individual parameters: This plot displays the estimated population distributions of the individual parameters.
 Distribution of the random effects: This plot displays the distribution of the random effects.
 Correlation between random effects: This plot displays scatter plots for each pair of random effects.
 Individual parameters vs covariates: This plot displays the estimators of the individual parameters in the Gaussian space (and those for random effects) w.r.t. the covariates.
Predictive checks and predictions
 Visual predictive checks: This plot displays the Visual Predictive Check.
 Numerical predictive checks: This plot displays the numerical predictive check.
 BLQ predictive checks: This plot displays the proportion of censored data w.r.t. time.
 Prediction distribution: This plot displays the prediction distribution.
Convergence diagnosis
 SAEM: This plot displays the convergence of the population parameters estimated with SAEM with respect to the iteration number.
 MCMC: This plot displays the convergence of the Markov Chain Monte Carlo algorithm for the individual parameters estimation.
 Importance sampling: This plot displays the convergence of loglikelihood estimation by importance sampling.
Tasks results
 Likelihood contribution: This plot displays the contribution of each individual to the loglikelihood.
 Standard errors for the estimates: This plot displays the relative standard errors (in %) for the population parameters.
Saving plots
The user can choose to export each plot as an image with an icon on top of it, or all plots at once with the menu Export. It is also possible to export plots data as table, for example to build new plots with external tools.
Note that:
 the export starts after the display of the plots,
 the plots are exported in the result folder,
 only plot selected in Plots tasks are exported,
 legends and information frames are not exported.
Automatic exporting can be chosen in the project Preferences (in Settings), as well as the exporting format (png or svg):
4.1.Data
4.1.1.Observed data
It is always good to have a look first at the observed data before running the parameter estimation. Indeed, it is very convenient to see if all the data is consistent, or if some outliers appear. Moreover, looking at the plot can help to identify hypotheses about the model, such as covariate effects. Three types of data can be visualized in Monolix using the graphical interface: continuous data, dicrete data and timetoevent data.
Continuous data
The purpose of this plot, also called a spaghetti plot, is to display the observed data (from a dataset loaded in Monolix) w.r.t. time. It is available in the PLOTS tab after loading and accepting a dataset in the DATA tab. You can access data for different observation in the left panel.
Settings in the right panel allow to adapt the plots display, full description is here.
General settings
General settings include: legend, display of grid, information about the data and dosing times.
Information adds a box with summary of the data and is recomputed for each subplot after splitting by a covariate. It includes:
 The total number of subjects
 The average number of doses per subject
 The total, average, minimum and maximum number of observations per individual.
Option “dosing times” adds the individual dosing time as dashed vertical lines. You can make the lines always visible or only when you hover on individual data. Dosing times corresponding to doses with null amounts are not displayed.
Display
You can display data as dots and lines (by default), or only as dots or only as lines, as in the figure below. Toggle “Mean” overlays a trend line on the plot, based on mean values of the observed data pooled in bins: arithmetic or geometric mean. Toggle “error bars” adds error bars as either the standard deviations or standard errors. The mean and error values can be displayed as fixed labels next to the bars, or as tooltips when hovering on a bar.
It is possible to display the mean curves, split by covariate, in a single plot. After split by a covariate, switch on the “Merged splits” option. In the example below, the mean curves of the two weight groups, which were in separate plots, are merged into a single plot. This feature is available for versions Monolix 2023 and above.
Bins
This section contains settings related to the definition of bins used in the computation of plot statistics. Toggle “bin limits” add vertical lines corresponding to the limits. In this section you can change the number of bins as well as binning criteria.
Axes
This section contains settings related to the display of X and Y – axis.
 Log scale toggle applies logarithm on a selected axis, for example to have a better evaluation of the elimination part.
 Tick values “Auto” applies automatic axis ticks – if the toggle is enabled – or with a custom step – if the toggle is disabled.
 Label is an editable text displayed on each axis.
 Custom limits, when enabled, allows to choose the axis limits manually. They are applied to all subplots obtained by splitting by covariates.
Discrete data
Monolix uses also count or categorical data, see here for detailed examples. Observed data can be shown as continuous, stacked or grouped. Example below shows the evolution of scores, which are categories describing anxious disorders, from the zylkenedataset. The xaxis is time of measurements, while yaxis shows number of individuals in each category (different colors) at each time point. You can display the number of individuals for each category by hovering on it in the plot.
Settings, stratification and preferences work as for continuous data.
Timetoevent data
In Monolix, timetoevent data is displayed via a survival function, which describes the probability that an event happens after some time t. In general, this function is unknown and Monolix uses the nonparametric KaplanMeier estimator. It describes the probability that an individual survives until time t, knowing that it survived at any earlier time.
It is possible to display the Kaplan – Meier plot split by covariate in a single plot. After split by a covariate in the Stratify subtab, switch on the “Merged splits” option. In the example below, the curves of the four cell type groups, which were in separate plots, are merged into a single plot.
Kaplan – Meier estimator
For a single event data, the KaplanMeier estimator is given by the following formula
\( \hat{S}(t)=\sum_{i:t_i<t} \left(1d_i/n_i\right),\)
where
 (t_i) – times before t, when at least one event occurred,
 (d_i) – number of events at the time (t_i),
 (n_i) – number of individuals at risk, that is who did not experience an event until (t_i).
The probability that an event occurs ((p_e)) is the ratio between the number of events that has occurred ((d_i)) and the total number of individuals at risk ((n_i)). The complement of it, ((1p_e)), gives an estimation of the survival. For each time t the total number of individuals at risk changes, so the probabilities at all previous times (t_i), when at least one event occurred, are multiplied. It is similar to calculating the probability that a patient survives 2 days. It is a product of a probability that a patient survives the first day and a conditional probability that it survives the second day, knowing that it survived the first one.
Example:
A typical example of a timetoevent data set contains information about exact times when individuals experienced an event or when they left a study (dropout). In the following, there are five individuals, who have two observations: time when the observation starts, which is 0 for all, and time of an event. If a patient leaves a study, then the time of a dropout is given but instead of 1 in the column for the observation, there is 0. It indicates that this individual didn’t experience an event but survived until the dropout time.The advantage of the KaplanMeier estimate is that it takes into account situations when not all individuals continue the study. At the next event time, such individuals are not counted as individuals at risk (they are not counted in the denominator (n_i)).
A study starts at time (t_1). There are no events, so (d_1=0) and the value of the survival curve is 1. Until the next event time at (t_2=1), the survival remains constant. Then, one individual experienced an event, so (d_2=1), and all individuals survived until that time, so (n_2=5). The result is that the probability to survive decreases by 0.2, which corresponds to the height of the jump at (t=1) in the plot. Then again, until the next event, survival remains constant. At time (t=3), there are two events. The number (n_3) counts now only 4 individuals – it has decreased by 1 due to the previous event. To get the final value probability at time 3is multiplied by all earlier probabilities. At (t=4) there is a dropout. Patient 5 left the study and no event was registered. The survival curve remains constant, and the dropout is marked in red. The KaplanMeier estimator takes into account this situations, because at the next event time (t=5), this individual is not counted as an individual at risk – denominator n will be smaller. At time (t=5), there is only one individual left, and one event, so the survival equals 0.
Remarks
 KaplanMeier estimator handles correctly information about individuals who left the study, but there is a bias when the exact times of events are unknown.
 In data visualization, Monolix assumes that all events are exactly observed. For example: assume that an observation period started at (t=0) and at (t=1) an event is marked by 1 in the column for the observation. It is impossible to distinguish in the dataset, without any other information, if the event was exactly at (t=1) or before. The same problem is when a time of the beginning of the study and time interval limits of an event are given. Just looking at the data set, an exact and interval censored event type are indistinguishable. In other words, not knowing when an event happened, Monolix assumes that it happened at the end of the censored interval.
Mean number of events.
The KaplanMeier estimator can be used also for the analysis of repeated events. The survival curve is estimated for each kth event separately ( \hat{S}^{(k)}(t) = \sum_{i:t_i<t} \left(1\frac{d^{(k)}i}{n^{(k)}_i}\right)), and is used to calculate the mean number of events per individual as a function of time (\hat{m}(t) = \sum{k} \left(1\hat{S}^{(k)}(t)\right).) It can be visualized next to the Survival function by choosing this option from the Subplots settings:
Settings
 General: Add/remove the legend, the grid, data information and dosing times; constrains on zoom
 Display: add/remove data dots, lines, mean and error bars
 Bins: display bin limits, binning criteria, number of bins
 Axes: Add/remove logscale, modify labels, set tick values, set custom axis limits
 Stratify: Split, color and filter by covariates,
 Preferences: Add/remove elements or change colors and sizes for axes, observations, censored (BLQ) observations, highlighting.
Troubleshooting
If the Observed data plot is empty or with nonappropriate axis limits, follow the procedure below.
The problem appears after having clicked “Export > Export charts settings as default” on a previous project where the yaxis limits were different and this is now applied as default. It is possible to delete the default setting corresponding to the axis limits in the following way:

 Open the file C:/Users/<username>/lixoft/monolix/monolix2023R1/config/settings.default in a text editor
 Delete the following lines:
VPCContinuous\yInterval= ... outputPlot\yInterval= ...
 Save the file
 Reopen your project
4.2.Model for the observations
4.2.1.Individual fits
Purpose
The figure displays the observed data for each subject, as well as two curves from simulations using the design and the covariates of each subject:
 the predicted profile given by the estimated population model (Population fits),
 the predicted profile given by the estimated individual model (Individual fits). If the EBEs and/or the conditional distribution tasks were performed, the user can choose either the conditional means or the conditional modes, estimated by MCMC, as estimators. Otherwise, approximations of the conditional means from SAEM are used.
This is a good way to see on each subject the validity of the model, and the actual fit proposed, as well as the interindividual variability in the kinetics. It is possible to show the computed individual parameters on the figure. Moreover, it is also possible to display an individual predictive check: the median and a confidence interval for () estimated with a Monte Carlo procedure.
Examples

Individual fits and population fits
In the example below, the concentration for the theophylline data set is shown with simulations of a onecompartment model with firstorder absorption and linear elimination. For each subject, the data are displayed with blue points along with the individual fit and population fit (the prediction using the estimated individual and population parameters respectively).

Individual parameters
Information on individual parameters can be used in two ways, as shown below. By clicking on Information (marked in green on the figure) in the General panel, individual parameter values can be displayed on each individual plot. Moreover, the plots can be sorted according to the values for a given parameter, in ascending or descending order (Sorting panel marked in orange). By default, the individual plots are sorted by subject id, with the same order as in the data set.

Individual predictive check
Individual predictive checks can be added to the plots: for each individual a prediction interval is computed based on multiple simulations with the population parameters and the design structure of this individual. The median line of the interval is also drawn. The interval allows to check whether the observed data are compatible with the population prediction, taking into account the interindividual variability. The example below shows that the first subject in the theophylline data set show too much variability from the rest of the population to be correctly described by the population model.

Dosing times
Dosing times can also be overlayed, which is useful to visualize the effect of doses on the prediction. As an example, the following figure shows the observations of an individual from the tobramycin data set along with the corresponding individual fit and multiple dosing times. Starting in version 2021R1 on, dosing times corresponding to doses with null amounts are not displayed.

Special zoom
Userdefined constraints for the zoom are available. They allow to zoom in according to one axis only instead of both axes. Moreover, a link between plots can be set in order to perform a linked zoom on all individual plots at once. This is shown on the figure below with observations from the remifentanil example, and individual fits from a twocompartment model. It is thus possible to focus on the same time range or observation values for all individuals. In this example it is used to zoom on time on the elimination phase for all individuals, while keeping the Y axis in log scale unchanged for each plot.

Censored data
When a data is censored, this data is different to a “classical” observation and has thus a different representation. We represent it as a bar from the censored value specified in the data set and the associated limit.
If there is no limit column then is goes to Infinity as in the following example. However, in any case, the user can choose the limit of the plot.
Settings
 Grid arrange. The user can define the number of subjects that are displayed, as well as the number of rows and the number of columns. Moreover, a slider is present to be able to change the subjects under consideration.
 General
 Legend: hide/show the legend. The legends adapts automatically to the elements displayed on the plot. The same legend box applies to all subplots and it is possible to drag and drop the legend at the desired place.
 Grid : hide/show the grid in the background of the plots.
 Information: hide/show the individual parameter values for each subject (conditional mode or conditional mean depending on the “Individual estimates” choice is the setting section “Display”).
 Dosing times: hide/show dosing times as vertical lines for each subject.
 Link between plots: activate the linked zoom for all subplots. The same zooming region can be applied on all individuals only on the xaxis, only on the Yaxis or on both (option “none”).
 Display
 Observed data: hide/show the observed data.
 Censored intervals [if censored data present]: hide/show the data marked as censored (BLQ), shown as a rectangle representing the censoring interval (for instance [0, LOQ]).
 Split occasions [if IOV present]: Split the individual subplots by occasions in case of IOV.
 Individual fits: Model prediction for each individual using the subject’s design and the individual parameters. The individual parameters can be the conditional mode or the conditional mean depending on the choice in the “Individual estimates” section.
 Population fits [if no covariates in the model]: Model prediction for each individual using the subject’s design and the population parameters.
 Population fits (individual covariates) [if covariates present in the model]: Model prediction for each individual using the subject’s design, the population parameters and the individual covariates values.
 Population fits (population covariates) [if covariates present in the model]: Model prediction for each individual using the subject’s design, the population parameters and the median covariates values (median from all individuals of the data set).
 Individual estimates [if EBEs task has run]: depending on the tasks that have been calculated, choice between conditional mode (given by EBEs task), conditional mean (approximation given by the population parameter estimated task) or conditional mean (given by the conditional distribution task).
 Individual predictive check: For each individual, 500 (see “number of simulations” in the PLOTS task settings) data sets are simulated using the individual’s design (dose and regressor values). The parameter values used for the simulation include the population parameter values, the individual covariate values and random effects sampled from the population distribution. The simulated data sets include residual errors. The prediction interval represents the interval containing 90% (see “level” setting) of the simulated data points. The predicted median is the median of all simulated data points. The individual predictive check allows to visualize the interindividual variability (unexplained by covariates) and compare the population prediction to the individual observations.
 Sorting: Sort the subjects by ID or individual parameter values in ascending or descending order.
By default, only the observed data and the individual fits are displayed.
4.2.2.Observation versus Prediction
Purpose
This figure displays observations (\(y_{ij}\)) versus the corresponding predictions (\(\hat{y}_{ij}\)) computed using either the population parameters and covariate effects but random effects set to 0, or with the individual parameters. This figure is useful to detect misspecifications in the structural model. The 90% prediction interval, which depends on the residual error model, can be overlaid. Predictions that are outside of the interval are denoted as outliers. A high proportion of outliers suggest misspecifications in the model. Moreover, the distribution of the observations should be symmetrical around the corresponding predicted values.
 Population and individual predictions vs observations
 Visual guides
 Outliers proportion
 Individual estimates
 Highlight
 Log scale
 Settings
Population and individual predictions vs observations
The following example corresponds to the observations and predicted concentrations for the PK of warfarin, modeled by a onecompartment model with a firstorder absorption and a linear elimination.On the left, predictions are made using the population parameters while on the right they correspond to the individual parameters. More points appear with the individual predictions: for each observation point, ten predictions are displayed, corresponding to ten simulated individual parameters.
Visual guides
In addition to the line y = x, it is possible to display the 90% prediction interval, as well as a spline interpolation.
The 90% prediction interval represents the uncertainty of predictions due to the residual error model defined in the observation model. In the figure below, the shape of this interval can be seen for the four existing residual error models (constant, proportional, combined1, combined2) when the observation model is defined with a normal distribution:
The next figure corresponds to data that follow a lognormal distribution. The combination of constant error model and lognormal distribution corresponds to an exponential error model. Error models with a proportional term can cause numerical issues with the lognormal distribution for small observations because the error becomes very small as well.
Choosing an observation model with a logitnormal distribution for the data is useful to take into account bounded data. The figure below shows the shape of the prediction intervals for the different error models associated with data that follow a logitnormal distribution in [0.110]:
The prediction interval for the same example as above on the PK of warfarin characterizes a residual error model that combines a constant and a proportional term:
On the figure above it can be noted that several zero observations measured at low times correspond to nonzero predictions that fall outside the 90% prediction interval, and thus cannot be explained by the residual error. This could be explained by a delay between the administration and absorption of warfarin, therefore a model with a delayed absorption might fit better the data.
Outliers proportion
The outliers proportion can be displayed: it is the proportion of residuals outside the 90% prediction interval.
Individual estimates
As for all diagnosis plots based on individual parameters, it is possible to choose the individual estimates that are used to compute the plot of observations vs individual predictions, among the different estimates computed during the individual parameter estimation: conditional modes (EBEs) or means of the conditional distributions, or simulated individual parameters drawn from the conditional distributions (by default). In the latter case, each observation is associated with a set of individual predictions derived from a set of individual parameters simulated from the same individual conditional distribution. On the two figures below, one can compare the plot based on simulated parameters from the conditional distribution (top) and the same plot based on conditional modes (bottom).
