1 Date

Lecture 4 Non-Linear and Generalized Mixed Effects Models

Ziad Taib Biostatistics, AZ MV, CTH April 2011

2 Date

Part I Generalized Mixed Effects Models

Outline of part I

1.

Generalized Mixed Effects Models 1.

2.

Formulation Estimation 3.

Inference 4.

Software 2.

Non-linear Mixed Effects Models in Pharmacokinetics 1.

Basic Kinetics 2.

3.

4.

Compartmental Models NONMEM Software issues Name, department 3 Date

Various forms of models and relation between them

Classical statistics (Observations are random, parameters are unknown constants) LM : Assumptions: 1.

independence, 2.

3.

normality, constant parameters GLM : assumption 2) Exponential family LMM : Assumptions 1) and 3) are modified Repeated measures : Assumptions 1) and 3) are modified GLMM : Assumption 2) Exponential family and assumptions 1) and 3) are modified Maximum likelihood Longitudinal data

LM - Linear model GLM - Generalised linear model LMM - Linear mixed model GLMM - Generalised linear mixed model

Name, department 4 Date Bayesian statistics Non-linear models

Example 1

Toenail 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.

5 Date

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.

Name, department 6 Date

Example 2 The Analgesic Trial

 

Analgesic treatment for pain caused by chronic

non malignant disease.

   

Single-arm trial with 530 patients recruited (491

selected for analysis).

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.

Name, department 7 Date

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.

Observed frequencies Name, department 8 Date

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

http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap46/index.htm

            

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

these statements.

;

The following sections provide a detailed description of each of 23 Date

     24        

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 Statements

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; 25 Date

 Suppose

n ij

denotes the number of trials for the and the

j

th treatment (

i

= 1, ... ,8

j

= 0,1), and

x ij i

th clinic 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

t j

indicates the

j

th treatment, and the

u i

assumed to be iid .

are 26 Date

 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; Name, department 27 Date

   The PROC NLMIXED statement invokes the procedure, and the PARMS statement defines the parameters and their starting values. The next three statements define

p ij

, and the MODEL defines the conditional distribution of

x ij

statement 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

u i

. The ESTIMATE statement requests an estimate of the reciprocal of .

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Parameter Estimates

Paramet er Estimate Standar d Error beta0 beta1

-1.1974

0.5561

0.7385

0.3004

s2u

1.9591

1.1903

Label

1/beta1

Estimate Standar d Error

1.3542

0.5509

DF t Value Pr > |t| Alpha Lower Upper Gradient

7 7 7 -2.15

2.46

1.65

0.0683

0.0436

0.1438

0.05 -2.5123

0.1175

-3.1E-7 0.05 0.02806

1.4488 -2.08E-6 0.05 -0.8554

4.7736 -2.48E-7

DF t Value Pr > |t|

7 2.46

0.0436

Alpha Lower

0.05 0.05146

Upper

2.6569

Name, department 29 Date

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 0.3004/0.7385

2 = 0.5509 by the delta method (Billingsley 1986). Note this particular approximation produces a

t

statistic identical to that for the estimate of .

30 Date

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) Results Name, department 37 Date

Name, department 38 Date

39

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 .

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.  40 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. Date

        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. Name, department 41 Date

       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.  42 Date

        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 43 Date

Name, department 44 Date End of Part I Any Questions ?

Part II Introduction to non-linear mixed models in Pharmakokinetics

Typical data

180 180 120 120 One curve per patient 40 40 0 0 0 0 5 5 Time

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 time trajectories or mechanisms

of the response and how these

vary

that underlie

individual

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 exogenous compound in the body such as alcohol, toxins etc). (or any other

PHARMACOKINETICS

 

Pharmacokinetics

is about what happens to the drug in the body. It involves the kinetics of drug a bsorption, d istribution, and elimination i.e. m etabolism and e xcretion ( 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 Concentrati on Tim e

Pharmacokinetic models

Various types of models

One-compartment model with rapid intravenous administration: The pharmacokinetics parameters     Half life Distribution volume AUC T max and C max •

D

: Dose •

V D

: Volume •

k

: Elimination rate •

Cl

: Clearance i.v.

k D, V D

 V in k a One compartment model  General model

dC dt



v in



v out

 Tablet

C

(

t

) 

