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1
Generalized Estimating Equations (GEEs)
Purpose: to introduce GEEs
These are used to model correlated data from
• Longitudinal/ repeated measures studies
• Clustered/ multilevel studies
2
Outline• Examples of correlated data• Successive generalizations
– Normal linear model– Generalized linear model– GEE
• Estimation• Example: stroke data
– exploratory analysis– modelling
3
Correlated data1. Repeated measures: same subjects, same measure,
successive times – expect successive measurements to be correlated
Subjects, i = 1,…,n
A
C
B
Randomize i1Y i2Y i3Y i4Y
Treatment groups Measurement times
4
Correlated data2. Clustered/multilevel studies
Level 3
Level 2
Level 1
E.g., Level 3: populations
Level 2: age - sex groups
Level 1: blood pressure measurements in sample of people in each age - sex group
We expect correlations within populations and within age-sex groups due to genetic, environmental and measurement effects
5
Notation
i
i1
i2i
in
y
y Vector of measurements for unit i
y
y
• Repeated measurements: yij, i = 1,… N, subjects;
j = 1, … ni, times for subject i
• Clustered data: yij, i = 1,… N, clusters; j = 1, … ni,
measurements within cluster i
• Use “unit” for subject or cluster
1
2
N
Vector of measurements for all units
y
yy
y
6
Normal Linear Model
For all units: E(y)==X, y~N(,V)
NNN
V
V
V
X
X
X
X
μ
μ
μ
μ
0
0
00
,,1
2
1
2
1
This V is suitable if the units are independent
For unit i: E(yi)=i=Xi; yi~N(i, Vi)
Xi: nip design matrix
: p1 parameter vector
Vi: nini variance-covariance matrix,
e.g., Vi=2I if measurements are independent
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Normal linear model: estimation
T 1
T 1
T 1i i i i
log-likelihood function ( ) ( )
Score = U( ) = ( )
( )
y μ V y μ
β X V y μβ
X V y X β 0
Solve this set of score equations to estimate β
We want to estimate and V
Use
β
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Generalized linear model (GLM)
ij i
ij i j
ij ij i
T 1i i i i
i
Y 's (elements of ) are not necessarily Normal
(e.g., Poisson, binomial)
E(Y ) μ
g(μ ) = η = ; g is the function
Score = U( ) = ( )
where is matrix of derivatives with e
y
xβ link
β D V y μ 0
D
i iik
k k
i ij
i i
lements
μ μ = x
β η
and is diagonal with elements var(Y )
(If link is identity then = )
V
D X
9
Generalized estimating equations (GEE)
ij
ij
i ij
1/ 21/ 2i i i
i
Y 's are not necessarily Normal
Y 's are not necessarily independent
is correlation matrix for Y 's
Variance-covariance matrix can be written as
where is diagonal with elements var
R
A R A
A
ij
1Ti i i i
1/ 21/ 2i i i i
(Y )
Score = U( ) = ( )
where ( ) ( allows for over-dispersion)
β D V y μ 0
V A R A
10
Generalized estimating equations
Di is the matrix of derivatives i/j
Vi is the ‘working’ covariance matrix of Yi
Ai=diag{var(Yik)}, Ri is the correlation matrix for Yi
is an overdispersion parameter
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Overdispersion parameter
Estimated using the formula:
i j ij
ijijypN )var(
1ˆ
Where N is the total number of measurements and p is the number of regression parameters
The square root of the overdispersion parameteris called the scale parameter
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Estimation (1)
T T 1 Ti i i
T 1 1
ˆSolve U( ) = ( ) to get ( )
ˆwith var( ) = ( )
β X y Xβ 0
For Normal linear mod
β X X X y
β
l
V X
e
X
More generally, unless Vi is known, need iteration to solve
1. Guess Vi and estimate by b and hence
2. Calculate residuals, rij=yij-ij
3. Estimate Vi from the residuals
4. Re-estimate b using the new estimate of Vi
Repeat steps 2-4 until convergence
0)()( 1 iiiTiU μyVDβ
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Estimation (2) – For GEEs
Liang and Zeger (1984) showed if is correctly
ˆspecified, is consistent and asymptotically Normal.
ˆ is fairly robust, so correct specification of
('working correlation matrix') is not critical
R
β
β
R
-1 1 T -1s
T -1 T -1
.