Highlight
As shown on the figures below, hovering on a point of observed data reveals the subject id and time corresponding to this point. All the points corresponding to this subject are highlighted in yellow. On the left, there are several predictions per observation, and the ten points corresponding to the hovered observation are indicated with a bigger diameter. On the right, there is only one prediction per observation, and all points corresponding to the same individual are linked with segments to visualize the time chronology.
Log scale
A log scale is useful to focus on low observation values. It can be set for each axis separately or both together.
A second example below displays the predicted concentrations of remifentanil, modeled by a twocompartments model with a linear elimination. In this example, the loglog scale reveals a clear misspecification of the model: the small observations are underpredicted. These observations correspond to high times: this means that the elimination is not properly captured by the twocompartment model. A threecompartment model might give better results.
that here 10 predictions are displayed for each observation, corresponding to different simulated parameters drawn from the conditional distribution during the individual parameter estimation task.
Settings
 General
 Legend and grid : add/remove the legend or the grid. There is only one legend for both plots.
 Outliers proportion: display/hide the proportion of points outside the 90% prediction interval.
 Subplots
 Population prediction: add/remove the figure with the comparison between the population predictions and the observations.
 Individual prediction: add/remove the figure with the comparison between the individual predictions and the observations.
 Display
 Observed data: Add/remove the points corresponding to pairs of observations and predictions.
 BLQ data : show and put in a different color the data that are BLQ (Below the Limit of Quantification)
 Individual estimates: select the estimates condition mean or mode, or simulated estimates from the conditional distribution (by default).
 Visual cues: add/remove visual guidelines such as the line y = x, a spline interpolation, and the 90% prediction interval indicated with dotted lines.
By default, only the individual predictions are displayed.
4.2.3.Scatter plot of the residuals
Purpose
These plots display the PWRES (population weighted residuals), the IWRES (individual weighted residuals), and the NPDEs (normalized prediction distribution errors) as scatter plots with respect to the time or the prediction.
The PWRES and NPDEs are computed using the population parameters and the IWRES are computed using the individual parameters. For discrete outputs, only NPDEs are used.
These plots are useful to detect misspecifications in the structural and residual error models: if the model is true, residuals should be randomly scattered around the horizontal zeroline.
Definition
Population Weighted Residuals \(\text{PWRES}_{ij}\)
\(\text{PWRES}_{ij}\) are defined as the normalized difference between the observations and their expected mean. Let \(y_i = (y_{ij}, 1 \leq j \leq n_i)\) be the vector of observations for subject i. The mean of \(y_i\) is the vector \(\mathbb{E}(y_i)=(\mathbb{E}(f(t_{ij};\psi_i), 1 \leq j \leq n_i)\). Let \( \textrm{V}_i\) be the \(n_j \times n_j\) variancecovariance matrix of \(y_i\). Then, the ith vector of the population weighed residuals \( \text{PWRES}_i = \{\text{PWRES}_{ij}, 1\leq j \leq n_i\} \) is defined by
$$\text{PWRES}_i = V_i^{1/2}(y_i\mathbb{E}(y_i))$$
The population residuals \((y_i\mathbb{E}(y_i))\) are correlated within each individual. The population weighted residuals PWRES standardize and decorrelate the population residuals using the modelpredicted variancecovariance matrix of observations \(V_i\).
\(\mathbb{E}(y_i) \) and \(V_i\) are not known in practice but are estimated empirically by MonteCarlo simulation without any approximation of the model. In the formula above, \(y_i\) represents the observations from the dataset. \(\mathbb{E}(y_i)\) is calculated as the mean of simulated observations using individual parameters sampled from the population distribution to obtain the model prediction plus a sample from the residual error model. The number of simulated observations depends on the Plot task setting “Number of simulations”. \(V_i\) is calculated on the same simulated observations.
When the PWRES are plotted w.r.t the model prediction, the population prediction popPred (i.e with random effects eta equal to zero but keeping the covariate effects) are used on the xaxis.
Individual weighted residuals \(\text{IWRES}_{ij}\)
\(\text{IWRES}_{ij}\) are estimates of the standardized residual (\(\epsilon_{ij}\)) based on individual predictions, with \(g\) the function defining the residual error model:
$$\text{IWRES}_{ij} = \dfrac{ y_{ij}f(t_{ij};\hat{\psi}_i)}{g(t_{ij};\hat{\psi}_i)}$$
If the residual errors are assumed to be correlated, the individual weighted residuals can be decorrelated by multiplying each individual vector \(\text{IWRES}_i = (\text{IWRES}_{ij} ; 1\leq j\leq n_i)\) by \(\hat{\text{R}}_i^{1/2}\), where \(\hat{\text{R}}_i\) is the estimated correlation matrix of the vector of residuals \((\epsilon_{ij}; 1\leq j \leq n_i)\).
When the IWRES are plotted w.r.t the model prediction, the individual prediction using the conditional mode (EBEs), conditional mean or samples from the conditional distribution is used, depending on the choice of the “Individual estimates” in the settings panel on the right.
Normalized prediction distribution errors \(\text{NPDE}_{ij}\)
\(\text{NPDE}_{ij}\) are a nonparametric version of \(\text{PWRES}_{ij}\) based on a rank statistic. For any (i,j), let \(\text{F}_{ij} = \text{F}_{\text{PWRES}_{ij}}(\text{PWRES}_{ij})\) where \(\text{F}_{\text{PWRES}_{ij}}\) is the cumulative distribution function (cdf) of \(\text{PWRES}_{ij}\). NPDEs are then obtained from \(\text{F}_{ij}\) by applying the inverse of the standard normal cdf \(\Phi\).
In practice, one simulates a large number \(K\) of simulated data set \(y^{(k)}\) using the model, and estimate \(\text{F}_{ij}\) as the fraction of simulated data below the original data, i.e:
$$\hat{\text{F}}_{ij}=\frac{1}{K}\sum_{k=1}^K 1_{y_{ij}^{(k)}\leq y_{ij}^{\text{obs}}}$$
By definition, the distribution of \(\text{F}_{ij}\) is uniform on [0,1], we thus rather use \(\Phi^{1}(\text{F}_{ij})\), which follows a standard normal distribution (with \(\Phi\) the cdf of the standard normal distribution). NPDEs are defined as an empirical estimation of \(\Phi^{1}(\text{F}_{ij})\), i.e \(\text{NPDE}_{ij}=\Phi^{1}(\hat{\text{F}}_{ij})\).
When the NPDE are plotted w.r.t the model prediction, the population prediction popPred (i.e with random effects eta equal to zero but keeping the covariate effects) are used on the xaxis.
Below is a correspondence table of the Nonmem and Monolix terms used for residuals:
For count and categorical data:
FAQ: Is there no CWRES in Monolix? No, PWRES are given instead. They are defined with the same formula but are obtained with simulations rather than FOCE approximation, so without any approximation of the model. The PWRES in Monolix are equivalent to the EWRES in Nonmem.
Examples
In the following example, the parameters of a twocompartment model with iv unfusion and linear elimination are estimated on the remifentanil data set. One can see the PWRES, the IWRES and the NPDE w.r.t. the time (on top), and the prediction (at the bottom).
Since the points are clearly scattered unevenly around the horizontal zeroline, these plots suggest a misspecifcation of the structural model.
The corresponding distributions can be seen on this page.
It is possible to select some of the subplots to focus on, with the panel Subplots in Settings:
Presets
A number of element can be overlaid or hidden from the plots in the panel Display. Only the horizontal zeroline, representing the theoretical mean, is always displayed. Two presets with predefined selections of displayed elements are available: the first one called “Scatter” hides all elements except the points for residuals, while the second called “VPC” displays instead empirical and predicted percentiles for the residuals as lines, as well as prediction intervals as colored areas. This figure is detailed below.
Predictive checks
The preset “VPC” displays prediction intervals for the median, 10th and 90th percentiles, obtained with simulations of the residuals, as well as the empirical percentiles to compare the behavior of the model to the data. Residual points are hidden, but the trend is represented with a spline interpolation.
Misspecification in the structural model, the error model, and the covariate model can be detected by discrepancies between the observed percentiles and their prediction intervals, as can be seen for example on the plots of IWRES vs time and NPDE vs time below, with logscale on the xaxis. Population residuals greatly depart from the data at all time points, while individual residuals show better predictions for low times only.
Outliers (empirical percentiles outside the prediction intervals) can be marked with red points or red areas:
Comparing PWRES and NPDEs
NPDEs are quite similar to PWRES, but are simulationbased, and therefore account for the heterogeneity in study design by comparing the observations with their own distribution. NPDEs are thus displayed by default rather than PWRES.
Comparing IWRES and NPDEs
The IWRES are based on individual predictions, therefore the values on the X axis with respect to predictions are not the same as for NPDEs and PWRES, as can be seen on the plots below. If the tasks EBEs and Conditional distribution have been run, several different individual estimates are available to be used for the individual predictions. The next section shows how to choose the estimates.
Preventing shrinkage in IWRES
The individual estimates used to compute the IWRES can be chosen in the Display panel:
By default, the individual estimates are drawn from the conditional distributions rather than coming from usual estimators such as conditional modes (EBEs) or conditional means. This choise is recommended in order to prevent shrinkage, a phenomenon that occurs when the individual data are not sufficiently informative with respect to one or more parameters. If overfitting occurs, IWRES computed from biased estimators might thus shrink toward 0.
Highlight
Hovering on a point highligths all the points from the same individual in yellow on all plots, and reveals the corresponding subject id and time. If the individual estimates selected in Display are the simulated condition distribution, each observation corresponds to a set of IWRES computed from a set of simulated individual parameters. When the observation is hovered, the points from this set are indicated with a bigger diameter.
If the individual estimates selected in Display are condition modes (EBEs) or conditional means, there is only one residual per observation, and all points corresponding to the same individual are linked with segments to visualize the time chronology.
Binning
As for VPC, data binning used to compute percentiles can be changed. Several strategies exist to segment the data: equalwidth binning, equalsize binning, and a leastsquares criterion. The number of bins can also be either set by the user, or automatically selected to obtain a good trade off.
On the three figures below where NPDEs are displayed with respect to logscaled time, 5 bins are selected with equal width on the left, equal size in the center, and the leastsquares criteria on the right. Observations are overlaid in light purple to visualize the data density in each bin. Equal width in particular shows low density for some bins, and result in a less informative plot for low times were data density is high.
On the figure below, the number of bins for leastsquares criteria is automatically set, allowing a more precise display.
Censored data
The residuals for censored data appear in a different color. They are by default based on simulated observations that take into account the censoring interval.
An option available in the panel “Display” can be used to select the method of calculation for the residuals corresponding to censored data: either based on simulated observations (by default), or based on LOQ (values from the observation column in the dataset).
Discrete data
For categorical or count data, only NPDEs are used. Here again, NPDEs correspond to the rank of each observation among a set of simulations based on the model. However, to prevent problems with discrete values, both observations and simulations are slightly perturbed with a uniform distribution before computing the ranks.
Settings
 Subplots
 Residuals
 Population residuals: Add/remove scatterplots for PWRES. Hidden by default.
 Individual residuals: Add/remove scatterplots for IWRES, using the individual parameter estimated using the conditional mode or the conditional mean. By default, individual parameters come from the conditional mode estimation.
 NPDE: Add/remove scatterplots for NPDE.
 Xaxis
 time: Add/remove the scatterplots w.r.t. the time.
 prediction: Add/remove the scatterplots w.r.t. the prediction.
 Residuals
 Display
 Presets: apply the preselections of elements for scatter plots or VPC
 Residuals: Add/remove observed data.
 Censored data: Add/remove BLQ data (with a different color) if present.
 Empirical percentiles: Add/remove empirical percentiles for the 10%, 50% and 90% quantiles.
 Predicted percentiles: Add/remove theoretical percentiles for the 10% and 90% quantiles.
 Prediction interval: Add/remove prediction intervals given by the model for the 10% and 90% quantiles (in blue) and the 50% quantile (in pink), with userdefined level (by default, 90).
 Outliers: Add/remove dots or areas to mark outliers.
 Individual estimates: Choose the individual estimates among conditional modes (EBEs), conditional means (computed with SAEM), or simulated parameters from the conditional distributions.
 Calculations – linear interpolation: Choose the display for prediction intervals: by default linear interpolation is used, otherwise the display is piecewise.
 Calculations – Use censored data: Choose the display for censored data: by default simulated BLQ observations are used, otherwise the LOQ from the observation column in the data set can be used.
 Visual cues: Add/remove spline interpolation.
 Bins
 Bin values: Add/remove vertical lines on the scatterplots to indicate the bins.
 Binning criteria: Choose the bining criteria among equal width (default), equal size or leastsquares.
 Number of bins: Choose a fixed number of bins or a range, with the range for the number of data points per bin.
4.2.4.Distribution of the residuals
Purpose
These plots display the empirical distributions of the residuals: the PWRES (population weighted residuals), the IWRES (individual weighted residuals), and the NPDEs (normalized prediction distribution errors) for continuous outputs, together with the standard Gaussian probability density function and cumulative distribution function.
If the model is true, the PWRES, IWRES and NPDEs should behave as independent standardized normal random variables. These plots are thus useful to detect misspecifications in the structural and residual error models.
Example
In the following example, the parameters of a twocompartment model with iv unfusion and linear elimination are estimated on the remifentanil data set.
Below, one can see on top the comparison between the empirical and theoretical probability density function (PDF) of the PWRES, IWRES and NPDE, and at the bottom the comparison between the empirical and theoretical cumulative distribution function (CDF).
The corresponding scatter plots can be seen on this page.
Settings
By default, all the residuals and all the plots are displayed.
 Subplots
 Residuals: choose the plots to display according to residuals
 Population residuals: PWRES
 Individual residuals: IWRES, using the individual parameter estimated using the conditional distribution, the conditional mode, or the conditional mean.
 NPDE
 Xaxis
 PDF: Probability density function of residuals and empirical distribution as histograms.
 CDF: Theoretical and empirical cumulative distribution functions.
 Residuals: choose the plots to display according to residuals
 Display
 Individual estimates: choose the estimator for individual parameters as parameters drawn from the conditional distributions. or the modes or means of the conditional posterior distributions.
4.3.Model for the individual parameters
4.3.1.Distribution of the individual parameters
Purpose
This figure can be used to compare:
 the empirical distribution of the individual parameters, estimated with the conditional means, the conditional modes, or simulated from the conditional distributions,
 the theoretical distribution defined in the statistical model, with the estimated population parameter.
Further analysis such as stratification by covariate or shrinkage assessment can be performed and will be detailed below.
PDF and CDF
In the warfarinPK_project, several parameters are estimated. It is possible to display the theoretical distribution and the histogram of the empirical disitribution as proposed below.
The distributions are represented as histograms for the probability density function (PDF). Hovering on the histogram also reveals the density value of each bin as shown on the figure below
Notice that the theoretical pdf is a pure lognormal distribution. However, in case of covariate use with the parameters, it is not a pure lognormal but rather a combinaison of lognormal distribution. If for example, on set the SEX covariate on the parameter V, a parameter beta_V_SEX_1 is created and the individual parameter distribution becomes as the following.
Cumulative distribution functions (CDF) is proposed too.
Again, overlaying the plots display the information concerning the parameter value and its empirical and theoretical cdf.
Getting away with shrinkage using simulated individual parameters
If the data does not contain enough information to estimate correctly some individual parameters, individual estimates that come from the means or the modes of the individual conditional distributions are shrunk towards the same population value, which is respectively the mean and the mode of the population distribution of the parameter. For a parameter \(\psi_i\) which is a function of a random effect \(\eta_i\), this phenomenon can be quantified by defining the \(\eta\)shrinkage as:
$$\eta\text{shrinkage} = 1 \frac{Var(\hat{\eta})}{\hat{\omega}^2} $$
where \(\text{Var}\left(\hat{\eta}_i\right)\) is the empirical variance of the estimated random effects \(\hat{\eta}_i\)’s. It is possible to display the shrinkage value on top of the histograms, as can be seen below:
The “simulated individual parameters” option uses instead individual parameters drawn from the conditional distribution, simulated by the MCMC procedure. This method is recommended as it permits to obtain unbiased estimators that are not affected by possible shrinkage, and leads to more reliable results. For more details see the page Understanding shrinkage and how to circumvent it.
In the same example, the simulated individual parameters provide much better shrinkage as can be seen below.
The following table compiles the shrinkage calculation (in %) for all methods
Method\Parameters  Tlag  ka  V  Cl 

Conditional mean  71.5  69.8  8.87  0.23 
Conditional mode  74.2  74.7  10.3  0.2 
Simulated individual parameters  17.1  3.66  2.63  1.01 
Example of stratification
It is possible to stratify the population by some covariate values and obtain the distributions of the individual parameters in each group. This can be useful to check covariate effect, in particular when the distribution of a parameter exhibits two or more peaks for the whole population. On the following example, the distribution of the parameter k from the same example as above has been split for two groups of individuals according to the value of the continuous covariate AGE, allowing to visualize two clearly different distributions.
Settings
 General: add/remove the legend, the grid, and the shrinkage in %.
