Lecture 4
Non-Linear and Generalized
Mixed Effects Models
Ziad Taib
Biostatistics, AZ
MV, CTH
Mars 2009
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Part I
Generalized Mixed Effects
Models
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Outline of part I:
Generalized Mixed Effects Models
1.Formulation
2.Estimation
3.Inference
4.Software
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Various forms of models and relation between them
Classical statistics (Observations are random, parameters are unknown constants)
LM: Assumptions:
1.
independence,
2.
normality,
3.
constant parameters
LMM:
Assumptions 1)
and 3) are modified
GLM: assumption 2)
Exponential family
Repeated measures:
Assumptions 1) and 3)
are modified
GLMM: Assumption 2) Exponential
family and assumptions 1) and 3) are
modified
Longitudinal data
Maximum likelihood
LM - Linear model
Non-linear models
GLM - Generalised linear model
LMM - Linear mixed model
GLMM - Generalised linear mixed model
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Bayesian statistics
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.
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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?
• 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.
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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 nonmalignant disease.
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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, . . .
Observed
frequencies
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Generalized linear Models:
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The Bernoulli case
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Generalized Linear Models
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Longitudinal Generalized Linear Models
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Generalised Linear Mixed Models
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Empirical
Bayes
estimates
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Example 1 (cont’d)
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Syntax for NLMIXED
http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap46/index.htm
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PROC NLMIXED options ;
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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PROC NLMIXED Statement
BY Statement
CONTRAST Statement
ESTIMATE Statement
ID Statement
MODEL Statement
PARMS Statement
PREDICT Statement
RANDOM Statement
REPLICATE Statement
Programming Statements
Example
data infection;
input clinic t x n;
datalines;
 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.
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
7115
7019
8146
8067
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 estimates .
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Parameter Estimates
Paramet
Standar
er
Estimate d Error
DF
t Value Pr > |t|
Alpha
Lower
-2.5123
Upper Gradient
beta0
-1.1974
0.5561
7
-2.15
0.0683
0.05
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
Estimate
Standar
d Error
DF
t Value Pr > |t|
Alpha
Lower
Upper
1.3542
0.5509
7
0.05 0.05146
2.6569
Label
1/beta1
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2.46
0.0436
0.1175
-3.1E-7
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 tstatistic 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)
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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End of Part I
Any Questions
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?