Also is estimated so need ' '
ˆfor var( )
ˆ ˆ( ) = where =
sandwich est
and
ˆ ˆˆ ˆ= ( - )( - )
imatorV
β
V β C D V D
C D V y μ y μ V D
I I I
14
Iterative process for GEE’s
• Start with Ri=identity (ie independence) and =1: estimate
• Use estimates to calculated fitted values:
• And residuals:
• These are used to estimate Ai, Ri and
• Then the GEE’s are solved again to obtain improved estimates of
)(gμ̂ 1 ii X
ii μ̂Y
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Correlation
12 1n
212i
..
n1 ..
1 ρ ρ
ρ 1=
ρ
ρ ρ 1
V
For unit i
For repeated measures = correl between times l and m
For clustered data = correl between measures l and m
For all models considered here Vi is assumed to be same for all units
lmρ
lmρ
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Types of correlation
1. Independent: Vi is diagonal
2. Exchangeable: All measurements on the same unit are equally correlated
Plausible for clustered data
Other terms: spherical and compound symmetry
lmρ ρ
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Types of correlation3. Correlation depends on time or distance between
measurements l and m
e.g. first order auto-regressive model has terms , 2, 3 and so on
Plausible for repeated measures where correlation is known to decline over time
4. Unstructured correlation: no assumptions about the correlations
Lots of parameters to estimate – may not converge
- |l-m|lm lmρ is a function of |l - m|, e.g. ρ e
lmρ
18
Missing Data
For missing data, can estimate the working correlation using the all available pairs method, in which all non-missing pairs of data are used in the estimators of the working correlation parameters.
19
Choosing the Best Model
Standard Regression (GLM)
AIC = - 2*log likelihood + 2*(#parameters)
Values closer to zero indicate better fit and greater parsimony.
20
Choosing the Best ModelGEE
QIC(V) – function of V, so can use to choose best correlation structure.
QICu – measure that can be used to determine the best subsets of covariates for a particular model.
the best model is the one with the smallest value!
21
Other approaches – alternatives to GEEs1. Multivariate modelling – treat all
measurements on same unit as dependent variables (even though they are measurements of the same variable) and model them simultaneously
(Hand and Crowder, 1996)
e.g., SPSS uses this approach (with exchangeable correlation) for repeated measures ANOVA
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Other approaches – alternatives to GEEs2. Mixed models – fixed and random effects
e.g., y = X + Zu + e
: fixed effects; u: random effects ~ N(0,G)
e: error terms ~ N(0,R)
var(y)=ZGTZT + R
so correlation between the elements of y is due to random effects
Verbeke and Molenberghs (1997)
23
Example of correlation from random effectsCluster sampling – randomly select areas (PSUs) then
households within areas
Yij = + ui + eij
Yij : income of household j in area i
: average income for population
ui : is random effect of area i ~ N(0, ); eij: error ~ N(0, )
E(Yij) = ; var(Yij) = ;
cov(Yij,Ykm)= , provided i=k, cov(Yij,Ykm)=0, otherwise.
So Vi is exchangeable with elements: =ICC
(ICC: intraclass correlation coefficient)
2u 2
e22eu
2u
22
2
eu
u
24
Numerical example: Recovery from strokeTreatment groups
A = new OT interventionB = special stroke unit, same hospitalC= usual care in different hospital
8 patients per groupMeasurements of functional ability – Barthel index
measured weekly for 8 weeks
Yijk : patients i, groups j, times k • Exploratory analyses – plots• Naïve analyses• Modelling
25
Numerical example: time plots
Individual patients and overall regression line
19
8642
100
80
60
40
20
0
week
score
26
Numerical example: time plots for groups
8642
80
70
60
50
40
30
week
score
A:blue
B: black
C: red
27
Numerical example: research questions
• Primary question: do slopes differ (i.e. do treatments have different effects)?
• Secondary question: do intercepts differ (i.e. are groups same initially)?
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Numerical example: Scatter plot matrix
Week1
Week2
Week3
Week4
Week5
Week6
Week7
Week8
29
Numerical example
Correlation matrix
week 1 2 3 4 5 6 7
2 0.93
3 0.88 0.92
4 0.83 0.88 0.95
5 0.79 0.85 0.91 0.92
6 0.71 0.79 0.85 0.88 0.97
7 0.62 0.70 0.77 0.83 0.92 0.96
8 0.55 0.64 0.70 0.77 0.88 0.93 0.98
30
Numerical example 1. Pooled analysis ignoring correlation within patients
; ijk j j ijk
ijk
Y α β k e j for groups, k for time
Different intercepts and different slopes for groups.