 Display
 Empirical: add/remove histogram of empirical distribution.
 Theoretical: add/remove curve of theoretical distribution.
 Distribution function: The user can choose to display either the probability density function (PDF) as histogram or the cumulative distribution function (CDF).
 Individual estimates: The user can define which estimator is used for the definition of the individual parameters.
Simulated individual parameters are used by default, otherwise the conditional mode is the default estimation if it has been computed with the “Individual parameters estimation” task.
4.3.2.Distribution of the random effects
Purpose
This plot displays the distribution of the standardized random effects with boxplots or with histograms. Since random effects should follow normal probability laws, it is useful to compare the distributions to standard Gaussian distributions.
Example
In the following example, one can see the distributions of two parameters of a twocompartment bolus model with linear elimination, estimated on the tobramycin example. On the left, the distributions are represented as boxplots, in the middle as histograms for the probability density function (PDF), and on the right as cumulative distribution functions (CDF). In each case, marks to compare the results to standard Gaussian distributions are overlaid: dotted horizontal lines indicate the interquartile interval of a standard Gaussian distribution for the boxplots, and black curves represent the PDF and CDF of a standard Gaussian distribution.
In boxplots, the dashed line represents the median, the blue box represents the 25th and 75th percentiles (Q1 and Q3), and
the whiskers extend to the most extreme data points, outliers excluded. Outliers are the points larger than Q1 + w(Q3 – Q1) or smaller than Q1 – w(Q3 – Q1) with w=1.5. Outliers are shown with red crosses.
On the figure below, the individuals have been split into two groups according to the value of the continuous covariate CLCR. One can notice differences on the boxplots for the distributions of random effects between both groups, in particular for the parameter k.
Settings
 Display
 Distribution function. The user can choose which type of plot is used to represent the distributions of the random effects: boxplots, pdf (probability density function) or cdf (cumulative distribution function).
 Individual estimates. The user can define which estimator is used for the definition of the individual parameters and thus for the random effects (conditional mean, conditional mode, or simulated random effects)
 Visual cues: If boxplot has been selected, the user can choose to add or hide dotted lines to mark the median or quartiles of a standard Gaussian distribution.
By default, the distributions of simulated random effects are displayed as boxplots.
Standarized random effects in case of IOV
Starting from the 2019 version, in case of IOV and if the conditional distribution is computed, we propose to display the standarized random effect by level of variability. In the presented example, we put IOV on both ka and V parameters. Thus, in the plot, we proposed to display the several levels of variability for all the parameters. We can then display the standarized random effects for
 the ID level,
 the OCC level (where only the parameters with variability on this level are displayed)
 the ID+OCC level corresponding to the addition of the levels
4.3.3.Correlation between the random effects
Purpose
This plot displays scatter plots for each pair of random effects. It allows to identify correlations between random effects, which can then be introduced in the models for the probability distributions for the individual parameters.
Example
In the following example, one can see pairs of random effects estimated for all parameters of a onecompartment model with delayed firstorder absorption and linear elimination estimated on the PK of warfarin data set. The estimators for random effects have been simulated from conditional distributions of individual random effects.
Visual guidelines
In addition to regression lines, Pearson correlation coefficients can been added to see the correlation between random effects, as well as spline interpolations.
Selection
It is possible to select a subset of random effects, whose pairs of correlations are then displayed, as shown below. In the selection panel, a set of contiguous rows can be selected with a single extended click, or a set of noncontiguous rows can be selected with several clicks while holding the Ctrl key.
Highlight
Similarly to other plots, hovering on a point provides information on the corresponding subject id, and highlights other points corresponding to the same individual, if multiple individual parameters have been simulated by sampling the conditional distribution.
On the figure below, we can see the same plot with only one parameter value per individual (left) or ten values (right). Notice that the correlation coefficient is more reliable when multiple parameters are available, as they take into account the uncertainty of individual parameters.
Stratification: coloring and filtering
Stratification can be applied by creating groups of covariate values. As can be seen below, these groups can then be colored (left) or filtered (right), allowing to check the effect of the covariate on the correlation between two parameters. The correlation coefficient is updated according to the filtering.
Settings
 General
 Legend and grid : add/remove the legend or the grid. There is only one legend for all plots.
 Correlation: display/hide the correlation coefficient associated with each scatter plot.
 Display
 Selection. The user can select some of the parameters to display only the corresponding scatter plots. A simple click selects one parameter, whereas multiple clicks while holding the Ctrl key selects a set of parameters.
 Individual estimates. The user can define which estimates are used for the definition of the individual parameters and thus for the random effects (conditional mean, conditional mode, simulated parameters)
 Visual cues. Add/remove the regression line or the spline interpolation.
By default, all scatter plots are proposed with simulated individual parameters.
Display in case of correlation in the statistical model
Starting from the 2019 version, when there is correlation in the statistical model, we propose to see the decorrelated random effects in order to see if some correlation remains. In that case, there is an option in the display to switch between the random effects and the decorrelated random effects as can be seen on the following figure. The methodology behind the calculation can be found here. The formula to calculate the decorrelated random effects \(\hat{\eta}_i\) from the random effects \(\eta_i\) and the estimated variancecovariance of the random effects \(\Omega\) is:
\(\hat{\eta}_i=\Omega^{1/2}\times \eta_i \)
Display in case of IOV
Starting from the 2019 version, in case of IOV and if the conditional distribution is computed, we propose to display the by level of variability as can be seen in the following figure.
We can then display the correlation of the random effects for
 the ID level,
 the OCC level (where only the parameters with variability on this level are displayed)
 the ID+OCC level corresponding to the addition of the levels
Notice that at least 2 parameters should have variability on the occasion level to have the correlation on the several level displayed.
4.3.4.Individual parameters versus covariates
Purpose
The figure displays the estimators of the individual parameters, and those for random effects, as a function of the covariates. It allows to identify correlation effects between the individual parameters and the covariates.
The estimators can be:
 simulated parameters: individual parameters and individual random effects are sampled from the distributions \(p(\psi_iy_i;\hat{\theta})\) and \(p(\eta_iy_i;\hat{\theta})\). These estimators lead to more reliable results, especially when individual data are sparse and the distributions of conditional modes and means of individual parameters are affected by shrinkage.
 the conditional means \(E(\psi_iy_i;\hat{\theta})\) for parameters \(\psi_i\)and \(E(\eta_iy_i;\hat{\theta})\) for random effects \(\eta_i\),
 the conditional modes of the same distributions.
On the boxplots for categorical covariates, red “+” are outlier points below Q11.5*IQR (interquartile range) or above Q3+1.5*IQR.
Identifying correlation effects
In the example below, we can see the parameters of a onecompartment PK model with delayed firstorder absorption and linear elimination estimated on the warfarin data set. The simulated individual parmeters of the 4 parameters of the PK model are displayed with respect to the covariates: the weight wt, a transformed version of the weight (lw70=log(wt/70)) and the sex category.
Visual guidelines
In order to help identifying correlations, regression lines, spline interpolations and Pearson correlation coefficients can be overlaid on the plots for continuous covariates. Here we can see a strong correlation between the parameter V and the covariate lw70.
Highlight
Hovering on a point reveals the corresponding individual and, if multiple individual parameters have been simulated from the conditional distribution for each individual, highlights all the points points from the same individual. This is useful to identify possible outliers and subsequently check their behavior in the observed data.
Selection
It is possible to select a subset of covariates or parameters, as shown below. In the selection panel, a set of contiguous rows can be selected with a single extended click, or a set of noncontiguous rows can be selected with several clicks while holding the Ctrl key. This is useful when there are many parameters or covariates. In particular, it is frequent to introduce transformed covariates, the selection allows to focus on the transformed versions rather than the original.
Comparing individual parameters and random effects
By default, the values on the Yaxis are computed with the individual parameters. One can choose to display the random effects instead. If some individual parameters are already modelled with covariates, this is taken into account by the random effects values, thus allowing to focus on remaining correlations.
The figures below show the diagnosis plots with individual parameters or random effects when the models for parameters V includes the covariate lw70. On the top, one can identify the correlations between individual parameters and covariates: the logvolume (log(V)) clearly increases with the logtransformed weight, as well with sex. On the other hand, the random effects on the bottom allow to focus on correlations that are not yet taken into account in the covariate model. Because the model already includes a linear relationship between the logvolume and the logtransformed weight, shows no correlation with lw70. There is no correlation either between and sex, because of an existing correlation between lw70 and sex.
Stratification
Stratification can be applied by creating groups of covariate values. As can be seen below, these groups can then be split, colored or filtered, allowing to check the effect of the covariate on the correlation between two parameters. The correlation coefficient is updated according to the split or filtering.
Settings
 General
 Legend and grid : add/remove the legend or the grid. There is only one legend for all plots.
 Correlation: display/hide the correlation coefficient associated with each scatter plot.
 Display
 Yaxis. The user can choose to see either the individual parameters or the random effects.
 Selection. The user can select some of the parameters or covariates to display only the corresponding plots. A simple click selects one parameter (or covariate), whereas multiple clicks while holding the Ctrl key selects a set of parameters.
 Individual estimates. The user can define which estimators are used for the definition of the individual parameters and thus for the random effects (conditional mean, conditional mode, simulated parameters)
 Visual cues. Add/remove a regression line or a spline interpolation.
By default, all plots are proposed with simulated individual parameters.
4.4.Predictive checks and predictions
4.4.1.Visual predictive checks
 Purpose
 Examples
 Details
 Settings
 Generating predictive checks on an external data set
 Exporting VPC simulations
 Correcting the VPC for missing observations
 Troubleshooting
Purpose
The VPC (Visual Predictive Check) offers an intuitive assessment of misspecification in structural, variability, and covariate models. The principle is to assess graphically whether simulations from a model of interest are able to reproduce both the central trend and variability in the observed data, when plotted versus an independent variable (typically time). It summarizes in the same graphic the structural and statistical models by computing several quantiles of the empirical distribution of the data after having regrouped them into bins over successive intervals.
More precisely, the goal is to compare the two following elements:
 Empirical percentiles: percentiles of the observed data, calculated either for each unique value of time, or pooled by adjacent time intervals (bins). By default, the 10th, 50th and 90th percentiles are displayed as green lines. These quantiles summarize the distribution of the observations.
 Theoretical percentiles: percentiles of simulated data are computed from multiple Monte Carlo simulations with the model of interest and the design structure of the original dataset (i.e., dosing, timing, and number of samples). For each simulation, the same percentiles are computed across the same bins as for empirical percentiles. Prediction intervals for each percentile are then estimated across all simulated data and displayed as colored areas (pink for the 50th percentile, blue for the 10th and 90th percentiles). By default, prediction intervals are computed with a level of 90%.
If the model is correct, the observed percentiles should be close to the predicted percentiles and remain within the corresponding prediction intervals.
Examples
VPCs vary slightly for different types of data. For joint models for multivariate outcomes, VPCs are available for each outcome.
 Continuous outcomes
warfarinPK_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)
In the following example, the parameters of a onecompartment model with delayed firstorder absorption and linear elimination are estimated on the warfarin dataset. A constant residual error model was used. The figure presents the VPC with the prediction intervals for the 10th, 50th and 90th percentiles. Outliers are highlighted with red dots and areas. Here the three quantiles appear closer together than the model would suggest, therefore the VPC suggests that a proportional component should be added to the error model.
For joint models for continuous PK and timetoevent data, VPCs are available for each type of data. However it is important to note that dropout events are not taken into account in the VPC corresponding to the continuous data. Therefore, in the case of nonrandom dropout events in the dataset, this can result in discrepancies between observed and simulated data and thus hamper the diagnosis value of the VPC. Correcting this bias would require to include the simulated dropout in VPC, as well as adapt the design structure to compensate observed dropouts, an approach that is problematic when the design structure is complex. More details on this approach are given here.
 Noncontinuous outcomes: count data and categorical data
VPCs for count data and categorical data compare the observed and predicted frequencies of the categorized data over time. The predicted frequency is associated with a blue prediction interval.
The following figure shows the VPC for a project with a continuous time Markov chain model and time varying transition rates.
 markov3b_project (data = ‘markov3b_data.txt’, model = ‘markov3b_model.txt’)
In addition to the categorization over time (binning on X), count data are also binned into groups of count values on the VPC (binning on Y). The number of bins and binning method can be set in Settings under “Y Bins”.
As an example, the VPC below corresponds to a project where a Poisson model is used for fitting the data. Observations are binned in 3 groups on the Y axis and 20 bins on the X axis.
 count1a_project (data = ‘count1_data.txt’, model = ‘count_library/poisson_mlxt.txt’)
 Timetoevent data
In case of timetoevent data, two visual predictive checks are available, survival function based on the KaplanMeier plot for exactly observed events and the Turnbull estimator for the interval censored data, and the mean number of events per individual using Turnbull estimation (see here and here for reference papers).
Details on the VPC for TTE generation in Monolix are presented here.
The example below shows these two figures, computed with a model for the survival of patients with advanced lung cancer from the Veterans’ Administration Lung Cancer study. Censored data has been selected and displayed on the KaplanMeier plot. Note that censored data also cause an over prediction bias in the VPC based on the mean number of events per individual, because censored individuals contribute to the prediction interval but not to the empirical curve.
Sometimes numerical errors can appear in the simulations used for the TTE VPC. For example, when simulating a joint model with tumor growth and survival over a long time scale, the simulated hazard for death can become too high for some individuals at high times. In Monolix2021R1, the appearance of NaNs in the VPC simulations does not stop the generation of the VPC. The simulated individuals with NaNs in their simulations are not used in the prediction interval, and the percentage of simulated individuals with NaNs is then displayed in the Warnings.
Details
 Binning criteria
Correctly defining the intervals (or bins) into which the data are grouped is crucial to construct a VPC that avoids distortion between the original and approximated distributions. Several strategies exist to segment the data: equalwidth binning, equalsize binning, and a leastsquares criterion. The number of bins can also be either set by the user, or automatically selected to obtain a good tradeoff. Indeed, a small number of bins leads to a poor approximation but a good estimation of the data’s distribution, while a large number of bins leads to a good approximation but poor estimation.
As an example, the VPCs below are computed on the PK model built for remifentanil pharmacokinetics, a dataset that involves a large variability in doses. The bins are delimited with vertical lines. The first VPC on the left is computed with 5 bins, the number automatically selected for this dataset. On the other hand, the second VPC on the right is computed with 15 bins. We notice that in this case the heterogeneity of the data results in a poor estimation of the data’s distribution. To keep a good estimation, a small number of bins is required, but the approximation then prevents from visualizing the kinetics in details. The absorption phase is for example not visible.
 Corrected predictions
As shown above, VPCs can be misleading if applied to data that include a large variability in dose and/or influential covariates, or that follow adaptive designs such as dose adjustments. The predictioncorrected VPC (pcVPC), with prediction correction, was developed to maintain the diagnosis value of a VPC in these cases. In each bin, the observed and simulated data are normalized based on the typical population prediction for the median time in the bin. Formulae are applied as in the publication from Bergstrand et al. In the simple case of normally distributed observations and a zero lower bound for observations, we get:
\[ pcY_{ij} = Y_{ij} \frac{\tilde{PRED}_{bin}}{PRED_{ij}} \]
where
 Yij : observation or prediction for the ith individual and jth time point,
 pcYij : predictioncorrected observation or prediction,
 PREDij: typical population prediction for the ith individual and jth time point
 PRẼDij: median of typical population predictions for the specific bin of independent variables.
This removes the variability coming from binning across independent variables.
The example below shows the pcVPC computed on the PK model built for remifentanil pharmacokinetics with 15 bins: the figure now gives a good estimation of the data’s distribution, including the absorption phase.
 Stratification
When possible, another useful approach to deal with heterogeneous data can be to split the VPC into groups of subjects that are more homogeneous. As an example, the VPCs below are computed again on the PK model built for remifentanil pharmacokinetics, with 15 bins, but the data was first split by a categorical covariate that characterizes groups of similar doses.
 VPC based on time after last dose (continuous data only)
Option in the display panel “Time after last dose” displays the VPC with data for times (the Xaxis) after last dose (for Monolix versions 2021 and above). Below is an example with result for the demo project multidose_project.mlxtran (Monolix demo folder 6.3). Observed data overlaid the VPC. On the left image the Xaxis is “Time”, and on the right the Xaxis is “Time after last dose”. The exported charts data for the VPC include two columns for time and time after last dose – independently of the option selected in the interface. Doses with amount=0 are handled as the others. For observations before the first dose (or if no doses at all), time remains unchanged (i.e time after last dose = time). Finally, all administration ids are handled together (it is not possible to define time after dose for adm id = 2 only for instance).
 Axis limits
From the 2023 version on, the user can choose the axis limits manually. This is particularly convenient when the prediction intervals go into negative values and logscale is used on the yaxis. In that case, the VPC is shown by default from 1e16 to the maximum value, which leads to an unreadable plot. After toggling the setting “custom limits” on, the axis limits can be selected by the user. As the other plot settings, the axis limits are copied to the placeholders when preparing a report.
