Ziad Taib
Biostatistics, AZ
MV, CTH
April 2011
Lecture 4
Non-Linear and Generalized Mixed Effects Models
1 Date
Part I
Generalized Mixed Effects Models
2 Date
Outline of part I
1. Generalized Mixed Effects Models1. Formulation
2. Estimation
3. Inference
4. Software
2. Non-linear Mixed Effects Models in Pharmacokinetics1. Basic Kinetics
2. Compartmental Models
3. NONMEM
4. Software issues
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Various forms of models and relation between them
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LM: Assumptions:
1. independence,
2. normality,
3. constant parameters
GLM: assumption 2) Exponential family
LMM: Assumptions 1) and 3) are modified
GLMM: Assumption 2) Exponential family and assumptions 1) and 3) are modified
Repeated measures: Assumptions 1) and 3) are modified
Longitudinal dataMaximum likelihood
Classical statistics (Observations are random, parameters are unknown constants)
Bayesian statistics
LM - Linear model
GLM - Generalised linear model
LMM - Linear mixed model
GLMM - Generalised linear mixed model
Non-linear models
Example 1Toenail Dermatophyte Onychomycosis
Common toenail infection, difficult to treat, affecting more than 2% of population. Classical treatments with antifungal compounds need to be administered until the whole nail has grown out healthy.
New compounds have been developed which reduce treatment to 3 months.
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Example 1 :
• Randomized, double-blind, parallel group, multicenter study for the comparison of two such new compounds (A and B) for oral treatment.
Research question:
Severity relative to treatment of TDO ?
• 2 × 189 patients randomized, 36 centers
• 48 weeks of total follow up (12 months)
• 12 weeks of treatment (3 months)
measurements at months 0, 1, 2, 3, 6, 9, 12.Date
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Example 2 The Analgesic Trial Single-arm trial with 530 patients recruited (491 selected
for analysis).
Analgesic treatment for pain caused by chronic non- malignant disease.
Treatment was to be administered for 12 months.
We will focus on Global Satisfaction Assessment (GSA).
GSA scale goes from 1=very good to 5=very bad.
GSA was rated by each subject 4 times during the trial, at months 3, 6, 9, and 12.
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Questions Evolution over time.
Relation with baseline covariates: age, sex, duration of the pain, type of pain, disease progression, Pain Control Assessment (PCA), . . .
Investigation of dropout.
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Observedfrequencies
Generalized linear Models:
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The Bernoulli case
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Generalized Linear Models
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Longitudinal Generlized Linear Models
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Generalized Linear Mixed Models
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Empirical bayes estimates
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Example 1 (cont’d)
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Types of inference
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Syntax for NLMIXED
PROC NLMIXED options ;
ARRAY array specification ;
BOUNDS boundary constraints ;
BY variables ;
CONTRAST 'label' expression <,expression> ;
ESTIMATE 'label' expression ;
ID expressions ;
MODEL model specification ;
PARMS parameters and starting values ;
PREDICT expression ;
RANDOM random effects specification ;
REPLICATE variable ;
Program statements ; The following sections provide a detailed description of each of these statements.
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http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap46/index.htm
PROC NLMIXED Statement
ARRAY Statement
BOUNDS Statement
BY Statement
CONTRAST Statement
ESTIMATE Statement
ID Statement
MODEL Statement
PARMS Statement
PREDICT Statement
RANDOM Statement
REPLICATE Statement
Programming Statements24
Example
This example analyzes the data from Beitler and Landis (1985), which represent results from a multi-center clinical trial investigating the effectiveness of two topical cream treatments (active drug, control) in curing an infection. For each of eight clinics, the number of trials and favorable cures are recorded for each treatment. The SAS data set is as follows.
data infection;
input clinic t x n;
datalines;
1 1 11 36
1 0 10 37
2 1 16 20
2 0 22 32
3 1 14 19
3 0 7 19
4 1 2 16
4 0 1 17
5 1 6 17
5 0 0 12
6 1 1 11
6 0 0 10
7 1 1 5
7 0 1 9
8 1 4 6
8 0 6 7
run;
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Suppose nij denotes the number of trials for the ith clinic and the jth treatment (i = 1, ... ,8 j = 0,1), and xij denotes the corresponding number of favorable cures. Then a reasonable model for the preceding data is the following logistic model with random effects:
The notation tj indicates the jth treatment, and the ui are assumed to be iid .
