Ziad Taib

Biostatistics, AZ

MV, CTH

April 2011

Lecture 4

Non-Linear and Generalized Mixed Effects Models

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Part I

Generalized Mixed Effects Models

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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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-5

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Time

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idua

l

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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

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expexp

The naïve approach does not allow to estimate inter-individual variation.

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2. Two stages method: stage 1Within individual variability

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jijii

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iji

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iji t

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Cl

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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

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ed c

once

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tions Variance reduction

Graphical analysis

Time

ji ,

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-1

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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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