Settings
 General: Add/remove legend or grid
 Subplots (for TTE data)
 Add/remove plot for survival function (KapanMeier plot) or plot for mean number of events per individual
 Add/remove plot for survival function (KapanMeier plot) or plot for mean number of events per individual
 Display
 Observed data
 Observed data: Add/remove observed data.
 BLQ: Add/remove BLQ data if present.
 Use BLQ: Choose to use BLQ data or to ignore it to compute the VPC. BLQ data can be simulated, or can be equal to the limit of quantification (LOQ). The latter case induces strong bias .
 Empirical percentiles: Add/remove empirical percentiles for the 10%, 50% and 90% quantiles.
 Predicted percentiles: Add/remove theoretical percentiles for the 10% and 90% quantiles.
 Prediction interval: Add/remove prediction intervals given by the model for the 10% and 90% quantiles (in blue) and the 50% quantile (in pink).
 Set interpercentile level and higher percentile for prediction intervals (for continuous data by default the level is 90 and the higher percentile is 90%), or number of bands for TTE data
 Outliers
 Dots: Add/remove red dots indicating empirical percentiles that are outside prediction intervals
 Areas: Add/remove red areas indicating empirical percentiles that are outside prediction intervals
 Calculations
 Observed data


 Corrected predictions: compute the pcVPC using Uppsala prediction correction (see details above)
 Linear interpolation: Set piece wise display for prediction intervals (by default the display is linear)
 Time after last dose: Use time after last dose instead of time on the Xaxis.

 Bins – for categorical data, X Bins and DV Bins (for Y axis) can be specified
 Bin limits: Add/remove vertical lines on the scatter plots to indicate the bins.
 Binning criteria: Choose the bining criteria among equal width (default), equal size or leastsquares
 Number of bins: Choose a fixed number of bins or a range for automatic selection, and a range for the number of data points per bin.
All colors, points and lines can be modified by the user.
Generating predictive checks on an external data set
This video shows how to generate an external VPC in Monolix: a VPC that compares the simulations based on a population model estimated on a first data set to a second data set, for example to check whether a population model estimated on a single dose study is also valid on a new multiple dose study for the same molecule.
Exporting VPC simulations
When exporting the charts data, by default the charts data for the VPC include the observed data and predicted percentiles, but not the detailed simulations, which are usually quite large. Starting from the 2020 version, the user can nonetheless choose to include the VPC simulations by clicking on “Export > Export VPC simulations” in the application menu:
The simulated values are saved in <result folder>/ChartsData/VisualPredictiveCheck/XXX_simulations.txt. They can be used to replot the VPC in R for instance.
In Monolix2021 it is also possible to include systematically the VPC simulations in the exported charts data with an option in the Preferences:
Correcting the VPC for missing observations
Missing observations can cause a bias in the VPC and hamper its diagnosis value. This page discusses why missing censored data or censored data replaced by the LOQ can cause a bias in the VPC, and how Monolix handles censored data to prevent this bias. It also explains the bias resulting from nonrandom dropout, and how this can be corrected with Simulx.
Troubleshooting
If the VPC plot is emtpy or with nonappropriate axis limits, follow the procedure below.
The problem appears after having clicked “Export > Export charts settings as default” on a previous project where the yaxis limits were different and this is now applied as default. It is possible to delete the default setting corresponding to the axis limits in the following way:

 Open the file C:/Users/<username>/lixoft/monolix/monolix2023R1/config/settings.default in a text editor
 Delete the following lines:
VPCContinuous\yInterval= ... outputPlot\yInterval= ...
 Save the file
 Reopen your project
4.4.2.Numerical predictive checks
Purpose
This plot displays the numerical predictive check (NPC). The NPC is a model diagnosis tool for continuous data which is closely related to the VPC procedure: the principle is similar, with a different way to visualize the resulting information. While the VPC maintains the time dimension and can be used to point out at which time points the model overpredicts or underpredicts the data, the NPC allows to compare the empirical cumulative distribution function (CDF) of the observations, computed on the original data set, with the theoretical cumulative distribution, computed from data simulated with the model of interest and the design structure of the original data set.
Note that since the NPC compares each observation with its own simulated distribution, there is no concern of data binning like for the VPC.
Examples
In the following example, the parameters of a twocompartment model with iv infusion and linear elimination are estimated on the remifentanil data set.
One can see the empirical CDF of remifentanil concentration in blue, compared to the theoretical CDF based on simulated data in black. The 90% prediction interval corresponding to the theoretical CDF is visualized as a light blue area. Discrepancies between the empirical CDF and this area are marked in red. The logscale on the xaxis allows to focus on small observations. The plot shows that the model underpredicts small observations, and tends to overpredict some observations between 20 and 40 units.
Settings
 General: Add/remove legend or grid.
 Display
 Empirical distribution: Add/remove empirical CDF.
 Predicted median: Add/remove theoretical CDF.
 Prediction interval: Add/remove the prediction interval for the theoretical CDF, and set the interpercentile level for the prediction interval (by default the level is 90) and its associated level.
 Outliers (area): Add/remove red areas indicating where the empirical CDF is outside the prediction interval.
 Calculations:
 Set the number of evaluation points in the NPC.
 Use BLQ: Choose to use BLQ data or to ignore it to compute the VPC. BLQ data can be simulated, or can be equal to the limit of quantification (LOQ). The latter case induces strong bias.
4.4.3.BLQ predictive checks
Purpose
This plot displays the proportion of censored data w.r.t. time. It is possible to choose the censoring interval. This plot is only available for projects with censored data.
Examples
The figure presents the simulated and empirical BLQ frequencies w.r.t.time (example taken from the censored1_project of the demos).
Censored interval
The censored interval can be modified in Display. By default, the limit and the censored values are used. However, one can look at a smaller censored interval for example. This is the case with the example below: on the left, the BLQ predictive check uses the default censored interval wher Min=Infinity because there is no LIMIT column in the dataset. Since the observations are PK concentrations, Min=0 has been specified on the right, and no more outlier area is visible. This suggests that the outlier areas seen on the left are due to predicted negative values for censored observations, which means that it would be important in this case to specify a null lower limit for the censored observations by adding a column LIMIT to the dataset.
Settings
 General: Add/remove legend or grid
 Display
 Empirical proportion: Add/remove the blue line for empirical proportion of censored observations in cumulative observations.
 Predicted median : Add/remove the median proportion of censored observations in cumulative observations calculated by simulation.
 Prediction interval: Add/remove the prediction interval given by the model for the 90% quantile. The level (quantile) can be modified.
 Outliers (area): Add/remove red areas indicating time points for which the empirical proportion is outside the prediction interval.
 Calculations
 Number of point for the discretization
 Censored interval: min and max for censored data. By default, the limit and the censored values are used. However, one can look at a smaller censored interval for example.
By default, the censored area corresponds to the data set description and the BLQ frequency observation, the prediction interval, and the outliers are displayed.
4.4.4.Prediction distribution
Purpose
This plot displays the prediction distribution. It allows to compare the observations with the theoretical distribution of the predictions. It is based on multiple simulations of all individuals from the dataset, without the residual error. The simulations use the estimated population parameters, without considering their uncertainty.
Example
Prediction distribution plots vary slightly for different types of data. For joint models for multivariate outcomes, a separate plot is available for each type of data.
 Continuous outcomes
In the following example, the parameters of a one compartmental model with first order absorption and linear elimination are estimated on the theophylline data set. One can see the prediction distribution of the concentration overlayed with the data set.
 Noncontinuous outcomes: count data and categorical data
count2_project (data = ‘count2_data.txt’, model = ‘count_library/poissonTimeVarying_mlxt.txt’)
Prediction distribution plots for count data and categorical data show the predicted frequencies of the categorized data over time, computed by MonteCarlo. In the following example, predictions come from a Poisson distribution with a time varying intensit. Note that hovering on a band reveals the corresponding modality.
Settings
 General: Add/remove legend or grid
 Display (for continuous data)
 Observed data
 BLQ: Add/remove BLQ data if present.
 Median: Add/remove the median of predictions.
 Level: set the level (90 by default). The distribution corresponds to .
 Number of bands: set the number of bands (9 by default) and the associated percentile in case of a discrete representation
 X Bins (for discrete data)
 Bin values: Add/remove vertical lines on the plot to indicate the bins.
 Bining criteria: Choose the bining criteria among equal width (default), equal size or leastsquares
 Number of bins: Choose a fixed number of bins or a range for automatic selection, and a range for the number of data points per bin.
By default, only the prediction distribution and the median are displayed (for continuous data).
4.5.Tasks results
4.5.1.Likelihood contribution
Purpose
This plot displays the contribution of each individual to the loglikelihood. It is only available if the loglikelihood was previously computed. It can be sorted by index or by loglikelihood value (either via linearization of importance sampling, depending on which has been computed).
Example
In the following example, the parameters of a onecompartment model with firstorder absorption, linear elimination and a delay are estimated on the warfarin data set. The figure shows the top ten contributions from individuals to the loglikelihood, computed via linearization or importance sampling methods. All the contributions appear in small size on the miniplot on top, as well as a window indicating the selection of individuals for the main histogram.
Here we notice that the subject with id 8 has a much higher contribution to the loglikelihood than all other subjects, meaning that its reponse is less well captured by the model than others. It is worth checking this individual in the plots of individual fits, and remove it from the data set if its observations look abnormal.
Settings
 General. Add/remove the legend, grid, or a miniplot that allows to select a range of ranks to display.
 Methods. Add/remove histograms bins for values computed by linearization or importance sampling, if they have been computed.
 Sorting. The user can choose to sort the histogram by index, or by contribution value computed with linearization or importance sampling, if they have been computed.
 Label. The user can choose to display labels on top of the bins to indicate subject indices or names (ids).
4.5.2.Standard errors of the estimates
Purpose
This plot displays a histogram with the relative standard error of each parameter estimate, computed with the Fisher Information Matrix. It is only available if the standard errors were previously computed, and it provides a graphical representation of the information already available in the column “R.S.E (%)” in the Pop.Param tab of the Results frame.
Example
In the following example, the parameters of a onecompartment model with firstorder absorption, linear elimination and a delay are estimated on the warfarin data set. The figure shows the relative standard errors of all population estimates, computed via linearization and importance sampling methods, with the exact values written in front of each bin.
This figure allows here to highlight beta_Cl_tsex_F as an estimate associated with a high standard error. This suggests to check the relevance of this covariate effect, by looking at the estimate for this parameter and the result of the Wald test in the statistical tests.
Settings
 General. Add/remove the legend or grid.
 Methods. Add/remove histograms bins for values computed by linearization or stochastic approximation, if they have been computed.
4.6.Convergence diagnosis
4.6.1.SAEM
Purpose
This plot displays the sequence of estimates for population parameters computed after each iteration of the SAEM algorithm. The purpose is to check the convergence of the algorithm. In addition, a convergence indicator gives the estimation for 2 x loglikelihood along the iterations.
Example
In the following example, the parameters of a onecompartment model with firstorder absorption and linear elimination are estimated on the theophylline data set. The vertical line indicates where the algorithm switches from the first phase to the second. Notice also that the convergence indicator is displayed. More details on the convergence indicator can be found here.
Settings
 Select plots and arrange layout. It is convenient when there are many parameters and the user wants to focus on some particular parameter convergence for example.
By default, 12 parameters are displayed.
4.6.2.MCMC
Purpose
This plot displays the sequence of estimates for the conditional means and the conditional standard deviations along the iterations of the MH algorithm during individual parameters estimation by MCMC. The purpose is to check the convergence of the algorithm. The algorithm stops when these sequences remain in an interval of a given amplitude for a certain number of iterations: this interval is visualized on the figure with horizontal lines. The plot is shown and interactively updated while the task is running, and can be found after the end of the task in the Plots frame.
Example
In the following example, the parameters of a onecompartment model with firstorder absorption and linear elimination are estimated on the theophylline data set.
Settings
 Select plots and arrange layout. It is convenient when there are many parameters and the user wants to focus on some particular parameter convergence for example.
By default, 12 parameters are displayed.
4.6.3.Importance sampling
Purpose
This plot displays the sequence of estimates for observed loglikelihood computed by Monte Carlo approach. The purpose is to check the convergence of the algorithm.
Example
In the following example, the parameters of a threecompartments infusion model with linear elimination are estimated on the remifentanil data set. As explained on this page, the bias of the loglikelihood estimator decreases with the number of iterations, before the estimation value stabilizes. The number of points in the plot is usually smaller than the number of iterations, and depends on the total number of observations.
Settings
 Add/remove grid.
4.7.Export charts
All plots generated by Monolix can be exported
All plots generated by Monolix can be exported as a figure or as text files in order to be able to plot it in another way or with other software for more flexibility.
All the files can be exported in R for example using the following command
read.table("/path/to/file.txt", sep = ",", comment.char = "", header = T)
Remarks
 The separator is the one defined in the user preferences. We set “,” in this example as it is the one by default.
 The command comment.char = "" is needed for some files because to define groups or color, we use the character # that can be interpreted as a comment character by R.
The list of plots below corresponds to all the plots that Monolix can generate. They are computed with the task “Plots”, and the list of plots to compute can be selected by clicking on the button next to the task as shown below, prior to running the task.
Exporting the charts data can be made through the Export menu or through the preferences as described here.
In the following, we describe all the files generated by the export function
 Charts concerning the Data (Observed data).
 Charts concerning the model for the observations (Individual fits, Observations vs predictions, Scatter plot of the residuals, Distribution of the residuals
 Charts concerning the model for the individual parameters (Distribution of the individual parameters, Distribution of the random effects, Correlation between random effects, Individual parameters vs covariates)
 Charts concerning the predictive checks and predictions (Visual predictive checks, Numerical predictive checks, BLQ predictive checks, Prediction distribution)
 Charts concerning the convergence diagnosis (SAEM, MCMC) and the tasks results (Standard errors for the estimates)
Charts concerning the Data
Observed data (continuous, categorical, and count)
xxx_observations.txt
Description: observation values
Full output file description
Column  Description  Comment 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Observation time  – 
y  Observation value (loq)  The name of the column is the observation name 
censored  1 if the observation is censored, 0 otherwise  – 
split  Name of the split the subject belongs to  – 
color  Name of the color the observation is colored with  – 
filter  1 if the subject is filtered, 0 otherwise  – 
Observed data (event)
xxx_curves.txt
Description: observation values
Full output file description
Column  Description  Needed Task 
time  Observation times  – 
survivalFunction  Survival of first event  – 
averageEventNumber  Average number of event at that time  – 
split  Name of the split the subject occasion belongs to  – 
xxx_censored.txt
Description: censored values
Full output file description
Column  Description  Needed Task 
time  Observation times  – 
values  Survival of first event  – 
split  Name of the split the subject occasion belongs to  – 
Model for the observations
Individual Fits
xxx_observations.txt
Description: observation values
Full output file description
Column  Description  Comment 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Observation time  – 
y  Observation value (loq)  – 
median  Prediction interval median  – 
piLower  Lower percentile of the individual prediction interval  – 
piUpper  Upper percentile of the individual prediction interval  – 
censored  1 if the observation is censored, 0 otherwise  – 
xxx_fits.txt
Description: individual fits based on population parameters and individual parameters
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Continuous time grid used to compute fits  – 
pop  Prediction using population parameter values and average covariate value from the population (continuous) or reference covariate value (categorical).  – 
popPred  Prediction using population parameter values and individual covariates.  – 
indivPredMean  Prediction based on the individual parameter values estimated by conditional mean  Conditional distribution need to be computed 
indivPredMode  Prediction based on the individual parameter values estimated by conditional mode  EBEs need to be computed 
Observation vs Prediction
xxx_obsVsPred.txt
Description: observation and prediction (pop & indiv) values
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  time of the observation  
y  Observation value (loq)  – 
y_simBlq  Observation value (simulated blq)  
popPred  Predictions based on population parameter values  
indivPredMean  Predictions based on the individual parameter values estimated by conditional mean  Conditional distribution need to be computed 
indivPredMode  Predictions based on the individual parameter values estimated by conditional mode  EBEs need to be computed 
censored  1 if the observation is censored, 0 otherwise  
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise  – 
xxx_obsVsSimulatedPred.txt
Description: observation and simulated prediction values
Full output file description
Column  Description  Needed Task 
rep  Replicate id  
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  time of the observation  
y  Observation value (loq)  – 
y_simBlq  Observation value (simulated blq)  
indivPredSimulated  Predictions based on the simulated individual parameter values estimated by conditional distribution  Conditional distribution need to be computed 
censored  1 if the observation is censored, 0 otherwise  
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise 
xxx_visualGuides.txt
Description: splines and confidence intervals for predictions
Full output file description
Column  Description  Needed Task 
popPred  Continuous grid over population prediction values  – 
popPred_spline  Spline ordinates for population predictions  – 
popPred_piLower  Lower percentile of prediction interval for population predictions  – 
popPred_piUpper  Upper percentile of prediction interval for population predictions  – 
indivPred  Continuous grid over individual prediction values  – 
indivPred_spline  Spline ordinates for individual predictions  – 
indivPred_piLower  Lower percentile of prediction interval for individual predictions  – 
indivPred_piUpper  Upper percentile of prediction interval for individual predictions  – 
split  Name of the split the visual guides belong to 
Distribution of the residuals
xxx_pdf.txt
Description: probability density function of each residual type (pwres, iwres, npde)
Full output file description
Column  Description  Needed Task 
pwRes_abscissa  Abscissa for pwres pdf  – 
pwRes_pdf  Pdf of pwres  – 
iwRes_abscissa  Abscissa for iwres pdf  – 
iwRes_pdf  Pdf of the iwres  – 
npde_abscissa  Abscissa for npde pdf  – 
npde_pdf  Pdf of the npde  – 
split  – 
xxx_cdf.txt
Description: cumulative distribution function of each residual type (pwres, iwres, npde)
Full output file description
Column  Description  Needed Task 
pwRes_abscissa  Abscissa for pwres cdf  – 
pwRes_cdf  Cdf of pwres  – 
iwRes_abscissa  Abscissa for iwres cdf  – 
iwRes_cdf  Cdf of the iwres  – 
npde_abscissa  Abscissa for npde cdf  – 
npde_cdf  Cdf of the npde  – 
split  – 
theoreticalGuides.txt
Description: theoretical guides for the pdf and the cdf
Full output file description
Column  Description  Needed Task 
abscissa,pdf,cdf  Abscissa for the theoretical curves  – 
Theoretical value of the pdf  –  
cdf  Theoretical value of the cdf  – 
Scatter plot of the residuals
xxx_prediction_percentiles_iwRes.txt
Description: prediction percentiles of the iwREs to plot iwRes w.r.t. the prediction. The same files exists with the pwres and the npde.