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The PROC NLMIXED statements to fit this model are as follows:
proc nlmixed data=infection;
parms beta0=-1 beta1=1 s2u=2;
eta = beta0 + beta1*t + u;
expeta = exp(eta);
p = expeta/(1+expeta);
model x ~ binomial(n,p);
random u ~ normal(0,s2u) subject=clinic;
predict eta out=eta; estimate '1/beta1' 1/beta1; run;
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The PROC NLMIXED statement invokes the procedure, and the PARMS statement defines the parameters and their starting values. The next three statements define pij, and the MODEL statement defines the conditional distribution of xij to be binomial. The RANDOM statement defines U to be the random effect with subjects defined by the CLINIC variable.
The PREDICT statement constructs predictions for each observation in the input data set. For this example, predictions of and approximate standard errors of prediction are output to a SAS data set named ETA. These predictions include empirical Bayes estimates of the random effects ui.
The ESTIMATE statement requests an estimate of the reciprocal of .
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Parameter Estimates
Parameter
Estimate
Standard Error DF t Value Pr > |t| Alpha Lower Upper Gradient
beta0 -1.1974 0.5561 7 -2.15 0.0683 0.05 -2.5123 0.1175 -3.1E-7
beta1 0.7385 0.3004 7 2.46 0.0436 0.05 0.02806 1.4488 -2.08E-6
s2u 1.9591 1.1903 7 1.65 0.1438 0.05 -0.8554 4.7736 -2.48E-7
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LabelEstimat
eStandard Error DF t Value Pr > |t| Alpha Lower Upper
1/beta1 1.3542 0.5509 7 2.46 0.0436 0.05 0.05146 2.6569
Conclusions
The "Parameter Estimates" table indicates marginal significance of the two fixed-effects parameters. The positive value of the estimate of indicates that the treatment significantly increases the chance of a favorable cure.
The "Additional Estimates" table displays results from the ESTIMATE statement. The estimate of equals 1/0.7385 = 1.3541 and its standard error equals 0.3004/0.73852 = 0.5509 by the delta method (Billingsley 1986). Note this particular approximation produces a t-statistic identical to that for the estimate of .
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PROC NLMIXED
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PROC NLMIXED
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Example 2 (cont’d)
• We analyze the data using a GLMM, but with different approximations:
Integrand approximation: GLIMMIX and MLWIN (PQL1 or PQL2)
Integral approximation: NLMIXED (adaptive or not) and MIXOR (non-adaptive)
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Results
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PROC MIXED vs PROC NLMIXED
The models fit by PROC NLMIXED can be viewed as generalizations of the random coefficient models fit by the MIXED procedure. This generalization allows the random coefficients to enter the model nonlinearly, whereas in PROC MIXED they enter linearly.
With PROC MIXED you can perform both maximum likelihood and restricted maximum likelihood (REML) estimation, whereas PROC NLMIXED only implements maximum likelihood.
Finally, PROC MIXED assumes the data to be normally distributed, whereas PROC NLMIXED enables you to analyze data that are normal, binomial, or Poisson or that have any likelihood programmable with SAS statements.
PROC NLMIXED does not implement the same estimation techniques available with the NLINMIX and GLIMMIX macros. (generalized estimating equations). In contrast, PROC
NLMIXED directly maximizes an approximate integrated likelihood.
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References
Beal, S.L. and Sheiner, L.B. (1982), "Estimating Population Kinetics," CRC Crit. Rev. Biomed. Eng., 8, 195 -222.
Beal, S.L. and Sheiner, L.B., eds. (1992), NONMEM User's Guide, University of California, San Francisco, NONMEM Project Group.
