Version 2023

This documentation is for Monolix starting from 2018 version.
©Lixoft

Monolix

Monolix (Non-linear mixed-effects 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 non-linear 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 non-linear mixed effect models for advanced population analysis, PK/PD, pre-clinical and clinical trial modeling & simulation.

Objectives

The objectives of Monolix are to perform:

1. 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 Hastings-Metropolis algorithm
   • estimating standard errors for the maximum likelihood estimator
2. 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.
3. Easy description of pharmacometric models (PK, PK-PD, discrete data) with the Mlxtran language
4. Goodness of fit plots

An interface for ease of use

Monolix can be used either via a graphical user interface (GUI) or a command-line 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.

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, double-click 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.

• 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 step-by-step 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 step-by-step selection of its characteristics. An enriched PK, a PD, a joint PKPD, a target-mediated drug disposition (TMDD), and a time to-event (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 pre-written 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, first-order absorption, zero-order absorption, with or without Tlag), different number of compartments (1, 2 or 3 compartments), and different types of eliminations (linear or Michaelis-Menten). 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 pre-defined 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 target-mediated drug disposition (TMDD). It includes models with different administration routes (bolus, infusion, first-order absorption, zero-order absorption, bolus + first-order 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 time-to-event (TTE) data. TTE models are defined via the hazard function, in the library we provide exponential, Weibull, log-logistic, 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

Complete description of the TGI model library.

Step-by-step 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, first-order 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)}

• About the OUTPUT block
• Add additional outputs in 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 low-grade glioma treated with chemotherapy or radiotherapy. Clinical Cancer Research, 18(18), 5071-5080, 2012.)

DESCRIPTION: Tumor Growth Inhibition (TGI) model proposed by Ribba et al
A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. 
Clinical Cancer Research, 18(18), 5071-5080, 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*(1-PSTAR/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}

• 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/(1-y)). Thus, for a combined error model for example, the corresponding observation model writes \(\log(y/(1-y)) = \log(f/(1-f)) + (a + b\log(f/(1-f)))\varepsilon\). 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((y-y_min)/(y_max-y)). 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 model|Tasks 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:

• Introduction
• Theory
• Censoring definition in a data set
• PK data below a lower limit of quantification
  • Left censored data
  • Interval censored data
• 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 log-concentration in this example. The limit of quantification of 1.8 mg/l for concentrations becomes log(1.8)=0.588 for log-concentrations. The column of observations (Y) contains either the LLOQ for data below the limit of quantification (BLQ data) or the measured log-concentrations 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 log-concentrations 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 log-concentrations (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 left-censored 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’)

• 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:

• Between-subject 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.

• Within-subject 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 within-subject 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 1-p. 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, 1-p)

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 inter-patient variability on p: all the subjects have the same probability of being a non responder in this example. We use a logit-normal 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,M2,1-p1)

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 well-defined model from the mixture. We consider here that the mixture of models happens within each individual and use a WSMM: f = p*f1 + (1-p)*f2

[LONGITUDINAL]
input = {A1, A2, k, p}

EQUATION:
f1 = A1
f2 = A2*exp(-k*max(t,0))
f = wsmm(f1, p, f2, 1-p)

OUTPUT:
output = f

Remark: Here, writing f = wsmm(f1, p, f2, 1-p) is equivalent to writing f = p*f1 + (1-p)*f2
Important: Here, p is an individual parameter: the subjects have different proportions of non responder cells. We use a probit-normal 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.

• Introduction
• Formatting of time-to-event data in the MonolixSuite
• Single event
  • Single event exactly observed or right censored
  • Single event interval censored or right censored
• Repeated events
  • Repeated events exactly observed or right censored
  • Repeated events interval censored or right censored
• User defined likelihood function for time-to-event data

Objectives: learn how to implement a model for (repeated) time-to-event 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 one-off (e.g., death, hardware failure) or repeated (e.g., epileptic seizures, mechanical incidents, strikes). Several functions play key roles in time-to-event 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.

Time-to-event (TTE) models are thus defined in Monolix via the hazard function. Monolix also holds a TTE library that contains typical hazard functions for time-to-event data. More details and modeling guidelines can be found on the TTE dedicated webpage, along with case studies.