Full output file description
Column  Description  Needed Task 
prediction  Value of the prediction  – 
empirical_median  Empirical median of the iwRes  – 
empirical_lower  Empirical lower percentile of the iwRes  
empirical_upper  Empirical upper percentile of the iwRes  
theoretical_median  Theoretical median of the iwRes  
theoretical_lower  Theoretical lower of the iwRes  
theoretical_upper  Theoretical upper of the iwRes  
theoretical_median_piLower  Lower bound of the theoretical median prediction interval  
theoretical_median_piUpper  Upper bound of the theoretical median prediction interval  
theoretical_lower_piLower  Lower bound of the theoretical lower prediction interval  
theoretical_lower_piUpper  Upper bound of the theoretical lower prediction interval  
theoretical_upper_piLower  Lower bound of the theoretical upper prediction interval  
theoretical_upper_piUpper  Upper bound of the theoretical upper prediction interval  
split  Name of the split the subject occasion belongs to 
xxx_time_percentiles_iwRes.txt
Description: time percentiles of the iwREs to plot iwRes w.r.t. the time. The same files exists with the pwres and the npde.
Full output file description
Column  Description  Needed Task 
time  Value of the time  – 
empirical_median  Empirical median of the iwRes  – 
empirical_lower  Empirical lower percentile of the iwRes  
empirical_upper  Empirical upper percentile of the iwRes  
theoretical_median  Theoretical median of the iwRes  
theoretical_lower  Theoretical lower of the iwRes  
theoretical_upper  Theoretical upper of the iwRes  
theoretical_median_piLower  Lower bound of the theoretical median prediction interval  
theoretical_median_piUpper  Upper bound of the theoretical median prediction interval  
theoretical_lower_piLower  Lower bound of the theoretical lower prediction interval  
theoretical_lower_piUpper  Upper bound of the theoretical lower prediction interval  
theoretical_upper_piLower  Lower bound of the theoretical upper prediction interval  
theoretical_upper_piUpper  Upper bound of the theoretical upper prediction interval  
split  Name of the split the subject occasion belongs to 
xxx_residuals.txt
Description: residuals values (pwres, iwres, npde)
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Observation times  – 
prediction_pwRes  Predictions based on population parameter values  SAEM 
pwRes  PwRes (computed with observations)  SAEM 
pwRes_blq  PwRes (computed with simulated blq)  
prediction_iwRes_mean  Predictions based on the individual parameter values estimated by conditional mean (INDIVESTIM) if available, SAEM either  
iwRes_mean  IwRes (computed with observations and individual parameter values estimated by conditional mean (INDIVESTIM) if available, SAEM either)  
iwRes_mean_simBlq  IwRes (computed with simulated blq and individual parameter values estimated by conditional mean (INDIVESTIM) if available, SAEM either)  
prediction_iwRes_mode  Predictions based on the individual parameter values estimated by conditional mode (INDIVESTIM)  
iwRes_mode  IwRes (computed with observations and the individual parameter values estimated by conditional mode (INDIVESTIM))  
iwRes_mean_simBlq  IwRes (computed with simulated blq and the individual parameter values estimated by conditional mode (INDIVESTIM))  
prediction_npde  Predictions based on population parameter values  
npde  Npde (computed with observations)  
npde_simBlq  Npde (computed with simulated blq)  SAEM – If there are some censored data in the data set 
censored  1 if the observation is censored, 0 otherwise  
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise  – 
xxx_simulatedResiduals.txt
Description: simulated residuals values
Full output file description
Column  Description  Needed Task 
rep  replicate  – 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Observation times  – 
prediction_iwRes  Predictions based on the simulated individual parameter values based on the conditional distribution  
iwRes_simulated  IwRes (computed with observations and the simulated individual parameter values)  
iwRes_simulated_simBlq  IwRes (computed with simulated blq and the simulated individual parameter values)  
censored  1 if the observation is censored, 0 otherwise  – 
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise  – 
xxx_spline.txt
Description: splines (residuals values against time and prediction)
Full output file description
Column  Description  Needed Task 
time_pwRes  Time grid for pwRes spline  SAEM 
time_pwRes_spline  pwRes against time spline  SAEM 
time_iwRes  Time grid for iwRes spline  At least SAEM 
time_iwRes_spline  iwRes against time spline  At least SAEM 
time_npde  Time grid for npde spline  SAEM 
time_npde_spline  npde against time spline  SAEM 
prediction_pwRes  Prediction grid for pwRes spline  SAEM 
prediction_pwRes_spline  pwRes against population prediction spline  SAEM 
prediction_iwRes  Prediction grid for iwRes spline  At least SAEM 
prediction_iwRes_spline  iwRes against individual prediction spline  At least SAEM 
prediction_npde  Prediction grid for npde spline  SAEM 
prediction_npde_npde  npde against population prediction spline  SAEM 
split  Name of the split the visual guides belong to  If the chart is splitted 
xxx_{time,population,individual}Bins.txt
Description: bins values for the corresponding axis.
Full output file description
Column  Description  Needed Task 
binsValues  Abscissa bins values  – 
split  Name of the split the bins refer to  If the chart is splitted 
Model for the individual parameters
Distribution of the individual parameters
cdf.txt
Full output file description
Column  Description  Needed Task 
param_abscissa  Abscissa of the cdf of the individual parameter param  – 
param_cdf  Empirical cdf of the individual parameter param  – 
split  Name of the split the subject occasion belongs to 
pdf.txt
Full output file description
Column  Description  Needed Task 
param_abscissa  Abscissa of the pdf of the individual parameter param  – 
param_pdf  Empirical pdf of the individual parameter param  – 
split  Name of the split the subject occasion belongs to 
visualGuides.txt
Full output file description
Column  Description  Needed Task 
param_abscissa  Abscissa of the pdf and the pdf of the individual parameter param  – 
param_pdf  Theoretical pdf of the individual parameter param  – 
param_cdf  Theoretical cdf of the individual parameter param 
Distribution of the random effects
cdf.txt
Full output file description
Column  Description  Needed Task 
param_abscissa  Abscissa of the cdf of the standardized random effect of param  – 
param_cdf  Empirical cdf of the standardized random effect of param  – 
split  Name of the split the subject occasion belongs to 
pdf.txt
Full output file description
Column  Description  Needed Task 
param_abscissa  Abscissa of the pdf of the standardized random effect of param  – 
param_pdf  Empirical pdf of the standardized random effect of param  – 
split  Name of the split the subject occasion belongs to 
StandardizedEta.txt
Description: standardized random effects of the individual parameters
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
standEta_X_method  Standardized random effects values on each individual parameter with variability.
It can be StandEta_SAEM, StandEta_Mean, StandEta_Mode 
– 
filter  1 if the subject is filtered, 0 otherwise  – 
SimulatedStandardizedEta.txt
Description: simulated standardized random effects of the individual parameters
Full output file description
Column  Description  Needed Task 
rep  Replicate id  – 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
standEta_X_simulated  Standardized random effects values on each individual parameter with variability.  – 
filter  1 if the subject is filtered, 0 otherwise  – 
Correlation between Random Effects
eta.txt
Description: standard error on individual parameter predictions
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
eta_X_method  Random effects values on each individual parameter with variability. It can be
eta_SAEM, eta_Mean, or eta_Mode 
– 
color  Name of the color the ID is colored with  – 
filter  1 if the subject is filtered, 0 otherwise  – 
simulatedEta.txt
Description: standard error on individual parameter predictions
Full output file description
Column  Description  Needed Task 
rep  Replicate number  
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
eta_X_simulated  Simulated random effects values on each individual parameter  – 
color  Name of the color the ID is colored with  – 
filter  1 if the subject is filtered, 0 otherwise  – 
visualGuides.txt
Description: spline and linear regression for each couple of individual parameters plotted one against the other
This is done for each combination of parameter p1 and p2 to have p1 w.r.t. p2
Full output file description
Column  Description  Needed Task 
p1_vs_p2_abscissa  Abscissa  – 
p1_vs_p2_spline  Spline ordinates  – 
p1_vs_p2_regression  Linear regression ordinates  – 
Individual Parameters Vs Covariates
covariates.txt
Description: individual parameters and random effects and covariate value for each subject
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
trans_X_method  Individual parameter value in the transformed space. It can be X_SAEM, X_mean, or X_mode  – 
eta_X_method  Random effects values. It can be eta_X_SAEM, eta_Mean, or eta_Mode  – 
covariate  Covariates values  
split  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise 
simulatedCovariates.txt
Description: simulated individual parameters and random effects and covariate value for each subject
Full output file description
Column  Description  Needed Task 
rep  Replicate of the simulation  
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
trans_X_simulated  Simulated transformed individual parameter value.  – 
eta_X_simulated  Simulated random effects values.  – 
covariate  Covariates values  
split  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise  – 
visualGuides.txt
Description: spline and linear regression for each couple of individual parameters plotted against a covariate
This is done for each combination of parameter param and covariate cov to have param w.r.t. cov
Full output file description
Column  Description  Needed Task 
param_vs_cov_abscissa  Abscissa  – 
param_vs_cov_spline  Spline ordinates  – 
param_vs_cov_regression  Linear regression ordinates  – 
Predictive checks and prediction
Visual Predictive Checks (continuous)
xxx_observations.txt
Description: observation values
Full output file description
Column  Description  Needed Task 
ID  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Observation times  – 
timeAfterLastDose  Observation times after the last dose  – 
y  Observation values (loq)  – 
y_simBlq  Observation values (simulated blq)  
censored  1 if the observation is censored, 0 otherwise  – 
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise  – 
xxx_bins.txt
Description: bins values for the corresponding axis.
Full output file description
Column  Description  Needed Task 
binsValues  Abscissa bins values  – 
split  Name of the split the bins refer to  If the chart is splitted 
xxx_percentiles.txt
Description: empirical and theoretical percentiles values (lower, median & upper) + confidence interval on theoretical percentiles (lower & upper)
Full output file description
Column  Description  Needed Task 
bins_middles  Abscissa bin middles  – 
empirical_median  Empirical median  – 
empirical_lowerPercentile  Empirical lower percentile  – 
empirical_upperPercentile  Empirical upper percentile  – 
theoretical_median  Theoretical median  – 
theoretical_median_piLower  Lower percentile of prediction interval of theoretical median  – 
theoretical_median_piUpper  Upper percentile of prediction interval of theoretical median  – 
theoretical_lowerPercentile  Median of the prediction interval for the lower percentile  – 
theoretical_lower_piLower  Lower percentile of the prediction interval for the lower percentile  – 
theoretical_lower_piUpper  Upper percentile of the prediction interval for the lower percentile  – 
theoretical_upperPercentile  median of the prediction interval for the upper percentile  – 
theoretical_upper_piLower  Lower percentile of the prediction interval for the upper percentile  – 
theoretical_upper_piUpper  Upper percentile of the prediction interval for the upper percentile  – 
split  Name of the split the visual guides belong to  If the chart is splitted 
xxx_simulations.txt
Description: simulated values. This file is not exported by default with the charts data, but it can be exported by clicking on “Export > Export VPC simulations” in the application menu. Moreover, if the option “Export VPC simulations” is enabled in the Preferences, it is exported each time the VPC is generated.
Full output file description
Column  Description  Needed Task 
rep  Simulation replicate  – 
ID  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time / timeAfterLastDose  Observation times or times after last dose  timeAfterLastDose if the corresponding option is used in the VPC 
sim_y  Simulated observation values (loq)  – 
censored  1 if the observation is censored, 0 otherwise  – 
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject is filtered, 0 otherwise  – 
Visual Predictive Checks (discrete)
xxx_distribution.txt
Description: discrete observation modalities theoretical distribution (among continuous time grid)
Full output file description
Column  Description  Needed Task 
binsTimeBefore  Abscissa bins time value before this time  – 
binsTimeAfter  Abscissa bins time value after this time  
propCategory_empirical  Empirical proportion of the modality set represented by the subchart  – 
propCategory_median  Median of the prediction interval for the proportion of the modality set represented by the subchart  – 
propCategory_piLower  Lower percentile of the prediction interval for proportion of the modality set represented by the subchart  – 
propCategory_piUpper  Upper percentile of the prediction interval for proportion of the modality set represented by the subchart  – 
category  Label for modality sets  
split  Name of the split the distributions refer to  – 
xxx_xBins.txt
Description: bins values for the xaxis.
Full output file description
Column  Description  Needed Task 
binsValues  Abscissa bins values  – 
split  Name of the split the bins refer to  If the chart is splitted 
Visual Predictive Checks (event)
xxx_curves.txt
Description: observation values
Full output file description
Column  Description  Needed Task 
time  Observation times  – 
survivalFunction_empirical  Empirical survival of first event  – 
survivalFunction_median  Survival median  
survivalFunction_pXX  Survival percentile XX  
averageEventNumber_empirical  Empirical mean number of events  – 
averageEventNumber_median  Median mean number of events  
averageEventNumber_pXX  Percentile XX mean number of events  
split  Name of the split the subject occasion belongs to  If the chart is splitted 
xxx_censored.txt
Description: censored values
Full output file description
Column  Description  Needed Task 
time  Observation times  – 
values  Survival of first event  – 
split  Name of the split the subject occasion belongs to  – 
BLQ Predictive Checks
xxx_cumulatedBLQfrequencies.txt
Description: censored simulated observations cumulated frequency
Full output file description
Column  Description  Needed Task 
time  Time  – 
empiricalCumulatedFrequencies  Empirical fraction of data that is BLQ between time 0 and time t  – 
median  Prediction interval median  – 
piLower  Lower bound of prediction interval  – 
piUpper  Upper bound of prediction interval  – 
split  Name of the split the visual guides belong to 
Numerical Predictive Check
xxx_cdf.txt
Description: empirical and theoretical cumulative distribution function of observations
Full output file description
Column  Description  Needed Task 
time  Cdf continuous grid time  – 
empiricalCdf  Empirical cdf based on observations  – 
theoreticalCdf  Median of prediction interval for cdf  – 
piLower  Lower percentile of prediction interval for cdf  – 
piUpper  Upper percentile of prediction interval for cdf  – 
split  Name of the split the visual guides belong to 
Prediction distribution (continuous)
xxx_observations.txt
Description: observation values
Full output file description
Column  Description  Needed Task 
id  Subject identifier  – 
OCC  Occasion value  (optional) if there is IOV in the data set 
time  Observation times  – 
y  Observation values (loq)  – 
censored  1 if the observation is censored, 0 otherwise  – 
split  Name of the split the subject occasion belongs to  
color  Name of the color the observation is colored with  
filter  1 if the subject occasion is filtered, 0 otherwise 
xxx_percentiles.txt
Description: theoretical percentiles computed on continuous grid
Full output file description
Column  Description  Needed Task 
time  Continuous time grid used for simulation  – 
median  Median  – 
pPercentile  Percentile  – 
split  Name of the split the visual guides belong to  If the chart is splitted 
Prediction distribution (discrete)
xxx_distribution.txt
Description: discrete observation modalities theoretical distribution (among continuous time grid)
Full output file description
Column  Description  Needed Task 
binsTimeBefore  Abscissa bins time value before this time  – 
binsTimeAfter  Abscissa bins time value after this time  
propCategory_empirical  Empirical proportion of the modality set represented by the subchart  – 
propCategory_median  Median of the prediction interval for the proportion of the modality set represented by the subchart  – 
propCategory_piLower  Lower percentile of the prediction interval for proportion of the modality set represented by the subchart  – 
propCategory_piUpper  Upper percentile of the prediction interval for proportion of the modality set represented by the subchart  – 
category  Label for modality sets  
split  Name of the split the distributions refer to  – 
xxx_xBins.txt
Description: bins values for the xaxis.