F Dose V k a k a



k e

(

e



k e t



e



k a t

) C(t) , V V e k e  IV

dC

 

kC

0

dt C t



D

exp

V Cl t V

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 Fixed Fixed The concentrations The patients Fixed Random Random Fixed Analytical error Departure to kinetic model Population kinetics Classical kinetics

An example

One PCC per patients Time

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

140 120 100 80 60 40 20 0 0 10 20 30 40 50 60 70

One compartment model with constant intravenous infusion rate

C



C t

(

t

) 

C

 (

t

)

C

0 exp 

D D V V

  ;

C

0 exp exp 

D V V

  

t Cl

;

Cl t

   

kV

Step 1 : Write a deterministic (mean) model to describe the individual kinetics

140 120 100 80 60 40 20 0 0

C

(

t

) 

D

exp

V Cl t V

10 20 30 40 50 60 70

Step 1 : Write a deterministic (mean) model to describe the individual kinetics

140 120 100 80 60 40 20 0 0 residual 10 20 30 40 50 60 70

Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics

25 20 15 10 5 0 -5 0 -10 -15 -20 -25 10 20 30 40 50 60 70 Time

Step 1 : Write a model (variance) to describe the magnitude of departure to the kinetics

25 20 15 10 5 0 -5 0 -10 -15 -20 -25 10 20 30 40 50 60 70 Time

Step 1 : Describe the shape of departure to the kinetics

Residual 0 10 20 30 40 50 60 Time 70

Step 1 :Write an "individual" model

Y i

,

j t i

,

j j th

concentration measured on the

i th

patient

j th

sample time of the

i th

patient residual

Y i

,

j



D V i

exp   

Cl i V i t i

,

j

   

D V i

exp   

Cl i V i t i

,

j

  

i

,

j

Gaussian residual with unit variance

Step 2 : Describe variation between individual parameters

Population of patients 0 0.1

0.2

0.3

0.4

Clearance Distribution of clearances

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

Y i

,

j



D V i

exp   

Cl i V i t i

,

j

   

D V i

exp   

Cl i V i t i

,

j

  

i

,

j

A simple model : ln

Cl i

ln

V i

  

Cl



V

   

V i i Cl

ln

V

ln

Cl

Step 2 : Population parameters

ln

V



V



V

 

Cl

,

V

 

Cl

Mean parameters 

Cl



Cl



V

      2

Cl



Cl V

 

Cl



V



V

2    ln

Cl

Variance parameters : measure inter-individual variability

Step 2 : Parametric approach

Y i

,

j



D V i

exp   

Cl i V i t i

,

j

   

D V i

exp   

Cl i V i t i

,

j

  

i

,

j

A model including covariates ln

Cl i

ln

V i

  

Cl



V

 

θ

1

X

1

i



V i



θ

2

X

2

i

 

i Cl

Step 2 : A model including covariates

ln

Cl i

 

Cl

  1

X

1

i

  2

X

2

i

 

i Cl

ln

Cl



i Cl



Cl

  1

X

1   2

X

2 X 2i Age X 1i BMI

Step 3 :Estimate the parameters of the current model

Several methods with different properties 1. Naive pooled data 2. Two-stages 3. Likelihood approximations 1.

Laplacian expansion based methods 2.

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.

Y j



D V

exp   

Cl V t j

 

D V

exp   

Cl V t j



j

180 160 140 120 100 80 60 40 20 0 0 The naïve approach does not allow to estimate inter individual variation.

5 10 15 20 25 30 35 40 Time 45

2. Two stages method: stage 1 Within individual variability

Y i

,

j



D V i

exp   

Cl i V i t i

,

j

   

D V i

exp      

Cl i V i C

ˆ

l

1

C

ˆ

l t

, ,

i C

ˆ

l

3 2 , ,

j V

ˆ

V

ˆ 2

V

ˆ 3 1      

i

,

j

 .

.

