ENCODING NON LINEAR
MIXED EFFECTS MODEL
Marc Lavielle
INRIA Saclay
EBI, June 20th, 2011
Population approach & mixed effects model
Some examples of PK/PD data
Daily seizure counts (epilepsy)
Viral load
CD4 count
Some examples of PK/PD data
Daily seizure counts (epilepsy)
Viral load
CD4 count
The statistical model of the observations
Statistical model for continuous data
The model of the observations y is
completely defined by :
- The prediction f
-The standard deviation g
- The distribution of the residual errors e
Statistical model for continuous data
The statistical model
prediction = f
standard deviation = g
distribution = normal
Statistical model for continuous data
Any application dedicated to a given task should be able
to understand/interpret this description of the model
The statistical model
prediction = f
standard deviation = g
distribution = normal
Statistical model for continuous data
Any application dedicated to a given task should be able
to understand/interpret this description of the model
The statistical model
prediction = f
standard deviation = g
distribution = normal
Estimation
p( y ) 
1
2 g

e
1
y f
2g2
2
Statistical model for continuous data
Any application dedicated to a given task should be able
to understand/interpret this description of the model
The statistical model
Simulation
y ~ N( f , g2)
prediction = f
standard deviation = g
distribution = normal
Estimation
p( y ) 
1
2 g

e
1
y f
2g2
2
Statistical model for continuous data
Any application dedicated to a given task should be able
to understand/interpret this description of the model
The statistical model
Simulation
y ~ N( f , g2)
prediction = f
standard deviation = g
distribution = normal
Estimation
p( y ) 
1
2 g

e
1
y f
2g2
2
edition
y  f  ge
Statistical model for time-to-event data
The statistical model
hazard = l
Statistical model for time-to-event data
Simulation
The statistical model
t

 l ( u ) du
P(T  t )  e
hazard = l
0
Estimation
t

 l ( u ) du
p (t )  l (t )e
0
t

 l ( u ) du
P (T  t )  e
0
Statistical model for discrete data
Categorical data:
Y  1, 2 , ... , K 
Count data:
Y  0 ,1, 2 , ... , 
Y ~ parametric distribution
example: Y ~Poissonl
P(Y=k) , k=1,2,..K
distribution = poisson
parameter = lambda
The statistical model of the individual parameters
Statistical model of the individual parameters
General model:
Statistical model of the individual parameters
General model:
Linear model:
Statistical model of the individual parameters - Example
The statistical model
distribution = log-normal
standard deviation = omega
covariate = c
Statistical model of the individual parameters - Example
Simulation
edition
 ~ Log  N (log( pop )   c,  2 )
log( )  log( pop )   c  
The statistical model
distribution = log-normal
standard deviation = omega
covariate = c
Estimation
p( ) 
1
 2

e
1
2
2


log(

)

log(

)


c
pop
2
Coding non linear mixed effects models with MONOLIX
The main Graphical User Interface of MONOLIX
Defining the statistical model with the MONOLIX GUI
All the information related to the statistical model is stored:
- in a Matlab structure
- in a XML file
- in a « human-readable » script file
<project name="theophylline_project.xml">
<covariateDefinitionList>
<covariateDefinition columnName="WEIGHT" name="t_WEIGHT" transformation="log(cov/70)" type="continuous"/>
<covariateDefinition columnName="SEX" type="categorical">
<groupList>
<group name="F" reference="true"/>
<group name="M"/>
</groupList>
</covariateDefinition>
</covariateDefinitionList>
<data columnDelimiter="\t" headers="ID,DOSE,TIME,Y,COV,CAT" uri="%MLXPROJECT%/theophylline_data.txt"/>
<models>
<statisticalModels>
<parameterList>
<parameter name="ka" transformation="L">
<intercept initialization="1.000000"/>
</parameter>
<parameter name="V" transformation="L">
<intercept initialization="1.000000"/>
<betaList>
<beta covariate="t_WEIGHT" initialization="0"/>
</betaList>
<variability initialization="1.000000" level="1.000000" levelName="IIV"/>
</parameter>
<parameter name="Cl" transformation="L">
<intercept initialization="1.000000"/>
<variability initialization="1.000000" level="1.000000" levelName="IIV"/>
</parameter>
</parameterList>
<residualErrorModelList>
<residualErrorModel alias="const" output="1.000000" outputName="concentration">
<parameterList>
<parameter initialization="1.000000" name="a"/>
</parameterList>
</residualErrorModel>
</residualErrorModelList>
</statisticalModels>
Coding the (statistical) model with MLXTRAN
$DESCRIPTION PK of theophylline
$FILE
D:/Myproject/theophylline_data.txt
$VARIABLES
ID, TIME, AMT, OBS use=DV,WT, SEX use=cov type=cat,
LW70 = log(WT/70) use=cov
$INDIVIDUAL
default distribution=log-normal,
ka iiv=no, V cov=LW70, Cl,
$EQUATION
Cc=PKMODEL(ka,V,Cl)
$OBSERVATION
Concentration type=continuous pred=Cc err=constant