Formatting of time-to-event 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 interval-censoring. 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.

Examples for repeated events, and interval censored events are available on the data set documentation page.

Single event

To begin with, we will consider a one-off 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 “time-to-event”. The random variable representing the time-to-event 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 \(t_{i,j-1}\), given here in terms of the cumulative hazard from \(t_{i,j-1}\) to \(t_{i,j}\):

$$S(t_{i,j} | t_{i,j-1}; \psi_i) = \mathbb{P}(T_{i,j} > t_{i,j} | T_{i,j-1} = t_{i,j-1}; \psi_i) = \exp(-\int_{t_{i,j-1}}^{t_{i,j}}h(t,\psi_i) dt)$$

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 time-to-event 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)^(beta-1)

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).

• Introduction
• Formatting of categorical data in the MonolixSuite
• Ordered categorical data
• Ordered categorical data with regression variables
• Discrete-time Markov chain
• Continuous-time 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 K-1 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 \(k=1,\ldots ,K-1\), 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 \(y_{i,j-1}\)., i.e., for all \(k=1,2,\ldots ,K\),

$$\mathbb{P}(y_{ij} = c_k\,|\,y_{i,j-1}, y_{i,j-2}, y_{i,j-3},\ldots,\psi_i) = \mathbb{P}(y_{ij} = c_k | y_{i,j-1},\psi_i)$$

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 log-normal 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 Monte-Carlo:

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. time-varying covariates)

Discrete-time 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 discrete-time 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,

\(\begin{aligned}\mathbb{P}(y_{i,j} = 1 | y_{i,j-1} = 1) = 1 – \mathbb{P}(y_{i,j} = 2 | y_{i,j-1} = 1) = p_{i,11}\\ \mathbb{P}(y_{i,j} = 1 | y_{i,j-1} = 2) = 1 – \mathbb{P}(y_{i,j} = 2 | y_{i,j-1} = 2) = p_{i,12} \end{aligned}\)

[LONGITUDINAL]
input = {p11, p21}
DEFINITION:
State = {type = categorical,  categories = {1,2},  dependence = Markov
  P(State=1|State_p=1) = p11
  P(State=1|State_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=1|State_p=1) = p11
  P(State=1|State_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=1|State_p=1)) = lp11
  logit(P(State=1|State_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.

Continuous-time 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 continuous-time 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.

• Introduction
• Fitting first a PK model to the PK data
• Simultaneous PKPD modeling
• Sequential PKPD modelling
  • Using estimated population PK parameters
  • Using estimated individual PK parameters
• 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*(1-Imax*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*(1-Imax*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*(1-Imax*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’)

• 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:

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

• Log-normal distribution in ]0,+inf[:

In that case, \(h(\psi_i) = log(\psi_i)\). A log-normally random variable takes positive values only. A log-normal distribution looks like a normal distribution for a small variance \(\omega^2\). On the other hand, the asymmetry of the distribution increases when \(\omega^2\) increases.
Note: the two mathematical representations for log-normal 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 inter-individual variability. Note that \(\bar{\psi}_i\) is the median of the distribution (neither the mean, nor the mode).

• Logit-normal 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 logit-normal distribution takes its values in ]0,1[. The logit of \(\psi_i\) is normally distributed, i.e.,

\(\text{logit}(\psi_i) = \log \left(\frac{\psi_i}{1-\psi_i}\right) \ \sim \ \ {\cal N}( \text{logit}(\bar{\psi}_i), \omega^2) ~~\Leftrightarrow~~ \text{logit}(\psi_i) = \text{logit}(\bar{\psi}_i) + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2)\)

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 logit-normal 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 logit-normal 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:

\( m = \text{logit}_{(a,b)}(\psi_i) = \log \left(\frac{\psi_i – a}{b-\psi_i}\right) ~~\Leftrightarrow~~ \psi_i = \frac{b \exp(m)+a}{1+\exp(m)} \)

• Probit-normal distribution:

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

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

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

Remarks:

1. If you change your distribution and your population parameter is not valid, then an error message is thrown. Typically, when you want to change your distribution to a log normal distribution, make sure the associated population parameter is strictly positive.
2. When creating a project, the default proposed distribution is lognormal.
3. Logit transformations can be generalized to any interval (a,b) by setting \( \psi_{(a,b)} = a + (b-a)\psi_{(0,1)}\) where \(\psi_{(0,1)}\) is a random variable that takes values in (0,1) with a logit-normal 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” Logit-normal 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 log-normally distributed. LOGNORMAL distribution is then used for these four log-normal distributions in the main Monolix graphical user interface:

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

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

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

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

Remarks:

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

• warfarin_distribution2_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

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

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

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

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

Correlation structure of the random effects

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 inter-occasion 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 variance-covariance 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 variance-covariance 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 variance-covariance 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 inter-individual variability

By default, all parameters have inter-individual 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 pre-implemented in the Monolix GUI. This video shows how to implement a lognormal distribution with Box-Cox transformation and how to bound a parameter between two values using a transformed logit distribution (this latter case can be handled automatically from Monolix2019R1).

• Model with continuous covariates
• Model with categorical covariates
• Transforming categorical covariates
• Complex parameter covariate relationships (such as Michaelis-Menten or Hill dependencies, time-dependent covariates, or covariate-dependent 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 p-values 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: \( F_{SEX=1} = \frac{1}{1+ e^{-\left(  \log \left(\frac{F_{pop}}{1-F_{pop}} \right) + beta\_F\_SEX\_1 \right) }} \)

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 p-values concerning the transformed APGAR covariate are over .05.

• 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 intra-individual variability of the individual parameters by piecewise-constant covariates, i.e., occasion-dependent or occasion-varying (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 inter-individual variability of the individual parameters,
• \(\eta_{ik}^{(1)}\), the vector of random effects which describes the random intra-individual 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 inter-individual variability, i.e., variability at the individual level, while superscript (1) represents inter-occasion 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 column-type 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 inter-individual variability. On the figure below,  we see the OCC level. In the presented case, only the volume V has inter-study 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 inter-occasion information and is thus displayed with the occasion level. On the other hand, the SEX is constant for each subject. It contains then inter-individual information but no inter-occasion 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 inter-occasion variability. It can only be used with the parameter V that has inter-occasion variability. The two other parameters have only inter-individual 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 inter-individual variability. It can therefore be associated to any parameter that has inter-individual 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 subject-occasion 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.

• 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 between-subject 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 user-defined 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

• Combining iv and oral administrations – Example 1
• Combining iv and oral administrations – Example 2

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 first-order and zero-order absorptions, either sequentially or simultaneously, and fast and slow parallel first-order 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 first-order 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 logit-normal 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 zero-order absorption process. The 2 oral doses (adm=2,3) are absorbed into the central compartment following first-order 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, logit-normal 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.

• Multiple doses
• Additional doses (ADDL)
• Steady-state

Objectives: learn how to define and use a PK model with multiple doses or assuming steady-state.

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.

Steady-state

• 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 “steady-state 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 steady-state, 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 steady-state dose or not and a column INTERDOSE INTERVAL for the inter-dose 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 steady-state. 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’)

Steady-state and non steady-sates doses are combined in this project:

Individual fits display the predicted concentrations computed with this combination of doses:

• 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 column-type REGRESSOR. Within a given subject-occasion, string “.” will be interpolated (last value carried forward interpolation is used) for observation and dose-lines. 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

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
  • Introduction
  • Computing the Maximum a posteriori (MAP) estimate
• Fixing the value of a 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 so-called 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 trade-off 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 rule-book 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 variance-covariance matrices as they allow to fulfill the positive-definite 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.

• 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 low-grade glioma treated with chemotherapy or radiotherapy*. Clinical Cancer Research, 18(18), 5071-5080, 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 low-grade glioma treated with chemotherapy or radiotherapy. 
Clinical Cancer Research, 18(18), 5071-5080, 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*(1-PSTAR/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 one-dimensional component and T is the explicit delay. Therefore, DDEs with a nonconstant past of the form

$$ \begin{array}{ccl} \frac{dx}{dt} &=& f(x(t),x(t-T_1), x(t-T_2), …), ~~\text{for}~~t \geq 0\ x(t) &=& x_0(t) ~~~~\text{for}~~\text{min}(T_k) \leq t \leq 0 \end{array} $$

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 Recruitment-Death 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

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 log-likelihood. 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.