Beitler, P.J. and Landis, J.R. (1985), "A Mixed-effects Model for Categorical Data," Biometrics, 41, 991 -1000.
Breslow, N.E. and Clayton, D.G. (1993), "Approximate Inference in Generalized Linear Mixed Models," Journal of the American Statistical Association, 88, 9 -25.
Davidian, M. and Giltinan, D.M. (1995), Nonlinear Models for Repeated Measurement Data, New York: Chapman & Hall.
Diggle, P.J., Liang, K.Y., and Zeger, S.L. (1994), Analysis of Longitudinal Data, Oxford: Clarendon Press.
Engel, B. and Keen, A. (1992), "A Simple Approach for the Analysis of Generalized Linear Mixed Models," LWA-92-6, Agricultural Mathematics Group (GLW-DLO). Wageningen, The Netherlands.
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Fahrmeir, L. and Tutz, G. (2002). Multivariate Statistical Modelling Based on Generalized Linear Models, (2nd edition). Springer Series in Statistics. New-York: Springer-Verlag.
Ezzet, F. and Whitehead, J. (1991), "A Random Effects Model for Ordinal Responses from a Crossover Trial," Statistics in Medicine, 10, 901 -907.
Galecki, A.T. (1998), "NLMEM: New SAS/IML Macro for Hierarchical Nonlinear Models," Computer Methods and Programs in Biomedicine, 55, 107 -216.
Gallant, A.R. (1987), Nonlinear Statistical Models, New York: John Wiley & Sons, Inc.
Gilmour, A.R., Anderson, R.D., and Rae, A.L. (1985), "The Analysis of Binomial Data by Generalized Linear Mixed Model," Biometrika, 72, 593 -599.
Harville, D.A. and Mee, R.W. (1984), "A Mixed-model Procedure for Analyzing Ordered Categorical Data," Biometrics, 40, 393 -408.
Lindstrom, M.J. and Bates, D.M. (1990), "Nonlinear Mixed Effects Models for Repeated Measures Data," Biometrics, 46, 673 -687.
Littell, R.C., Milliken, G.A., Stroup, W.W., and Wolfinger, R.D. (1996), SAS System for Mixed Models, Cary, NC: SAS Institute Inc.
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Longford, N.T. (1994), "Logistic Regression with Random Coefficients," Computational Statistics and Data Analysis, 17, 1 -15.
McCulloch, C.E. (1994), "Maximum Likelihood Variance Components Estimation for Binary Data," Journal of the American Statistical Association, 89, 330 -335.
McGilchrist, C.E. (1994), "Estimation in Generalized Mixed Models," Journal of the Royal Statistical Society B, 56, 61 -69.
Pinheiro, J.C. and Bates, D.M. (1995), "Approximations to the Log-likelihood Function in the Nonlinear Mixed-effects Model," Journal of Computational and Graphical Statistics, 4, 12 -35.
Roe, D.J. (1997) "Comparison of Population Pharmacokinetic Modeling Methods Using Simulated Data: Results from the Population Modeling Workgroup," Statistics in Medicine, 16, 1241 - 1262.
Schall, R. (1991). "Estimation in Generalized Linear Models with Random Effects," Biometrika, 78, 719 -727.
Sheiner L. B. and Beal S. L., "Evaluation of Methods for Estimating Population Pharmacokinetic Parameters. I. Michaelis-Menten Model: Routine Clinical Pharmacokinetic Data," Journal of Pharmacokinetics and Biopharmaceutics, 8, (1980) 553 -571.
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Sheiner, L.B. and Beal, S.L. (1985), "Pharmacokinetic Parameter Estimates from Several Least Squares Procedures: Superiority of Extended Least Squares," Journal of Pharmacokinetics and Biopharmaceutics, 13, 185 -201.
Stiratelli, R., Laird, N.M., and Ware, J.H. (1984), "Random Effects Models for Serial Observations with Binary Response," Biometrics, 40, 961-971.
Vonesh, E.F., (1992), "Nonlinear Models for the Analysis of Longitudinal Data," Statistics in Medicine, 11, 1929 - 1954.