Full output file description
Column  Description  Needed Task 
binsValues  Abscissa bins values  – 
split  Name of the split the bins refer to  If the chart is splitted 
Convergence diagnosis
SAEM
CvParam.txt
Description: evolution of the parameters during SAEM iterations
Full output file description
Column  Description  Needed Task 
iteration  Iteration number  – 
convergenceIndicator  Convergence Indicator  – 
phase  1 for exploratory and 2 for smoothing  
X  Parameter X  – 
MCMC
convergences.txt
Description: evolution of the convergence with respect to the MCMC iterations
Full output file description
Column  Description  Needed Task 
iteration  Iteration number  – 
E_X  Conditional expectation of the parameter X  – 
sd_X  Conditional standard error of the parameter X 
bounds.txt
Description: bounds for each parameter corresponding to the bounds on the graph. The first line corresponds to the minimum and the second one corresponds to the maximum.
Full output file description
Column  Description  Needed Task 
E_X  Conditional expectation of the parameter X  – 
sd_X  Conditional standard error of the parameter X 
Standard errors of the estimates
rse.txt
Description: Standards errors for each parameter
Full output file description
Column  Description  Needed Task 
X  parameter  – 
rse_lin  Relative standard error with linearization method  – 
rse_sa  Relative standard error with linearization method 
5.FAQ
This page summarizes the frequent questions about Monolix.
 Download, Installation, Run and Display issues
 Regulatory
 What is needed for a regulatory submission using Monolix?
 How to cite Monolix, Simulx or PKanalix?
 Running Monolix
 On what operating systems does Monolix run?
 Is it possible to run Monolix using a simple command line?
 How to initialize my parameters?
 Results
 Can I define myself the result folder?
 What result files are generated by Monolix?
 Can I replot the plots using another plotting software?
 When I open a project, my results are not loaded (message “Results have not been loaded due to an old inconsistent project”). Why?
 Are the results of Monolix reproducible if I rerun the same analysis?
 My Observed data or VPC plot is empty or with inappropriate axis limits. What is going wrong?
 Tasks
 How are the censored data handled?
 How are the parameters without variability handled?
 What is the convergence indicator, displayed during SAEM?
 When estimating the loglikelihood via importance sampling, the loglikelihood does not seem to stabilize. What can I do?
 Model definition
 Is it possible to use timevarying covariates?
 Is it possible to define complex covariateparameter relationships such as MichaelisMenten for instance?
 Is it possible to define a categorical covariate effect on the standard deviation of a parameter?
 Is it possible to define mixture of structural models?
 Is it possible to define mixture of distributions?
 Can I set bounds on the population parameters for example between a and b?
 Can I put any distribution on the population parameters? Not directly through the interface.
 Can I set a custom error model?
 What are the units of estimated parameters? Can I define a scale factor?
 Tricks
 How to compute AUC, time interval AUC, … using Mlxtran in a Monolix project?
 How can I calculate the coefficient of variation?
 Export
 How to export a Monolix project to PKanalix and Simulx?
 I exported the chart settings as default, how can I come back to the default Monolix settings for all plots?
 Reporting
 How can I include plots and table from different Monolix projects into the same Word document?
 The report I generated via command line or via R does not include any plot.
 Feedback
 I would like to give a suggestion for the future versions of Monolix.
 I need more details in the documentation for a specific feature.
Regulatory

 What is needed for a regulatory submission using Monolix? Monolix is used for regulatory submissions (including the FDA and the EMA) of population PK and PK/PD analyses. The summary of elements needed for submission can be found here.
 How to cite Monolix, Simulx or PKanalix? Please reference each application as here (adjusting to the version you used):
Monolix 2023R1, Lixoft SAS, a Simulations Plus company Simulx 2023R1, Lixoft SAS, a Simulations Plus company PKanalix 2023R1, Lixoft SAS, a Simulations Plus company
Running Monolix
 On what operating systems does Monolix run? MonolixSuite runs on Windows, Linux and MacOS platform. The requirements for each platform can be found on our download page.
 Is it possible to run Monolix using a simple command line? Yes, see here. In addition, there is a complete R API providing the full flexibility on running and modifying a Monolix project.
Initialization
 How to initialize my parameters? There are several ways to initialize your parameters and visualize the impact. See here the different possibilities.
Results
 Can I define myself the result folder? By default, the result folder corresponds to the project name. However, you can define it by yourself. See here to see how to define it on the user interface.
 What result files are generated by Monolix? Monolix proposes a lot of different output files depending on the tasks done by the user. Here is a complete listing of the files along with the condition for creation. See here for more information.
 Can I replot the plots using another plotting software? Yes, if you go to the menu Export and click on “Export charts data”, all the data needed to reproduce the plots are stored in text files. See here for the description of all the files generated along with the plots.
 When I open a project, my results are not loaded (message “Results have not been loaded due to an old inconsistent project”). Why? When loading a project, Monolix checks that the project (i.e all the information saved in the .mlxtran file) being loaded and the project that has been used to generate the results are the same. If not, the error message is shown. For instance if one runs a project, then do “use last estimates”, save and try to reload the project, the saved project has the “last estimates” as initial values which are different from the initial values used to run and generate the results. In that case the results will not be loaded because they are inconsistent with the loaded project. This is also explained with some examples in this video.
It is possible to check what is preventing the load of the results by comparing the content of the .mlxtran file to load and the .mlxtran file located in the hidden .Internals folder in the result folder. To see the .Internals folder, the “show the hidden files/folders” must be activated on the machine.
 Are the results of Monolix reproducible if I rerun the same analysis? Yes, see here for more information.

The Observed data plot or VPC plot is emtpy or with nonappropriate axis limits
The problem appears after having clicked “Export > Export charts settings as default” on a previous project where the yaxis limits were different and this is now applied as default. It is possible to delete the default setting corresponding to the axis limits in the following way: Open the file C:/Users/<username>/lixoft/monolix/monolix2023R1/config/settings.default in a text editor
 Delete the following lines:
VPCContinuous\yInterval= ... outputPlot\yInterval= ...
 Save the file
 Reopen your project
Tasks
 How are the censored data handled? The handling of censored data is described here.
 How are the parameters without variability handled? The different methods for parameters without variability are explained here.
 What is the convergence indicator, displayed during SAEM? Its meaning and function is explained here.
 When estimating the loglikelihood via importance sampling, the loglikelihood does not seem to stabilize. What can I do? The loglikelihood estimator obtained by importance sampling is biased by construction (see here for details). To reduce the bias, the conditional distribution \(p_{\phi_iy_i}\) should be known as well as possible. For this, run the “conditional distribution” task before estimating the loglikelihood.
Model definition
 Is it possible to use timevarying covariates? Yes, however the covariates relationship must be defined in the model instead of the GUI. See here how to do that.
 Is it possible to define complex covariateparameter relationships such as MichaelisMenten for instance? Yes, this can be done directly in the model file. See here how to do it.
 Is it possible to define a categorical covariate effect on the standard deviation of a parameter? Yes, this can be done directly in the model file. See here how to do it.
 Is it possible to define mixture of structural models? Yes, it may be necessary in some situations to introduce diversity into the structural models themselves using betweensubject model mixtures (BSMM) or withinsubject model mixtures (WSMM). The handling of mixture of structural models is defined here. Notice that in the case of a BSMM, the proportion p between groups is a population parameter of the model to estimate. There is no interpatient variability on p: all the subjects have the same probability and a logitnormal distribution for p must be used to constrain it to be between 0 and 1 without any variability.
 Is it possible to define mixture of distributions? Yes, the handling of mixture of structural models is defined here.
 Can I set bounds on the population parameters for example between a and b? It is not possible to set bounds for the estimated population parameters. However it is possible to define bounded parameter distributions, which as a consequence also bound the estimated fixed effect parameter. See here how to do it.
 Can I put any distribution on the population parameters? Not directly through the interface. Using the interface, you can only put normal, lognormal, logitnormal and probitnormal. However, you can set any transformation of your parameter in the EQUATION: part of the structural model and apply any transformation on it. See here how to do it.
 Can I set a custom error model? No, this is not possible. It may however be possible to transform the data such that the error model can be picked from the list. For an example with a modelbased metaanalysis project, see here.
 What are the units of estimated parameters? Can I define a scale factor? The units of the estimated parameters depend on the units of the data set and are implicit. Check here to learn how to include a scale factor.
Tricks
 How to compute AUC, time interval AUC, … using Mlxtran in a Monolix project? See here.
 How can I calculate the coefficient of variation? The coefficient of variation is not outputted by Monolix but can easily be calculated manually. The coefficient of variation is defined as the ratio of the standard deviation divided by the mean. It is often reported for lognormally distributed parameters where it can be calculated as: \( \textrm{CV}=\sqrt{e^{\omega^2}1} \) with \( \omega \) the estimated standard deviation. See the video here.
Export
 How to export to Datxplore, Mlxpore and Simulx?
 I exported the chart settings as default, how can I come back to the default Monolix settings for all plots? Delete the file <home>/lixoft/monolix/monolix2020R1/config/settings.default . In windows, the path to the <home> folder is C:\Users\<username>.
Reporting
 How can I include plots and table from different Monolix projects into the same Word document? It is currently not possible to include plots and tables coming from different Monolix runs into the same Word document in one step. You can add placeholders to an already generated report and use it as a template for another project, however we are aware that this is not a very convenient solution. If generating a single report for several projects is important for you, please let us know so that we take it into account for future versions.
 The report I generated via command line or via R do not contain plots. This is a known limitation of the current implementation of the reporting feature. If generating reports with plots from R or from the command line is important for you, please let us know so that we take it into account for future versions.
5.1.Evolutions from Monolix2016R1 to Monolix2018R1
Monolix had a complete transformation to have a better interface and plots, better performance and be easier to use.
 Better performance thanks to the parallelization
 Better performance in structural model evaluation
 Algorithms settings
 SAEM algorithm
 Conditional distribution
 Conditional mode
 Standard error calculation
 Loglikelihood calculation
 Simulation computation for plots
 Plots during algorithms
 Tests computation
 Stratify
 Observed data enhancements
 Individual fits enhancements
 Observation vs Prediction enhancements
 Scatter plots of the residuals enhancements
 Distribution of the residuals enhancements
 Distribution of the individual parameters enhancements
 Distribution of the random effects enhancements
 Correlation between random effects enhancements
 Individual parameters vs covariates
 Visual Predictive Checks enhancements
 Prediction distribution
 Loglikelihood contribution
 New plots
Monolix project definition, settings, and outputs
Monolix Interface
The Monolix user interface is fresh new with a new javascript technology. It is not only one single frame anymore. There are now frames
Welcome frame
In this frame, it is possible to
– create a new project
– load a project
– load a recent project
– load a demo
– look at Monolix web documentation
Data frame
In this frame, the user defines its data set and tag each column of its data set. The possible column are the same but lot of names were changed to be more intuitive. Notice that, when the user defines the observation column, it should define its type (continuous/discrete/event)
When clicking on OK, it validates the data set and provide the possible use of it. When the data set is validated, a DATA VIEWER button appears providing the possibility to explore the data set parallel to the project.
Data frame enhancements
– error messages pop up when there is an error in the data set or in the consistency between the data set and the header definition.
– warning messages pop up when there is a warning in the interpretation of the data set.
– it is possible to scroll down the data set while keeping the header visible
– it is possible to sort the data set by any column
– loading a large data set is much more efficient
– it is possible to visualize the whole data
– number of doses can be chosen if there is steadystate
Structural model frame
In this frame, the user defines it structural model. The user can
– browse the file from any folder
– load a file from the library
– open in MlxEditor
– reload it (if it has been changed in the editor for example)
– error messages pop up when there is an error in the model or in the consistency between the data set and the proposed model.
– custom lixoft library browser to choose easily the model
Initial estimates frame
There are two possibilities
CHECK INITIAL ESTIMATES to see how the structural model fits each individual.
Enhancements
– it is possible to define the number of individuals showed and the associated layout
– it is possible to define the same xaxis and/or yaxis
– in case of bsmm, the two models are plotted in full and dotted red respectively
– the calculation is much faster and dedicated to the considered frame.
Evolution
– it is not possible to check the initial values of the beta’s anymore
– the grid for the prediction takes doses into account
Set values of the INITIAL ESTIMATES.
Enhancements
– there is a new link to fix all parameters.
– there is a new link to fix estimate all parameters (error model parameter ‘c’ is not affected by the ‘estimate all’ feature if its value is 1).
– there is a new link to use the last estimated values as initial estimates. Notice that this link is usable only if there has been no modification of the project.
– there is a new link to use only the last estimated fixed effects values as initial estimates.
– to define the estimation method is not a right click anymore, the user has to click on the wheel next to the parameter value.
– when the user clicks on the value, the associated constraint (typically: “Value must be >0”) is displayed to define the domain of definition of the parameter.
– there are error messages when the initial values are not set to a correct value (due to the associated distribution).
– in case of IOV, all the random effects are on the same frame.
– in case of use of a categorical covariate with several modalities in the statistical model, the user can initialize all associated beta’s independently
– in case of use of a categorical covariate with several modalities in the statistical model, the user can define the estimation method on all associated beta’s independently.
Evolution
– it is not possible to use the last estimate if there were any modification of the project
– for bayesian, only the MAP option is available.
– the methods color evolved: black for MLE, orange for fix and purple for MAP
Statistical model and tasks frame
Tasks
– The task for the calculation of the individual parameter was splitted in two tasks (Ebes referring to the conditional mode and conditional distribution allowing the conditional mean).
– The task for the calculation of the individual parameter is displayed before the other one to be consistent with a scenario usage.
– Use of the linearization method is now shared between the standard errors calculation and the loglikelihood calculation.
– The convergence assessment is now using a user defined scenario and not the current one. Three scenario are proposed (computed the se and the LL and if linearization method is used). Notice that the plots are not run.
– Assessment: new plot last convergences (dot) for each run.
– Assessment: ‘Stop’ button stops the current run and keep only the previous ones.
– Assessment: interactivity with graphs in real time (zoom, layout, selected subplots).
– Assessment: there is a summary provided in the Assessment folder in the project result folder.
– Assessment: the scenario of the assessment is now independent of the scenario of the project. The user can choose between three scenari.
– The settings for each tasks is now available with a button on next to each task.
– It is not possible to reload a previous convergence assessment using the interface. However, all the results are in the result folder in an Assessment folder.
– The list of plots is now arranged in categories to increase readability
– Lists of plots can be selected (All, none) by categories or for all the plots.
Observation model
– A button formula was added to show the formula associated to the error model in case of continuous error model in real time
– Additional and customizable error model are proposed. Now, the user can choose in a list of both distribution (normal/lognormal/logitnormal) and error models (constant/proportional/combined1/combined2)
– Generalization of error models: parameter c is always a parameter of proportional and combined1/combined2 models (fixed to 1 by default)
– it is possible to choose the minimum and the maximum of the logit function when chosen.
– there are error messages when the minimum and maximum values are set to a correct value
– there is an error message if the user try to set the distribution as lognormal and it is not possible (in case of negative observations for example)
– Type of discrete models display
Individual model
– The display is very different and more synthetic
– A button formula was added to show the formula associated to the error model in case of continuous error model in real time
– The names of the parameters and the covariates are displayed
– In case of IOV, all levels are displayed in the same frame
– The choice of the parameter distribution is done by choosing in a list
– The choice of adding or removing variability is performed by clicking in the column “Random Effects”
– There are two buttons to add and remove variability on all parameters at the same time
– The correlation is not defined as a matrix anymore, the user must define groups and add parameters on those groups
– Adding a covariate on an individual parameter is performed by clicking in the covariate name column
– In case of IOV, the covariates are arranged by level of variability
– There is dedicated buttons to add a transformed continuous covariate, transformed categorical covariate and mixture
– To add a transformed continuous covariate, the user click on the button CONTINUOUS and the user can define a Name and a Formula.
— a Name is proposed
— the list of available covariates is proposed
— by clicking on one available covariates write it in the Formula
— overlaying an available covariate show the min, mean, median, and max of the covariate
— the formula can be any Mlxtran compatible expression
— the Formula can contains several covariates
– To add a transformed categorical covariate, the user click on the button CATEGORICAL and the user can define a Name and a groups.
— a Name is proposed
— the list of modalities is proposed
— one can allocate, reset an allocation, and modify the allocation
— the user can choose the reference category
— the user can choose the name of the groups
– To add a transformed categorical covariate, the user click on the button MIXTURE and the user can define the name and the number of modalities
– a magnifying glass icon is proposed to be able to locate the covariate when there are several covariates
– For each transformed covariate, there is a possibility to edit and remove this covariate
– there are errors displayed explaining the reason of the error if the action is not possible
Results
– New section so see all the tasks results
– better representation
– It contains a section for Population parameter estimates
– It contains a section for Individual parameter estimates with the conditional mode and conditional mean
– It contains a section for Correlation matrix of the estimates (and RSE) with the linearization method or the stochastic approximation
– The values of the elements of the correlation matrix and the rse are colored to improve readability and faster diagnosis
– A selected correlation in the matrix set a focus on both associated population parameters
– It contains a section for the Estimated loglikelihood and information criteria with the linearization method or the importance sampling method
– It contains a section for all the statistical tests
– The values of the elements of the tests are colored to improve readability and faster diagnosis
– It is possible to open the output folder directly from here
– Results display is loaded if the project has results
Monolix calculation engine
Better performance thanks to the parallelization
It is now possible to parallelize the calculation of Monolix over several machines using open mpi.