Monolix-R 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.

Parameter initial estimates and associated methods

• Initialization of the estimates
  • Initialization of the “Fixed effects”
  • Initialization of the “Standard deviation of the random effects”
  • Initialization of the “Residual error parameters”
• What method can I use for the parameters estimations?
• How to initialize your parameters?
  • Use last estimates
  • Check initial estimates

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 non-reference 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.

• Purpose
• Calculations: the SAEM algorithm
• Running the population parameter estimation task
  • Overview
  • Dependencies between tasks
  • The convergence indicator
  • The simulated annealing
  • Methods for parameters without variability
  • Other estimation methods: Fixing population parameters or Bayesian estimation
• 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 non-linear mixed effect modeling. In Monolix, it is performed using the Stochastic Approximation Expectation-Maximization (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], time-to-event 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 pop-up window which permits to follow the progress of the task is shown below. The algorithm starts with a small number (5 by default) of burn-in 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 auto-stop 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 log-likelihood 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 (k-1), 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 \(p(\psi_i|y_i;\hat{\theta})\) 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 Nelder-Mead simplex algorithm (Matlab’s fminsearch method). Other options are also available in the SAEM settings:

• No variability (default): optimization via Nelder-Mead 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 1e-5. 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 1e-5 (same as above). In the smoothing phase, the Nelder-Mead 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 Nelder-Mead 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 Nelder-Mead 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 log-normally 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 Monte-Carlo. 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 (“auto-stop”), or if the algorithm arrived at the maximum number of iterations (“stopped at the maximum number of iterations/auto-stop 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.

Burn-in phase:

The burn-in 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 burn-in phase in Monolix is different to what is called burn-in in Nonmem algorithms.

• Number of iterations (default: 5): number of iterations of the burn-in phase

Exploratory phase: 

• Auto-stop criteria (default: yes): if ticked, auto-stop 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 auto-stop ticked): maximum number of iterations for the exploratory phase. Even if the auto-stop 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 auto-stop criteria are not fulfilled.
• Minimum number of iterations (default: 150, if auto-stop ticked): minimum number of iterations for the exploratory phase. This value also corresponds to the interval length over which the auto-stop criteria are tested. A larger minimum number of iterations means that the auto-stop criteria are harder to reach.
• Number of iterations (default: 500, if auto-stop unticked): fixed number of iterations for the exploratory phase.
• Step-size 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: 

• Auto-stop criteria (default: yes): if ticked, auto-stop 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 auto-stop ticked): maximum number of iterations for the smoothing phase. Even if the auto-stop criteria are not fulfilled, the algorithm stops after this maximum number of iterations.
• Minimum number of iterations (default: 50, if auto-stop ticked): minimum number of iterations for the smoothing phase. This value also corresponds to the interval length over which the auto-stop criteria is tested. A larger minimum number of iterations means that the auto-stop criteria is harder to reach.
• Number of iterations (default: 200, if auto-stop unticked): fixed number of iterations for the smoothing phase.
• Step-size 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 Nelder-Mead simplex algorithm. The absolute tolerance (stopping criteria) is 1e-4 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 1e-5.
• Variability in the first stage: during the exploratory phase, an artificial variability is added and progressively forced to 1e-5 (same as above). In the smoothing phase, the Nelder-Mead 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 inter-individual variability can be removed.

Example: The dataset used in the tobramycin case study is quite sparse. In these conditions, we expect that estimating the inter-individual variability for all parameters will be difficult. In this case, the estimation can be done in two steps, as shown below for a two-compartments 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% inter-individual 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 inter-individual 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.