Vonesh, E.F. and Chinchilli, V.M. (1997), Linear and Nonlinear Models for the Analysis of Repeated Measurements, New York: Marcel Dekker.
Wolfinger R.D. (1993), "Laplace's Approximation for Nonlinear Mixed Models," Biometrika, 80, 791 -795.
Wolfinger, R.D. (1997), "Comment: Experiences with the SAS Macro NLINMIX," Statistics in Medicine, 16, 1258 -1259.
Wolfinger, R.D. and O'Connell, M. (1993), "Generalized Linear Mixed Models: a Pseudo-likelihood Approach," Journal of Statistical Computation and Simulation, 48, 233 -243.
Yuh, L., Beal, S., Davidian, M., Harrison, F., Hester, A., Kowalski, K., Vonesh, E., Wolfinger, R. (1994), "Population Pharmacokinetic/Pharmacodynamic Methodology and Applications: a Bibliography," Biometrics, 50, 566 -575
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End of Part I
Any Questions?
Part IIIntroduction to non-linear mixed
models in Pharmakokinetics
Typical data
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One curve per patient
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Common situation (bio)sciences:
A continuous response evolves over time (or other condition) within individuals from a population of interest
Scientific interest focuses on features or mechanisms that underlie individual time trajectories of the response and how these vary across the population.
A theoretical or empirical model for such individual profiles, typically non-linear in the parameters that may be interpreted as representing such features or mechanisms, is available.
Repeated measurements over time are available on each individual in a sample drawn from the population
Inference on the scientific questions of interest is to be made in the context of the model and its parameters
Non linear mixed effects models
Nonlinear mixed effects models: or hierarchical non-linear models
A formal statistical framework for this situation
A “hot” methodological research area in the early 1990s
Now widely accepted as a suitable approach to inference, with applications routinely reported and commercial software available
Many recent extensions, innovations
Have many applications: growth curves, pharmacokinetics, dose-response etc
PHARMACOKINETICS
A drugs can administered in many different ways: orally, by i.v. infusion, by inhalation, using a plaster etc.
Pharmacokinetics is the study of the rate processes that are responsible for the time course of the level of the drug (or any other exogenous compound in the body such as alcohol, toxins etc).
PHARMACOKINETICS
Pharmacokinetics is about what happens to the drug in the body. It involves the kinetics of drug absorption, distribution, and elimination i.e. metabolism and excretion (adme). The description of drug distribution and elimination is often termed drug disposition.
One way to model these processes is to view the body as a system with a number of compartments through which the drug is distributed at certain rates. This flow can be described using constant rates in the cases of absorbtion and elimination.
Plasma concentration curves (PCC)
The concentration of a drug in the plasma reflects many of its properties. A PCC gives a hint as to how the ADME processes interact. If we draw a PCC in a logarithmic scale after an i.v. dose, we expect to get a straight line since we assume the concentration of the drug in plasma to decrease exponentially. This is first order- or linear kinetics. The elimination rate is then proportional to the concentration in plasma. This model is approximately true for most drugs.
Plasma concentration curve
Concentration
Time
Pharmacokinetic models
Various types of models
One-compartment model with rapid intravenous
administration: The pharmacokinetics parameters
Half life
Distribution volume
AUC
Tmax and Cmax
D, VD
i.v. k
•D: Dose•VD: Volume•k: Elimination rate•Cl: Clearance
0kCdt
dC
One compartment model
General model Tablet
IV
dC
dtv in vout
)()( tktk
ea
a ae eekk
k
V
DoseFtC
Vin
C(t) , V
Ve
ka ke
t
V
Cl
V
DCt exp
Typical example in kinetics
A typical kinetics experiment is performed on a number, m, of groups of h patients.
Individuals in different groups receive the same formulation of an active principle, and different groups receive different formulations.
The formulations are given by IV route at time t=0.The dose, D, is the same for all formulations.
For all formulations, the plasma concentration is measured at certain sampling times.
Random or fixed ?