.

l n

,

V

ˆ

n

 Time

Two stages method : stage 2 Between individual variability

ln ln

C

ˆ

V i

ˆ

i l

  

Cl



V

   

V i i Cl

• Does not require a specific software • Does not use information about the distribution • Leads to an overestimation of animal increases.  which tends to zero when the number of observations per • Cannot be used with sparse data

3. The Maximum Likelihood Estimator

Let  

i

     

Cl i Cl

,  , 

V i V

 , 

Cl

2 , 

V

2 ,  ,  2 

l



y

,     ˆ 

i N

   1 ln exp  

h i

 

i

,

Arg

inf 

i N

   1 ln exp  

h i

 

i

,

y i

,   

d



i y i

,   

d



i

The Maximum Likelihood Estimator

 ˆ •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)

NONMEM

Linearisation about 0 Y i

,

j

 

D V i

exp    exp

D

 

V Cl i V i

exp   

t i

,

j

   

D V i

exp exp   

Cl

 

V t i

,

j

  exp

D

 

V

exp      exp exp   

Cl

 

V t i

,

j

exp    

Z

1   

i

,

j Cl i V i



i Cl t i

,

j

  

i

,

j



Z

2 

i V



Z

3   

i V



i Cl

Laplacian expansion based methods

First Order Conditional Estimation (FOCE) (

Beal, Sheiner)

NONMEM Non Linear Mixed Effects models (NLME) (

Pinheiro, Bates

)S+, SAS (

Wolfinger

)

Linearisation about the current prediction of the individual parameter Y i

,

j



D V i

exp   

Cl i V i t i

,

j

   

D V i

exp   

Cl i V i t i

,

j

  

i

,

j



D V

ˆ

i

exp   

C

ˆ

V

ˆ

i l i t i

,

j



Z

1   ,  ˆ

i

  

i Cl

  ˆ

i Cl

 

Z

2   ,  ˆ

i

  

i V

  ˆ

i V

 

Z

3   ,  ˆ

i

  

i Cl

  ˆ

i Cl

 

i V

  ˆ

i V

  

D V

ˆ

i

exp   

C

ˆ

V

ˆ

i l i t i

,

j



i

,

j

Gaussian quadratures

Approximation of the integrals by discrete sums l



y

,    

i N

   1 exp  

h i

 

i

,

i

 1 ln

k P N

   1 exp  

h i

 

i k

,

y i

,   

y i

,   

d



i

4. Simulations methods

Simulated Pseudo Maximum Likelihood (SPML)

Minimize

i

  1 2 

i

,

j



y i

1

K

 

i k K

  1 exp 2

V i

 1   ,

D

,    ln

V i

 

V D

 

Cl i

,

K

 exp        ,

D

,   exp exp    

V Cl

  

Cl i

,

K



V i

,

K

 

t i

,

j

   

V i

simulated variance

Properties Criterion Naive pooled data Two stages FO FOCE/NLME Gaussian quadrature SMPL When Never Rich data/ initial estimates Initial estimate Advantages Easy to use Does not require a specific software quick computation Rich data/ small Give quickly a result. intra individual available on specific variance softwares Always consistent and efficient estimates provided P is large Always consistent estimates Drawbacks Does not provide consistent estimate Overestimation of variance components Gives quickly a result Does not provide consistent estimate Biased estimates when sparse data and/or large intra The computation is long when P is large The computation is long when K is large

Model check: Graphical analysis

180 160 140 120 100 80 60 40 20 0 0 20 ln

Cl i

ln

V i

  

Cl



V

   

V i i Cl

40 60 120 100 80 60 40 20 0 0 160 ln

Cl i

ln

V i

  

Cl



V

  1

BW i

 

V i

  2

age i

 

Cl i

Variance reduction 140 20 40 60 80 100 120 140 80 100 120 140 Observed concentrations

Graphical analysis

3 2 1 0 0 -1 -2 -3 -4  ˆ

i

,

j

10 20 30 40 50 0 0 -1 -2 -3 3 2 1 5 10 15 20 25 30 35 40 45 Time The PK model seems good The PK model is inappropriate

Graphical analysis

 ˆ

V i

 ˆ

Cl i

under gaussian assumption Normality acceptable Normality should be questioned add other covariates or try semi-parametric model  ˆ

Cl i

 ˆ

V i

The Theophylline example

 An alkaloid derived from tea or produced synthetically; it is a smooth muscle relaxant used chiefly for its center stimulant effects.

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 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.

(

An outstanding overview 

)

“

Pharmacokinetics and pharmaco- dynamics

, ” by D.M. Giltinan, in

Encyclopedia of Biostatistics, 2nd edition.

Any Questions

?