• 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 \(p(\psi_i|y_i;\hat{\theta})\) 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 Markov-Chain Monte-Carlo 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 Metropolis-Hastings (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:

$$p(\psi_i|y_i)=\frac{p(y_i|\psi_i)p(\psi_i)}{p(y_i)}$$

\(p(y_i|\psi_i)\) 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 \(p(\psi_i|y_i)\). 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 back-transformed to the non-gaussian 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“.

Shrinkage and the use of random samples from the conditional distribution are explained in more details here.

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(X|y)), and the standard deviation of the conditional means over all individuals (noted sd(X|y)) are calculated and displayed in the pop-up 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 pop-ups, displaying the average conditional means over all individuals (noted E(X|y)), and the standard deviation of the conditional means over all individuals (noted sd(X|y)) 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 log-likelihood task without the linearization method (i.e log-likelihood 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 (“auto-stop”) 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 subject-occasion 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 subject-occasion 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.

• 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 \( p(\psi_i|y_i;\hat{\theta})\) 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 Markov-Chain Monte-Carlo 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:

$$ \hat{\psi}_i^{mode} = \underset{\psi_i}{\textrm{arg max }}p(\psi_i|y_i;\hat{\theta})$$

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 Nelder-Mead Simplex algorithm [1].

As the conditional distribution does not have a closed form solution (i.e \(p(\psi_i|y_i;\hat{\theta})\) 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):

$$p(\psi_i|y_i)=\frac{p(y_i|\psi_i)p(\psi_i)}{p(y_i)}$$

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 Nelder-Mead 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 pop-up 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 Log-likelihood 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 subject-occasion 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 subject-occasion 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 Nelder-Mead Simplex algorithm, for each individual. Even if the tolerance criteria is not met, the algorithm stops after that number of iterations.
• Tolerance (default: 1e-6): 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

• 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 log-likelihood:

$$ I(\hat{\theta}) = -\frac{\partial^2}{\partial\theta^2}\log({\cal L}_y(\hat{\theta})) $$

The log-likelihood 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, time-to-event, 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 (non-normal) 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 variance-covariance 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 variance-covariance matrix are calculated on the transformed normally distributed parameters. The variance-covariance 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 variance-covariance 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 pop-up 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 p-value and the RSE:

• For the p-value: 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 (“auto-stop”) 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, p-values, 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 p-values 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: variance-covariance matrix of the transformed normally distributed population parameter, method without linearization (stochastic approximation)
• FisherInformation/covarianceEstimatesLin.txt: variance-covariance 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 time-consuming – 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).

Purpose

The log-likelihood is the objective function and a key information. The log-likelihood cannot be computed in closed form for nonlinear mixed effects models. It can however be estimated.

• • Log-likelihood estimation
    • Importance sampling
    • Linearization
    • Best practices: When should I use the linearization and when should I use the importance sampling?
  • Display and outputs
  • Settings

Log-likelihood estimation

Performing likelihood ratio tests and computing information criteria for a given model requires computation of the log-likelihood

$$ {\cal L}{\cal L}_y(\hat{\theta}) = \log({\cal L}_y(\hat{\theta})) \triangleq \log(p(y;\hat{\theta})) $$

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 log-likelihood 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 log-likelihood – even for nonlinear models – whose variance can be controlled by the Monte Carlo size.

Two different algorithms are proposed to estimate the log-likelihood:

• by linearization,
• by Importance sampling.

Log-likelihood by importance sampling

The observed log-likelihood \({\cal LL}(\theta;y)=\log({\cal L}(\theta;y))\) 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 log-likelihood as the sum of these individual log-likelihoods. 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:

1. Draw M independent values \(\phi_i^{(1)}\), \(\phi_i^{(2)}\), …, \(\phi_i^{(M)}\) from the marginal distribution \(p_{\phi_i}(.;\theta)\).
2. 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:

1. Draw M independent values \(\phi_i^{(1)}\), \(\phi_i^{(2)}\), …, \(\phi_i^{(M)}\) from the proposal distribution \(\tilde{p_{\phi_i}}(.;\theta)\).
2. 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 \(p_{\phi_i|y_i}\). 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 \(p_{\phi_i|y_i}\) 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 Metropolis-Hastings algorithm and a practical proposal “close” to the optimal proposal \(p_{\phi_i|y_i}\) 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 \(p_{\phi_i|y_i}\) are estimated by Metropolis-Hastings for each individual i. Then, the \(\phi_i^{(m)}\) are drawn with a noncentral student t-distribution:

$$ \phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m}$$

where \(\mu_i\) and \(\sigma^2_i\) are estimates of \(\mathbb{E}\left(\phi_i|y_i;\theta\right)\) and \(\mbox{Var}\left(\phi_i|y_i;\theta\right)\), and \((T_{i,m})\) is a sequence of i.i.d. random variables distributed with a Student’s t-distribution with \(\nu\) degrees of freedom (see section Advanced settings for the log-likelihood 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 \(p_{\phi_i|y_i}\), which means having to estimate this conditional distribution before estimating the log-likelihood (i.e. run task  “Conditional distribution” before).

Log-likelihood by linearization

The likelihood of the nonlinear mixed effects model  cannot be computed in a closed-form. 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 log-likelihood 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 log-likelihood 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 time-consuming – 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 log-likelihood 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 non-random 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 log-likelihood has been computed by importance sampling, the number of degrees of freedom used for the proposal t-distribution (5 by default) is also displayed, together with the standard error of the LL on the individual parameters drawn from the t-distribution.

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 log-likelihood, the Akaike Information Criteria (AIC), the Bayesian Information Criteria (BIC), and the corrected Bayesian Information Criteria (BICc).
• individualLL.txt: containing the -2 x log-likelihood for each individual for each computed method.

Advanced settings for the log-likelihood

Monolix uses a t-distribution 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.

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
• Display and outputs
• Best practices

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 log-likelihood
    • 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 log-likelihood 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 log-likelihood 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 log-likelihood 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.

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

• • Model for the individual parameters
    • The covariate model
    • The model for the random effects
    • The model for the individual parameters
  • Model for the observations
    • The distribution of the residuals

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 log-volume log(V) is a linear function of the log-weight \({\rm lw70 }= \log({\rm wt}/70)\).
• 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 p-value 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 p-value 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 p-values are colored in yellow (p-value in [0.01-0.05]), orange (p-value in [0.05-0.10]) or red (p-value in [0.10-1]) to draw attention on parameter-covariate 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 log-normally distributed parameters or logit(F) for logit-distributed parameters) sampled from the conditional distribution (called replicates, index l). Here we will call covariates all continuous covariates and all non-referent 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 t-test 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 \(N-n_C-1\) 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 t-distribution 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 one-way ANOVA tests the following null-hypothesis:

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 p-value 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 parameter-covariate 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 p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]) to draw attention on parameter-covariate 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 log-transformed 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:

$$t=\frac{r}{\sqrt{1-r^2}}\sqrt{N-2}$$

and it is compared to a t-distribution with \(N-2\) degrees of freedom with \(N\) the number of individuals.

Categorical covariates: The random effects are first averaged over replicates for each individual and a one-way analysis of variance is performed (simplified to a t-test 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. Shapiro-Wilk tests are performed to test this hypothesis. The null hypothesis is:

H0: the random effects are normally distributed

If the p-value 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 p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

In our example, there is no reason to reject the null-hypothesis and no reason to question the chosen log-normal 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 Shapiro-Wilk 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 p-values, one p-value is calculated for each replicate, using the Shapiro-Wild table with \(N\) (number of individuals) degrees of freedom. The Benjamini-Hochberg (BH) procedure is then applied: the p-values 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 p-value’s rank, \(L\) the total number of p-values (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest p-value 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 null-hypothesis is:

H0: the expectation of the product of the random effects of the first and second parameter is zero

The null-hypothesis is assessed using a t-test.

Remark: In the 2018 version, a Pearson correlation test was used.