The formulation
Dose
The sampling times
The concentrations
The patients
Fixed
Fixed
Fixed
Random
Fixed
Random
Analytical errorDeparture to kinetic model
Population kinetics
Classical kinetics
An example
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One PCC per patients
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Step 1 : Write a (PK/PD) model
A statistical model
Mean model :functional relationship
Variance model :Assumptions on the residuals
Step 1 : Write a deterministic (mean) model to describe the individual kinetics
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One compartment model with constant intravenous infusion rate
tV
Cl
V
DtC
kVClV
DCktCtC
exp)(
; ;exp)( 00
t
V
Cl
V
DCt exp
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Step 1 : Write a deterministic (mean) model to describe the individual kinetics
t
V
Cl
V
DtC exp)(
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Step 1 : Write a deterministic (mean) model to describe the individual kinetics
residual
Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics
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Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics
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Time
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Step 1 : Describe the shape of departure to the kinetics
Time
Residual
Step 1 :Write an "individual" model
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
jiY ,
jit ,
jth concentration measured on the ith patient
jth sample time of the ith patient
residual
Gaussian residual with unit variance
Step 2 : Describe variation between individual parameters
Distribution of clearancesPopulation of patients
Clearance0 0.1 0.2 0.3 0.4
Step 2 : Our view through a sample of patients
Sample of patients Sample of clearances
Step 2 : Two main approaches:parametric and semi-parametric
Sample of clearances Semi-parametric approach
Step 2 : Two main approaches
Sample of clearances Semi-parametric approach(e.g. kernel estimate)
Step 2 : Semi-parametric approach
• Does require a large sample size to provide
results
• Difficult to implement
• Is implemented on “commercial” PK software
Bias?
Step 2 : Two main approaches
Sample of clearances
0 0.1 0.2 0.3 0.4
Parametric approach
Step 2 : Parametric approach
• Easier to understand• Does not require a large sample size to provide (good or poor) results• Easy to implement• Is implemented on the most popular pop PK software (NONMEM, S+, SAS,…)
Step 2 : Parametric approach
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
VVi
ClCli
i
i
V
Cl
ln
ln
CllnVln
A simple model :
Cl
V
ln Cl
ln V
Cl
V VCl,
Step 2 : Population parameters
Cl VMean parameters
2
2
VVCl
VClCl
Variance parameters :
measure inter-individualvariability
Step 2 : Parametric approach
jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
VVi
CliiCli
i
i
V
XθXθCl
ln
ln 2211
A model including covariates
Clln
BMI
Age
X2i
X1i
2211 XXCl
Cl
i
CliiCli i
XXCl 2211ln
Step 2 : A model including covariates
Step 3 :Estimate the parameters of the current model
Several methods with different properties
1. Naive pooled data2. Two-stages3. Likelihood approximations
1. Laplacian expansion based methods2. Gaussian quadratures
4. Simulations methods
1. Naive pooled data : a single patient
Naïve Pooled Data combines all the data as if they came from a single reference individual and fit into a model using classical fitting procedures. It is simple, but can not investigate fixed effect sources of variability, distinguish between variability within and between individuals.
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jjjj tV
Cl
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Dt
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Cl
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DY
expexp
The naïve approach does not allow to estimate inter-individual variation.
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2. Two stages method: stage 1Within individual variability
Con
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rati
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jijii
i
iji
i
i
iji t
V
Cl
V
Dt
V
Cl
V
DY ,,,, expexp
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11ˆ,ˆ VlC
22ˆ,ˆ VlC
33ˆ,ˆ VlC
nn VlC ˆ,ˆ
.
.
.
Two stages method : stage 2 Between individual variability
• Does not require a specific software
• Does not use information about the distribution
• Leads to an overestimation of which tends to zero when the number of observations per animal increases.
• Cannot be used with sparse data
VVi
ClCli
i
i
V
lC
ˆln
ˆln
3. The Maximum Likelihood Estimator
i
N
iiii dyhyl
1
,,expln,
VCl
iii ,
Let 222 ,,,,, VClVCl
i
N
iiii dyhArg
1
,,explninfˆ
The Maximum Likelihood Estimator
•Is the best estimator that can be obtained among the consistent estimators
•It is efficient (it has the smallest variance)
•Unfortunately, l(y,) cannot be computed exactly
•Several approximations of l(y,) are used.