Assume all Y are independent and same variance
(i.e. ignore the correlation between observatio
' 'j j
ns).
Use multiple regression to compare α s and β s
To model different slopes use interaction terms
group time
31
Numerical example 2. Data reduction
ˆ
ijk ij ij ijk
ij ij
ij
Fit a straight line for each patient
Y α β k e
assume independence and constant variance
use simple linear regression to estimate α and β
Perform ANOVA using estimates α
'
ˆj
ij j
as data
and groups as levels of a factor in order to compare α s.
Repeat ANOVA using β 's as data and compare β 's
32
Numerical example 2. Repeated measures analyses using various variance-covariance structures
For the stroke data, from scatter plot matrix and correlations, an auto-regressive structure (e.g. AR(1)) seems most appropriate
Use GEEs to fit models
ijk j j ijk
j j
ijk
Fit Y α β k e
with α and β as the parameters of interest
Assuming Normality for e but try
various forms for variance-covariance matrix
33
Numerical example 4. Mixed/Random effects model
Use model
Yijk = (j + aij) + (j + bij)k + eijk
(i) j and j are fixed effects for groups
(ii) other effects are random
and all are independent
Fit model and use estimates of fixed effects to compare j’s and j’s
),0(~,),0(~,),0(~ 222eijkbijaij NeNbNa
34
Numerical example: Results for intercepts
Intercept A Asymp SE Robust SE
Pooled 29.821 5.772
Data reduction 29.821 7.572
GEE, independent 29.821 5.683 10.395
GEE, exchangeable 29.821 7.047 10.395
GEE, AR(1) 33.492 7.624 9.924
GEE, unstructured 30.703 7.406 10.297
Random effects 29.821 7.047
Results from Stata 8
35
Numerical example: Results for intercepts
B - A Asymp SE Robust SE
Pooled 3.348 8.166
Data reduction 3.348 10.709
GEE, independent 3.348 8.037 11.884
GEE, exchangeable 3.348 9.966 11.884
GEE, AR(1) -0.270 10.782 11.139
GEE, unstructured 2.058 10.474 11.564
Random effects 3.348 9.966
Results from Stata 8
36
Numerical example: Results for intercepts
C - A Asymp SE Robust SE
Pooled -0.022 8.166
Data reduction -0.018 10.709
GEE, independent -0.022 8.037 11.130
GEE, exchangeable -0.022 9.966 11.130
GEE, AR(1) -6.396 10.782 10.551
GEE, unstructured -1.403 10.474 10.906
Random effects -0.022 9.966
Results from Stata 8
37
Numerical example: Results for slopes
Slope A Asymp SE Robust SE
Pooled 6.324 1.143
Data reduction 6.324 1.080
GEE, independent 6.324 1.125 1.156
GEE, exchangeable 6.324 0.463 1.156
GEE, AR(1) 6.074 0.740 1.057
GEE, unstructured 7.126 0.879 1.272
Random effects 6.324 0. 463
Results from Stata 8
38
Numerical example: Results for slopes
B - A Asymp SE Robust SE
Pooled -1.994 1.617
Data reduction -1.994 1.528
GEE, independent -1.994 1.592 1.509
GEE, exchangeable -1.994 0.655 1.509
GEE, AR(1) -2.142 1.047 1.360
GEE, unstructured -3.556 1.243 1.563
Random effects -1.994 0.655
Results from Stata 8
39
Numerical example: Results for slopes
C - A Asymp SE Robust SE
Pooled -2.686 1.617
Data reduction -2.686 1.528
GEE, independent -2.686 1.592 1.502
GEE, exchangeable -2.686 0.655 1.509
GEE, AR(1) -2.236 1.047 1.504
GEE, unstructured -4.012 1.243 1.598
Random effects -2.686 0.655
Results from Stata 8
40
Numerical example: Summary of results
• All models produced similar results leading to the same conclusion – no treatment differences
• Pooled analysis and data reduction are useful for exploratory analysis – easy to follow, give good approximations for estimates but variances may be inaccurate
• Random effects models give very similar results to GEEs
• don’t need to specify variance-covariance matrix
• model specification may/may not be more natural