Better performance in structural model evaluation
– Faster analytical solutions
– Faster calculation for ODEs
– No restrictions to use analytical solutions if regressors are constant over the subject time. Sequential models (using a PK model and its associated analytical solution) will be much faster.
– Less restrictive conditions to use analytical solutions when IOV occurs
Bug fixed:
– A time varying initial condition (for DDE models) is now well taken into account
– A regressor as initial condition is now well taken into account
Algorithms settings
– Constraints for settings
– Names and reorganization modified for a better comprehension
– all the settings are now available through a button next to the task.
SAEM algorithm
– Addition of new error models. The user can now defined both the distribution and the error model.
– Optimization of SAEM strategy when the error model has several parameters (typically for combined1 and combined2 model).
– Strategy with simulated annealing for conbined1 and combined2 (improve convergence)
– Evolution of SAEM strategy when the error model is proportional (there were issues when the prediction was very close to zero).
– CPU time optimization of SAEM strategy.
– When latent covariate are used, the methodology to estimate the probabilities and the associated betas is now based on the mixing law and not on a individual probability draw. It allows a better evaluation of the Loglikelihood and better convergence properties.
– When there are parameters without variability,
— With the novariability method, the maximum number of iterations depends on the number of parameters without variability
— With the novariability method, the optimization is much faster.
— With the decreasing variability methods, the decreasing speed of the artificial variance is lower
— For normal law, better strategy to initialize variance (more consistent)
— when there is a latent covariate on the parameter, all methods can be used.
– When no parameter has variability and the novariability method is used. Only one iteration of SAEM is done.
– Two settings of SAEM were updated to provide a better convergence
— The minimum number of iterations in the exploratory phase is now at 150 (it was 100 previously)
— The step size exponent in the smoothing phase is now at .7 (it was 1 previously)
– Constraints for settings
– If SAEM reaches the maximum number of iterations during the exploratory phase, a warning arises.
Removed feature
– We removed the possibility to add different variances depending on the modality of a categorical covariate
– We removed the possibility to choose to work either with standard errors or variances. Only standard errors are proposed. However, variance project can be loaded.
– We removed the possibility to have a bayesian posterior distribution
– We removed the possibility to have a custom distribution of the individual parameters
– Autocorrelation can not be added anymore in graphical interface. However, it can be loaded or added by the connectors
Conditional distribution
– Conditional distribution can now be computed for discrete and event models.
– New setting: number of simulations by individual parameters
– Adaptative number of simulations value according to the data size
– If the Fisher Information Matrix by stochastic approximation has already been computed, all the MCMC draws are reused and providing a much faster calculation.
Conditional mode
– The calculation is now tremendously faster. (between 20 to 100 time faster)
Standard error calculation
– Fisher information matrix can now be computed with discrete and event models and IOV
– Improvement of the calculation for the linearization
– Improvement of the calculations for S.A. if there are nans
– Faster calculation for the linearization
– Decrease of the maximum of iterations to 200 (it was 300 previously)
– Settings are modified for S.A.: min and max iterations
– Warnings if there are numerical issues for linearization
– If the conditional distribution has already been computed, all the MCMC draws are reused and providing a much faster calculation.
Loglikelihood calculation
– Improvement of the calculation for the linearization
– Faster calculation in case of censored data
– Faster calculation in case of importance sampling
– When the calculation by linearization has issues, then warning is provided to the users.
– The number of MonteCarlo size in the importance sampling is now at 10000 (it was previously at 20000)
Simulation computation for plots
The simulation are much faster than in the previous version. It impacts a lot the time needed for the generation of the VPCs and the prediction interval for example.
In addition, a deep effort was done on the discrete and event models where the simulation is now tremendously faster. A progression bar is proposed too.
For the simulation on a grid, the doses and the regressors were added.
Plots during algorithms
– large interactivity (zoom, layout, coordinates)
– Possibility to switch between different frames during the algorithms calculation
– List of elements to compute for plots (‘Stop’ button keeps the done computations)
Tests computation
Tests are computed when the conditional distribution task is performed and the plots are launched. The following tests are computed
– Pearson’s correlation test on the individual parameter and the covariates used in the statistical model
– Pearson’s correlation test on the individual random effects and the covariates
– Fisher test for discrete covariates
– Shapiro Wilk test on the random effects
– Pearson’s correlation test on the random effects
– Shapiro Wilk test on the individual parameters that have no covariate
– Kolmogorov Smirnov adequacy test on the individual parameters that have covariates
– Van Der Waerden test on the residuals
– Shapiro Wilk test on the residuals
– For all tests associated to individual (parameters, random effects, NPDEs), the BenjaminiHochberg procedure is applied to avoid bias
Monolix plots
All the plots were updated with a new technology and new features. In addition, all the color/graphical can be changed in the Preferences frame.
Notice that
– When you save the projects, your current graphical settings are conserved
– you can export your settings to be your defaults in the Export menu.
Stratify
The user can now define all the stratifications needed in a Stratify frame in a very easy way and can split, color and filter bay any defined covariate.
Enhancements
– Large simplification of the usage
– For a continuous covariate, possibility to define groups with either equal number of individuals, or equal size
– Possibility to change all the color
– Possibility to highlight a full group when clicking on the covariate category
– Buttons to add and remove categories
– Better performance
Observed data enhancements
This plot contains all the observations and can be used with all types of observations. It produces
– the spaghetti plot for continuous observations.
– the spaghetti plot or histogram for discrete observations (the user has the possibility to switch).
– the kaplanMeier plot for event observations along with the mean number of events per individual
Enhancements
– When overlaying a curve, the ID is displayed and all the points of the subject are highlighted.
– When splitting, the information for each group is computed.
– When splitting, the user can choose to adapt the xaxis and or yaxis to each group or to have the same for all groups.
– Possibility to display the dosing times when overlaying an individual.
Individual fits enhancements
– It is possible to sort the individuals by individual parameters values.
– When there are censored data, the full interval is displayed
– the yscale management is better performed.
– Possibility to display the dosing times.
– The user can choose to share the xaxis and/or yaxis .
– Possibility to zoom on all the individual at the same time with a linked zoom.
– Population fits (population covariate)
– Grid takes doses (and regressors) into account
– Color is added for a better representation of IOV when occasions are joined (according the presence of washout or not)
Observation vs Prediction enhancements
– The conditional distribution can be used for this plots.
– 90% prediction interval is now available.
– Information on the outliers proportions.
– Overlaying a point will display both the ID and the time of the points (and its replicates if the conditional distribution is chosen). In addition, the other points corresponding to the same ID are also highlighted.
– The loglog scale management is more efficiently done.
Scatter plots of the residuals enhancements
– Possibility to have Scatter plot for event.
– IWRES can be computed with the conditional distribution.
– Overlaying a point will display both the ID and the time of the points (and its replicates if the conditional distribution is chosen). In addition,
— the other points corresponding to the same ID are also highlighted.
— the same points are overlaid on the other plots.
– 2 predefined configurations (VPC and scatter).
– In case of discrete models, the scatter plot w.r.t time was added.
Distribution of the residuals enhancements
– By overlaying a bar in the pdf plots, we have the percentage of individual in this bar.
– By overlaying in the cdf plot, the theoretical and empirical cdf are displayed along with the xaxis value.
– The qqplot representation was replaced by a cdf representation.
– The empirical pdf is not computed anymore.
Distribution of the individual parameters enhancements
– The non parametric pdf is not proposed anymore.
– By hovering over a bar in the pdf plots, we have the percentage of individual in this bar.
– The empirical and theoretical cdf of the individual parameters are now computed.
– By hovering over the cdf plot, the theoretical and empirical cdf are displayed along with the xaxis value.
– When splitting, the shrinkage information is computed.
Distribution of the random effects enhancements
– The empirical and theoretical pdf of the individual parameters are now computed.
– By hovering over a bar in the pdf plots, we have the percentage of individual in this bar.
– The empirical and theoretical cdf of the individual parameters are now computed.
– By hovering over the cdf plot, the theoretical and empirical cdf are displayed along with the xaxis value.
Correlation between random effects enhancements
– Correlation information is proposed.
– Hovering over a point will display the ID of the point (and its replicates if the conditional distribution is chosen). In addition, the same ID is overlaid in the other figures.
– Possibility to select the parameters to look at.
– Possibility to split the graphic.
– Optimized layout. At the beginning, a maximum of 6 random effects is displayed. However, the user can choose any number afterward.
Individual parameters vs covariates
– Overlaying a point will display the ID of the point (and its replicates if the conditional distribution is chosen). In addition, the same ID is overlaid in the other figures.
– Possibility to split and have all figures
– Possibility to select the parameters to look at.
– Possibility to select the covariates to look at.
– Possibility to split and color at the same time.
Visual Predictive Checks enhancements
– This plot contains all the observations and can be used with all types of observations.
– In case of categorical projects, there are no bins management on the yaxis. All the categories are displayed with the good ylabel
– In case of count projects, the ylabel is well defined
Prediction distribution
– Possibility to color the observations
– Possibility to differentiate the censored data and the non censored data
– Overlaying a point will display the ID of the point. In addition, the other points of the same ID are also overlaid.
– Overlaying a band will display the range of the band.
Loglikelihood contribution
– Possibility to zoom on part of the individuals
New plots
– Standards errors of the estimates
– MCMC convergence plot
– Importance Sampling convergence plot
Monolix project definition, settings, and outputs
Project evolution
In terms of project, there are only few modifications
– The definition of the number of doses in the STEADY STATE definition is now in the project and not in the user configuration
– In case of several outputs in the data set, the names of the type of outputs described in the observation is now named yname and not ytype anymore.. Retro compatibility is ensured.
– In case of a single output in the data set, the name of the type of output described in the observation was ytype=1. It is now removed as it is useless. Retro compatibility is ensured.
– The list of plots is now defined in the project file and not in the associated .xmlx anymore.
– The name of the tasks in the Mlxtran project evolved a little bit to be more consistent to the user interface.
Settings
In terms of project settings, there are only few modifications
– The flexibility to use or not the analytical solutions is now defined in the Mlxtran structural model and not in the user configuration
– The project settings are now available via the menu Settings/Project settings
– The user has the possibility to save the data and the model next to the project
– The preferences interface has evolved to be in javascript
– The working directory is not available through the interface but only with the user configuration file
– The change of the number of threads does not imply a restart of Monolix anymore
– We propose to automatically exports all the charts data after the run
– The charts export format are now .svg and .png
– The timestamping option is now called ‘Save History’. The project and its results is saved after each run now.
Configuration of the plots
It is now saved in a .properties associated to the project. It is not a xmxl anymore. It is not xmlx anymore but still readable. Retrocompatibility is only performed on the list of plots. This .properties
– overlay the default settings (default.settings in the user/lixoft folder)
– contains all the information for the display of the plots in terms of what is displayed
– contains all the information for the display of the plots in terms of the covariate stratification in the plot
– contains all the information for the display of the plots in terms of the colors and preferences for each plot
When saving a project, a .properties is generated ensuring to replot exactly the same figures after a reload.
It is possible to export all the settings to define it as the global settings.
Outputs
In terms of outputs, all the files and folder are reorganized. We now have
– summary.txt: providing a summary of the run
– populationparameter.txt with all the estimated population parameters
– the output predictions
– all the files concerning the Fisher Information Matrix are in a folder FisherInformation
– all the files concerning the individual parameters and the random effects are in a folder IndividualParameters
– all the files concerning the logLikelihood are in a folder logLikelihood
– all the files concerning the results of the Tests are in a folder Tests
– a part of the Lixoft files needed to reload is in a private folder .Internals
– when the charts data are exported, the data are exported in a folder ChartsData
– when the figures are exported, the data are exported in a folder ChartsFigures
– all figures can be exported independently
In terms of export, we can
– export all the charts data in Settings/Export charts data
– export all the figures in Settings/Export plots
– export the project in Mlxplore in Settings/Export in Mlxplore
Monolix Connectors
There is a R package associated to Monolix where the user has all the functions available through the interface. The following functions are available
– abort Stop the current task run
– addCategoricalTransformedCovariate: Add Categorical Transformed Covariate
– addContinuousTransformedCovariate: Add Continuous Transformed Covariate
– addMixture: Add Mixture Add a new latent covariate to the current model giving its name and its modality number.
– computePredictions: Compute predictions from the structural model
– getConditionalDistributionSamplingSettings: Get conditional distribution sampling settings
– getConditionalModeEstimationSettings: Get conditional mode estimation settings
– getContinuousObservationModel: Get continuous observation models information
– getCorrelationOfEstimates: Get the inverse of the Fisher Matrix
– getCovariateInformation: Get Covariates Information
– getData: Get project data
– getEstimatedIndividualParameters: Get last estimated individual parameter values
– getEstimatedLogLikelihood: Get LogLikelihood Values
– getEstimatedPopulationParameters: Get last estimated population parameter value
– getEstimatedRandomEffects: Get estimated the random effects
– getEstimatedStandardErrors: Get standard errors of population parameters
– getGeneralSettings: Get project general settings
– getIndividualParameterModel: Get Individual Parameter Model
– getLastRunStatus: Get last run status
– getLaunchedTasks: Get tasks with results
– getLogLikelihoodEstimationSettings: Get LogLikelihood algorithm settings
– getMCMCSettings: Get MCMC algorithm settings
– getMlxEnvInfo: Get information about MlxEnvironment object
– getObservationInformation: Get observations information
– getPopulationParameterEstimationSettings: Get population parameter estimation settings
– getPopulationParameterInformation: Get Population Parameters Information
– getPreferences: Get project preferences
– getProjectSettings: Get project settings
– getSAEMiterations: Get SAEM algorithm iterations
– getScenario: Get current scenario
– getSimulatedIndividualParameters: Get simulated individual parameters
– getSimulatedRandomEffects: Get simulated random effects
– getStandardErrorEstimationSettings: Get standard error estimation settings
– getStructuralModel: Get structural model file
– getVariabilityLevels: Get Variability Levels
– initializeMlxConnectors: Initialize MlxConnectors API
– isRunning: Get current scenario state
– loadProject: Load project from file
– mlxDisplay: Display Mlx API Structures
– newProject: Create new project
– removeCovariate: Remove Covariate
– runConditionalDistributionSampling: Sampling from the conditional distribution
– runConditionalModeEstimation: Estimation of the conditional modes (EBEs)
– runLogLikelihoodEstimation: LogLikelihood estimation
– runPopulationParameterEstimation: Population parameter estimation
– runScenario: Run Current Scenario
– runStandardErrorEstimation: Standard error estimation
– saveProject: Save current project
– setAutocorrelation: Set autocorrelation
– setConditionalDistributionSamplingSettings: Set conditional distribution sampling settings
– setConditionalModeEstimationSettings: Set conditional mode estimation settings
– setCorrelationBlocks: Set Correlation Block Structure
– setCovariateModel: Set Covariate Model
– setData: Set project data
– setErrorModel: Set error model
– setGeneralSettings: Set common settings for algorithms
– setIndividualParameterDistribution: Set Individual Parameter Distribution
– setIndividualParameterVariability: Individual Variability Management
– setInitialEstimatesToLastEstimates: Initialize population parameters with the last estimated ones
– setLogLikelihoodEstimationSettings: Set loglikelihood estimation settings
– setMCMCSettings: Set settings associated to the MCMC algorithm
– setObservationDistribution: Set observation model distribution
– setObservationLimits: Set observation model distribution limits
– setPopulationParameterEstimationSettings: Set population parameter estimation settings
– setPopulationParameterInformation: Population Parameters Initialization and Estimation Method
– setPreferences: Set preferences
– setProjectSettings: Set project settings
– setScenario: Set scenario
– setStandardErrorEstimationSettings: Set standard error estimation settings
– setStructuralModel: Set structural model file
5.2.Submission of Monolix analysis to regulatory agencies
Monolix is used for regulatory submissions (including the FDA and the EMA) of population PK and PK/PD analyses. Monolix analyses for first in human dose estimation, dosefinding studies and registration studies have been routinely and successfully submitted to the FDA [1], EMA [2] and other agencies. The FDA and the EMA do have access to Monolix and the modelling experts to understand, review and run Monolix. Regulators are also taking part in publishing research articles with Monolix [3].
Regulatory guidelines [4] provide only little information on the required electronic files for submission. Based on exchanges with regulatory agencies and confirmed through past regulatory submissions using Monolix, the following listed files (in Table 1) are required for a complete Monolix analysis submission package. The Monolix analysis submission package listed in Table 1 has all the files needed to run Monolix without any implementation work and to reproduce the results. Attention must be payed to use relative file path definitions to facilitate the project transfer from one computer system to another (Monolix documentation). Further, it is important to consider that Table 1 represents the Monolix submission package for a typical analysis. There might be modelling cases that require the submission of additional material. Each submission should be treated separately and carefully reviewed for completeness cases by cases.