3.1 Laplacian expansion based methods
First Order (FO) (Beal, Sheiner 1982) NONMEMLinearisation about 0
jiji
V
Cl
V
Cli
Vi
Vi
Cliji
V
Cl
V
jijii
i
iji
i
i
iji
tD
ZZZtD
tV
Cl
V
Dt
V
Cl
V
DY
,,
321,
,,,,
exp
expexp
exp
exp
expexp
exp
expexp
Laplacian expansion based methods
First Order Conditional Estimation (FOCE) (Beal, Sheiner) NONMEM Non Linear Mixed Effects models (NLME) (Pinheiro, Bates)S+, SAS (Wolfinger)
jiji
i
i
i
Vi
Vi
Cli
Clii
Vi
Vii
Cli
Cliiji
i
i
i
jijii
i
iji
i
i
iji
tV
lC
V
DZ
ZZtV
lC
V
D
tV
Cl
V
Dt
V
Cl
V
DY
,,3
21,
,,,,
ˆ
ˆexp
ˆˆˆˆ,
ˆˆ,ˆˆ,ˆ
ˆexp
ˆ
expexp
Linearisation about the current prediction of the individual parameter
Gaussian quadratures
N
i
P
ki
kii
i
N
iiii
yh
dyhyl
1 1
1
,,expln
,,expln,
Approximation of the integrals by discrete sums
4. Simulations methods
Simulated Pseudo Maximum Likelihood (SPML)
,,ln,1 2
,,2 1 DVDy ii
DViii
K
kjiV
V
ClCl
ClV
ji tD
KKi
Ki
Ki1,,
,
,
,exp
expexp
exp
1
iV simulated variance
Minimize
Properties
Naive pooled data Never Easy to use Does not provide consistent estimate
Two stages Rich data/ Does not require Overestimation of initial estimates a specific software variance components
FO Initial estimate quick computation Gives quickly a resultDoes not provideconsistent estimate
FOCE/NLME Rich data/ small Give quickly a result. Biased estimates whenintra individual available on specific sparse data and/orvariance softwares large intra
Gaussian Always consistent and The computation is long quadrature efficient estimates when P is large
provided P is large
SMPL Always consistent estimates The computation is longwhen K is large
Criterion When Advantages Drawbacks
Model check: Graphical analysis
VVi
ClCli
i
i
V
Cl
ln
ln
VVi
CliiCli
i
i
V
ageBWCl
ln
ln 21
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Observed concentrations
Pre
dict
ed c
once
ntra
tions Variance reduction
Graphical analysis
Time
ji ,
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The PK model seems good The PK model is inappropriate
Graphical analysis
Normality acceptable
Cl
iV
i
under gaussian assumption
Cl
i
V
i
Normality should be questioned
add other covariatesor try semi-parametric model
The Theophylline example
An alkaloid derived from tea or produced synthetically; it is a smooth muscle relaxant used chiefly for its bronchodilator effect in the treatment of chronic obstructive pulmonary emphysema, bronchial asthma, chronic bronchitis and bronchospastic distress. It also has myocardial stimulant, coronary vasodilator, diuretic and respiratory center stimulant effects.
http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap46/sect38.htm
References
Davidian, M. and Giltinan, D.M. (1995). Nonlinear Models for Repeated Measurement Data. Chapman & Hall/CRC Press.
Davidian, M. and Giltinan, D.M. (2003). Nonlinear models for repeated measurement data: An overview and update. Journal of Agricultural, Biological, and Environmental Statistics 8, 387–419.
Davidian, M. (2009). Non-linear mixed-effects models. In Longitudinal Data Analysis, G. Fitzmaurice, M. Davidian, G. Verbeke, and G. Molenberghs (eds). Chapman & Hall/CRC Press, ch. 5, 107–141.
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