Generalized Estimating Equations (GEEs)

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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 data

1. Repeated measures same subjects, same measure,
   successive times expect successive measurements
   to be correlated

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Correlated data

1. 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

• 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

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Normal Linear Model
For unit i E(yi)?iXi? yiN(?i, Vi) Xi
ni?p design matrix ? p?1 parameter
vector Vi ni?ni variance-covariance
matrix, e.g., Vi?2I if measurements are
independent
For all units E(y)?X?, yN(?,V)
This V is suitable if the units are independent
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Normal linear model estimation
We want to estimate and V Use
Solve this set of score equations to estimate
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Generalized linear model (GLM)
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Generalized estimating equations (GEE)
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Generalized estimating equations
Di is the matrix of derivatives ??i/??j Vi is the
working covariance matrix of Yi Aidiagvar(Yik)
, Ri is the correlation matrix for Yi ? is an
overdispersion parameter
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Overdispersion parameter

• Estimated using the formula

Where N is the total number of measurements and
p is the number of regression parameters The
square root of the overdispersion parameter is
called the scale parameter
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Estimation (1)

• More generally, unless Vi is known, need
  iteration to solve
• Guess Vi and estimate ? by b and hence ?
• Calculate residuals, rijyij-?ij
• Estimate Vi from the residuals
• Re-estimate b using the new estimate of Vi
• Repeat steps 2-4 until convergence

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Estimation (2) For GEEs
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Iterative process for GEEs

• Start with Riidentity (ie independence) and ?1
  estimate ?
• Use estimates to calculated fitted values
• And residuals
• These are used to estimate Ai, Ri and ?
• Then the GEEs are solved again to obtain
  improved estimates of ?

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Correlation
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
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Types of correlation

• 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

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Types of correlation
3. 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
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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 - 2log likelihood 2(parameters)
• Values closer to zero indicate better fit and
  greater parsimony.

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Choosing the Best Model

• GEE
• 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 GEEs

• 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 GEEs

• 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 effects

• Cluster 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 ik, cov(Yij,Ykm)0,
  otherwise.
• So Vi is exchangeable with elements
  ICC

(ICC intraclass correlation coefficient)
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Numerical example Recovery from stroke

• Treatment groups
• A new OT intervention
• B special stroke unit, same hospital
• C usual care in different hospital
• 8 patients per group
• Measurements 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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Numerical example time plots for groups
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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
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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
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Numerical example 2. Data reduction
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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
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Numerical example 4. Mixed/Random effects model

• Use model
• Yijk (?j aij) (?j bij)k eijk
• ?j and ?j are fixed effects for groups
• other effects are random
• and all are independent
• Fit model and use estimates of fixed effects to
  compare ?js and ?js

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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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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
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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
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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
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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
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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
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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
• dont need to specify variance-covariance matrix
• model specification may/may not be more natural