generalized estimating equations (gees)

1

Generalized Estimating Equations (GEEs)

Purpose: to introduce GEEs

These are used to model correlated data from

• Longitudinal/ repeated measures studies

• Clustered/ multilevel studies

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Outline• Examples of correlated data• Successive generalizations

– Normal linear model– Generalized linear model– GEE

• Estimation• Example: stroke data

– exploratory analysis– modelling

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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

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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

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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

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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

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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

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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

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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

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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.

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Choosing the Best Model

Standard Regression (GLM)

AIC = - 2*log likelihood + 2*(#parameters)

Values closer to zero indicate better fit and greater parsimony.

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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!

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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)

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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

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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

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Numerical example: time plots

Individual patients and overall regression line

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8642

100

80

60

40

20

0

week

score

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Numerical example: time plots for groups

8642

80

70

60

50

40

30

week

score

A:blue

B: black

C: red

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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

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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

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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

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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

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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

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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

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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

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B - A Asymp SE Robust SE

Pooled 3.348 8.166


GEE, independent 3.348 8.037 11.884


GEE, AR(1) -0.270 10.782 11.139


Random effects 3.348 9.966


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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


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Numerical example: Results for slopes

Slope A Asymp SE Robust SE

Pooled 6.324 1.143


GEE, independent 6.324 1.125 1.156


GEE, AR(1) 6.074 0.740 1.057


Random effects 6.324 0. 463


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B - A Asymp SE Robust SE

Pooled -1.994 1.617


GEE, independent -1.994 1.592 1.509


GEE, AR(1) -2.142 1.047 1.360




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C - A Asymp SE Robust SE

Pooled -2.686 1.617


GEE, independent -2.686 1.592 1.502


GEE, AR(1) -2.236 1.047 1.504




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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

generalized estimating equations (gees)

Documents

correlation matrix

successive measurements

new estimate

subject i clustered

mfor clustered data

working correlation

independentfor unit

residuals reestimate