For correlations not yet included in the model, a small p-value 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 p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).
For correlations already included in the model, a large p-value 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 p-values are colored in yellow (p-value in [0.01-0.05]), orange (p-value in [0.05-0.10]) or red (p-value in [0.10-1])

In our example, we have assumed in the model that \(\eta_V\) and \(\eta_{Cl}\) are correlated. The high p-value (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 non-significant 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 t-distribution with \(N-1\) 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 Shapiro-Wilk 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 p-value 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 p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

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  Shapiro-Wilk 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 log-normally distributed parameters and logit(F) for logit-distributed parameters) sampled from the conditional distribution (called replicates, index \(l\) ) for individual \(i\). The Shapiro-Wilk 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 p-values, one p-value is calculated for each replicate, using the Shapiro-Wild table with \(N\) (number of individuals) degrees of freedom. The Benjamini-Hochberg (BH) procedure is then applied: the p-values 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 p-value’s rank, \(L\) the total number of p-values (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest p-value 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 Kolmogorov-Smirnov test is used for testing the distributional adequacy of these individual parameters. The null-hypothesis 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 p-value indicates that the null hypothesis can be rejected. Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

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 p-value indicates that the null hypothesis can be rejected. Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

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:

$$T=\frac{\sqrt{n}}{0.9468922}\frac{\overline{R}-M_R}{ \sum_{i=1}^{n}|R_i-M|}$$

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 p-value 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 p-value), 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 Shapiro-Wilk 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 p-values, one p-value is calculated for each replicate, using the Shapiro-Wild table with \(N\) (number of individuals) degrees of freedom. The Benjamini-Hochberg (BH) procedure is then applied: the p-values 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 p-value’s rank, \(L\) the total number of p-values (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest p-value that is smaller than the corresponding critical value is selected.

Troubleshooting

If the Observed data plot is empty or with non-appropriate axis limits, follow the procedure below.

The problem appears after having clicked “Export > Export charts settings as default” on a previous project where the y-axis 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

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 one-compartment model with a first-order 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 log-normal distribution. The combination of constant error model and log-normal distribution corresponds to an exponential error model. Error models with a proportional term can cause numerical issues with the log-normal distribution for small observations because the error becomes very small as well.

Choosing an observation model with a logit-normal 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 logit-normal distribution in [0.1-10]:

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 two-compartments model with a linear elimination. In this example, the log-log scale reveals a clear misspecification of the model: the small observations are under-predicted. These observations correspond to high times: this means that the elimination is not properly captured by the two-compartment model. A three-compartment 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.

• Purpose
• Examples
  • Continuous outcomes
  • Count data and categorical data
  • Time-to-event data
• Details
  • Binning criteria
  • Corrected predictions
  • Stratification
  • VPC based on time after last dose
  • Axis limits
• 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 one-compartment model with delayed first-order 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 time-to-event 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 non-random 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.

• Non-continuous 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’)

• Time-to-event data

In case of time-to-event data, two visual predictive checks are available, survival function based on the Kaplan-Meier 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 Kaplan-Meier 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: equal-width binning, equal-size binning, and a least-squares 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 prediction-corrected 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 : prediction-corrected 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 X-axis) 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 X-axis is “Time”, and on the right the X-axis 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 log-scale is used on the y-axis. In that case, the VPC is shown by default from 1e-16 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 (Kapan-Meier 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
    

• • • 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 X-axis.
• 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 least-squares
  • 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

Troubleshooting

If the VPC plot is emtpy or with non-appropriate axis limits, follow the procedure below.

The problem appears after having clicked “Export > Export charts settings as default” on a previous project where the y-axis 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

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

Observed data (event)

xxx_curves.txt

Description: observation values

xxx_censored.txt

Description: censored values

Model for the observations

Individual Fits

xxx_observations.txt

Description: observation values

xxx_fits.txt

Description: individual fits based on population parameters and individual parameters

Observation vs Prediction

xxx_obsVsPred.txt

Description: observation and prediction (pop & indiv) values

xxx_obsVsSimulatedPred.txt

Description: observation and simulated prediction values

xxx_visualGuides.txt

Description: splines and confidence intervals for predictions

Distribution of the residuals

xxx_pdf.txt

Description: probability density function of each residual type (pwres, iwres, npde)

xxx_cdf.txt

Description: cumulative distribution function of each residual type (pwres, iwres, npde)

theoreticalGuides.txt

Description: theoretical guides for the pdf and 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.

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.

xxx_residuals.txt

Description: residuals values (pwres, iwres, npde)