Table 1 Monolix analysis submission package
File  Explanation  File type, format 
Report (.pdf)  Report detailing all of the modelling according to the EMA or FDA guidelines [1]  
Data (.txt or .csv)  Data file containing all the observations, dosing history, patient specific information and other information provided via the data file  Text file, CSV 
Project file (.mlxtran)  Defines what data file, model file, algorithm settings, graphical settings and parameters have been used to generate the results  Text file, Mlxtran 
Model file (.txt)  The structural model in Mlxtran syntax (if not one of the MonolixSuite libraries)  Text file, Mlxtran 
In addition, some Monolix files define project independent parameters that are applied to all projects run on the same computer account (“Preferences”). In all but the most special cases users will have these parameters set to the default values. In case these parameters are modified, they should also be communicated.
Table 2 Optional but recommended files
File  Explanation  File type, format 
Properties file (.properties)  Settings of the plots. It allows the project to reload all the graphical settings to be able to exactly reproduce the same plots (in terms of color, split by categories, …)  Text file, .properties 
A Monolix run will automatically produce a large number of additional files. However, the files listed in Table 1 and Table 2 are sufficient to entirely reproduce the results. Note that all files are in a human readable format. Thus, the information contained in these files can also be included into the appendices of the report creating one single document that contains everything to reproduce the results.
References
[1] Examples of submissions to the FDA using Monolix
 ZALTRAP (Aflibercept) Clinical pharmacology review (link)
 CORLANOR (Ivabradine) Clinical pharmacology review (link)
 JEVTANA (Cabazitaxel) Clinical pharmacology review (link)
 PONVORY (Ponesimod) Clinical pharmacology review (link)
[2] Examples of submission to the EMA using Monolix
 Procedure No. EMEA/H/C/000402/II/0110/G – EMA/CHMP/186699/2015 – Tamiflu – International nonproprietary name: OSELTAMIVIR. (link)
 Procedure No. EMEA/H/C/000401/II/0066 – EMA/168487/2015 – Tracleer – International nonproprietary name: BOSENTAN. (link)
 Procedure No. EMEA/H/C/004977/II/0003 – EMA/CHMP/186236/2021 – Sarclisa – International nonproprietary name: ISATUXIMAB (link and link to statistical analysis plan)
[3] FDA publications using MonolixSuite (nonexhaustive)
 “Plasma pharmacokinetics of ceftiofur metabolite desfuroylceftiofur cysteine disulfide in holstein steers: application of nonlinear mixedeffects modeling.”, J Vet Pharmacol Ther 2016 Apr;39(2):14956, DOI: 10.1111/jvp.12245. O. A. Chiesa, S. Feng, P. Kijak,E. A. Smith, H. Li and J. Qiu.
 “Quantification of disease progression and dropout for Alzheimer’s disease.”, Int. Journal of Clinical Pharmacology and Therapeutics, Volume 51 – February (120 – 131),(Doi: 10.5414/CP201787). D. WilliamFaltaos, Y. Chen, Y. Wang, J. Gobburu,, and H. Zhu.
 “Estimation of Population Pharmacokinetic Parameters Using MLXTRAN Interpreter in MONOLIX 2.4”, D. William Faltaos, Acop 2009.
[4] Regulatory guidelines
 EMA “Guideline on reporting the results of population pharmacokinetic analyses” (CHMP/EWP/185990/06): details what should be contained in a PK or PK/PD analysis report.
 FDA “Population pharmacokinetics – Guidance for industry” (2022): guidance on how to present the results of a population pharmacokinetic
analysis and the information to be included. It also addresses “Electronic Files”.  FDA “ExposureResponse Relationships – Study Design, Data Analysis, and Regulatory Applications – Guidance for Industry”. (2003)
5.3.Running Monolix using a command line
Yes, it is possible to run Monolix using a simple command line:
monolixSuiteInstallationFolder/bin/Monolix.sh nogui p fullPathProjectName
(replace .sh into .bat for windows operating system)
Notice that the project name should be defined using a full path and not a relative path. The program options are:
 nogui: run without opening a window, mandatory in nodesktop environments.
 p, project: path to the project to run. It should be the absolute (not relative) path name of the project.
 thread, numberofthreads: number of threads to use for the run (integer)
 mode, consolemode: select the verbosity of the run information that will be log in console. It can be “none”, “basic” (default value), or “complete”.
 nosplash: no splash screen. Only used when opening with GUI. When the option –nogui is used, nosplash option is not read.
 o, outputdir: output directory
 t, tool: tool to launch. It can be “monolix” (default value) or “modelBuilding”.
 s, strategy: select the model building algorithm. It can be “cossac” (default value), “samba”, “covSamba” or “scm”.
 c, stoppingCriterion: Select the stopping criterion of model building. It can be “bic” or “lrt” (default value).
 a, LRTThresholdUp : pvalue threshold for the LRT in backward (default 0.01)
 r, LRTThresholdLower: pvalue threshold for the LRT in forward (default 0.01 for cossac and 0.05 for scm)
 lin, useLinearization: use linearization if possible. “true” (default value) or “false”.
Notes for Windows:
 The maximum number of arguments is 10. If more are needed, you can use the monolix.exe file in the /lib directory.
 The native terminals command prompt and PowerShell will not show an output while the project runs, whereas terminals for Windows with a bash shell will. A workaround to still get an output is to write the output to a text file (use tee console_output.txt at the end of your command).
Examples
monolix.bat nogui mode none thread 4 p "C:\Users\celliere\lixoft\monolix\monolix2021R1\demos\1.creating_and_using_models\1.1.libraries_of_models\theophylline_project.mlxtran"
monolix.bat nogui t modelBuilding s cossac a 0.06 r 0.001 lin false p "C:\Users\celliere\lixoft\monolix\monolix2021R1\demos\1.creating_and_using_models\1.1.libraries_of_models\theophylline_project.mlxtran"
monolix.bat nogui t modelBuilding s scm c bic p "C:\Users\celliere\lixoft\monolix\monolix2021R1\demos\1.creating_and_using_models\1.1.libraries_of_models\theophylline_project.mlxtran"
monolix.bat nogui t modelBuilding s covSamba p "C:\Users\celliere\lixoft\monolix\monolix2021R1\demos\1.creating_and_using_models\1.1.libraries_of_models\theophylline_project.mlxtran"
Exporting charts data
If the plots task is selected in the Monolix scenario, and if “Export charts data” is selected in Monolix preferences, the charts data are saved in the result folder. Generating the interactive plots requires to open the project in the GUI. From the 2021 version on, the plots are also be generated as ggplot object using R functions from the lixoftConnectors R package.
Running the covariate search (model building) with additional options
From the 2021 version on, the settings of the model building are saved and reloaded. It is thus possible to do the following to run the model building (covariate search) with more custom settings than provided with the command line arguments, for instance regarding which parametercovariate relationship to test:
 with the GUI, setup the model building settings you need
 launch the model building run and click on the “stop” button immediately. The model building setting are saved in the monolix result folder.
 launch the model building in command line using:
monolix.bat nogui t modelBuilding p yourProject
Note that the result folder of the monolix run needs to the present for the model building setting to load and be used by the run in command line.
5.4.How to compute AUC using in Monolix and Mlxtran
Computing additional outputs such as AUC and Cmax in the structural model is possible but requires a particular syntax, because it is not possible to redefine a variable in Mlxtran. For example, it would not be possible to directly update the variable Cmax = Cc when Cc>Cmax. But it is possible to compute Cmax via an ODE, where Cmax increases like Cc at each time where Cc>Cmax. This page gives detailed examples for the AUC on full time interval or specific interval, Cmax, nadir, and other derived variables such as duration above a threshold.
Often the Area under the PK curve (AUC) is needed as an important PK metric to link with the pharmacodynamic effect. We show here how to:
 compute the AUC within the mlxtran model file
 output the AUC calculations for later analysis.
Calculation of the AUC can be done in the EQUATION section of the Mlxtran model. If a dataset contains the AUC observations, then the calculation in the EQUATION section can be used as an output in the output={} definition (matched to observations of the data set). Or it can be saved for posttreatment using table={}, as described here.
AUClast from t=0 to t=tlast
The following code is the basic implementation of the AUC for a 1compartmental model with the absorption rate ka. It integrates the concentration profile from the start to the end of the observation period.
[LONGITUDINAL] input = {ka, V, Cl} PK: Cc=pkmodel(ka,V,Cl) EQUATION: odeType=stiff ddt_AUC = Cc OUTPUT: output = {Cc} table = {AUC}
AUC in a time interval
The following code computes the AUC between two time points t1 and t2. The idea is to compute the AUC_0_t1 and AUC_0_t2 using if/else statements and then do the difference between the two.
[LONGITUDINAL] input = {ka, V, Cl} PK: depot(ka, target=Ac) EQUATION: odeType = stiff useAnalyticalSolution=no Cc = pkmodel(ka,V,Cl) t1=24 t2=48 AUCt1_0 = 0 if(t < t1) dAUCt1 = Cc else dAUCt1 = 0 end ddt_AUCt1 = dAUCt1 AUCt2_0 = 0 if(t < t2) dAUCt2 = Cc else dAUCt2 = 0 end ddt_AUCt2 = dAUCt2 AUC_t1_t2 = AUCt2  AUCt1 OUTPUT: output = {Cc} table = {AUC_t1_t2}
Note that the t==tDose would not work because the integrator does not necessarily evaluate the time exactly at the times of doses. Thus the test t==tDose might not be tested at all during the computation.
Also note that, although partial AUC can be calculated by comparing time with the both boundaries in the if statement (e.g. if (t>20 and t<40)), this might not be the best practice when concentration is calculated using an analytical solution, because ODE solver used for calculating AUC could be deceived by AUC remaining 0 for a period of time and might increase the integration time step to a value larger than the difference between time boundaries.
Doseinterval AUC (AUCtau)
The following code compute the AUCtau for each dose interval. At each dose the AUC is reset to zero and the concentration is integrated until the next administration.
[LONGITUDINAL] input = {ka, V, Cl} PK: Cc = pkmodel(ka,V,Cl) ; Reset the ODE variable AUCtau to zero each time there is a dose empty(target=AUCtau)
EQUATION: odeType=stiff ddt_AUCtau = Cc OUTPUT: output = {Cc} table = {AUCtau}
Computing the Cmax or nadir in the structural model
Cmax
Cmax can be calculated directly in the structural model by integrating the increase of the concentration. The following example shows how to do it in case of a onecompartment model with firstorder absorption and linear elimination:
[LONGITUDINAL] input = {Cl, ka, V} PK: depot(target=Ad) EQUATION: ; initial conditions t_0 = 0 Ad_0 = 0 Ac_0 = 0 ddt_Ad = ka*Ad ddt_Ac = ka*Ad  k*Ac Cc = Ac/V ; Calculation of Cmax slope = ka*Ad k*Ac Cmax_0 = 0 if slope > 0 && Cc > Cmax x = slope/V else x = 0 end
ddt_Cmax = x
If the dose is administered as bolus, it is necessary to add in the model a very short infusion. This prevents from the instantaneous increase of the concentration. The following example shows this situations in case of a threecompartments model with linear elimination. The duration of the “short infusion” can be adapted with respect to the time scale by modifying dT=0.1.
If you prefer to keep strict bolus administrations, then you can resimulate the project with simulx and postprocess the result in R to extract the Cmax.
If the doses are administered via iv infusion, then dT=0.1 can be replaced by dT = inftDose, which reads the infusion duration from the data.
[LONGITUDINAL] input = {Cl, V1, Q2, V2, Q3, V3} EQUATION: ; Parameter transformations V = V1 k = Cl/V1 k12 = Q2/V1 k21 = Q2/V2 k13 = Q3/V1 k31 = Q3/V3 ; initial conditions t_0 = 0 Ac_0 = 0 A2_0 = 0 A3_0 = 0 ;short pseudoinfusion duration dT = 0.1 ; use dT=inftDose if the administration is infusion: inftDose is the infusion duration from the last dose read from the data ; infusion input to Ac if t < tDose+dT input = amtDose/dT ;amtDose is the last dose amount read from the data else input = 0 end dAc = input k*Ac  k12*Ac  k13*Ac + k21*A2 + k31*A3 ddt_Ac = dAc ddt_A2 = k12*Ac  k21*A2 ddt_A3 = k13*Ac  k31*A3 Cc = Ac/V ; Calculation of AUC AUC_0 = 0 ddt_AUC = Cc ; Calculation of Cmax Cmax_0 = 0 if dAc > 0 && Cc > Cmax x = dAc/V else x = 0 end ddt_Cmax = x OUTPUT: output = {Ac, Cc} table = {Cmax, AUC}
Nadir
This example shows how to compute tumor size (variable TS) at time of nadir:
[LONGITUDINAL] input = {ka, Vl, Cl, TS0, kge, kkill, lambda} EQUATION: odeType=stiff Cc = pkmodel(ka, V, Cl) ; ODE system for tumor growth dynamics t_0 = 0 TS_0 = TS0 TSDynamics = (kge*TS)kkill*TS*Cc*exp(lambda*t) ddt_TS = TSDynamics ; ===== computing TS at nadir if TSDynamics < 0 && TS < TS_atNadir x = TSDynamics else x = 0 end TS_atNadir_0 = TS0 ddt_TS_atNadir = x OUTPUT: output = {TS} table = {TS_atNadir}
Here is an example of simulation for TS and TS_atNadir with a multiple dose schedule:
Time above a threshold
Here we compute the time that a variable C spends above some threshold, which could be a toxicity threshold for example. This piece of code comes in the structural model, after the dynamics of the variable C has already been defined.
TimeAboveThreshold_0 = 0 if C > Cthreshold xTime = 1 else xTime = 0 end ddt_TimeAboveThreshold = xTime
Time since disease progression
In this full structural model example, tumor size TS is described with an exponential growth, and a constant treatment effect since time=0 with a logkill killing hypothesis and a Claret exponential resistance term. Several additional variables are computed:
 TS_atNadir: the tumor size at the time of nadir,
 PC_from_Nadir: the percent change of TS between the time of nadir and the current time,
 DP: predicted disease progression status (1 if TS has increased more than >20% and >5mm from last nadir, 0 otherwise),
 TDP: time since disease progression (duration since last time when DP is switched from 0 to 1)
[LONGITUDINAL] input = {TS0, kge, kkill, lambda} EQUATION: odeType=stiff ;initial conditions: ;t_0 = 0 ;this should be uncommented if the initial time is 0 for all subjects TS_0 = TS0 K = kkill*exp(lambda*t) ;Saturation for TS at 1e12 to avoid infinite values if TS>1e12 TSDynamics = 0 else if t<0 ; before treatment TSDynamics = (kge*TS) else ; after treatment TSDynamics = (kge*TS)K*TS end end ddt_TS = TSDynamics ; Computing time to nadir (TS decreases after treatment start at time=0, then increases again because of resistance) if TSDynamics<0 x1=1 else x1=0 end TimeToNadir_0=t0 ddt_TimeToNadir = x1 ; x1 is used as intermediate variable because it is not possible to define an ODE inside an if/else statement ; Computing TS at time of nadir if TSDynamics < 0 & TS < TS_atNadir x2 = TSDynamics else x2 = 0 end TS_atNadir_0 = TS0 ddt_TS_atNadir = x2 ; ; Computing DP: predicted disease progression status at the previous visit (1 if TS has increased more than >20% and >5mm from last nadir, 0 otherwise) PC_from_Nadir = (TS/TS_atNadir1)*100 if t>TimeToNadir & PC_from_Nadir > 20 & (TSTS_atNadir) > 5 DP = 1 else DP = 0 end ; Computing TDP = time since disease progression: duration since last time DP was switched from 0 to 1 if DP==1 x3 = 1 else x3 = 0 end TDP_0 = 0 ddt_TDP = x3 OUTPUT: output = {TS} table = {TS_atNadir, PC_from_Nadir, DP, TDP}
Simulx can be conveniently used to simulate each intermediate variable with typical parameters (the example model and Simulx project can be downloaded here):
5.5.How to export to Datxplore, Mlxpore and Simulx ?
MonolixSuite applications are interconnected and projects can be exported/imported between different applications. This interconnection is guaranteed by using the same model syntax (mlxtran language) and the same dataset format. You can export a Monolix project to PKanalix or Simulx in one click directly from the interface.
To export a project click EXPORT PROJECT TO in the top menu “Export”:
The export popup window will appear where you can choose:
 to which application you want to export your current project: PKanalix or Simulx
 which dataset you want to use in the export: original dataset, vpc dataset (if VPC plot has been generated), individual fits dataset (if individual fits plot has been generated)
By default, Monolix will copy all files, e.g. dataset and model, next to the new PKanalix/Simulx project. To keep current location of these files, switch the toggle “Generated files next to project” off.
To export a Monolix project click the “EXPORT” button at the bottom. PKanalix/Simulx will open automatically with a predefined project called “untitled”. You can save this project, edit and run. For more information about exported elements, see:
 Import from Monolix to Simulx documentation page
 Import from Monolix to PKanalix documentation page