1. GEE and Generalized Linear Mixed Models
Tom Greene

2. Outline
Subject specific and population average inference in generalized linear models
Review of classical generalized linear models with independent observations
Generalized Estimating Equations
Contrasts of GLMMs with GEEs
GEE example

3. Classes of Generalized Linear Models
(Linear regression, ANOVA, ANCOVA)
E(Y) = X β,
Responses Independent
Linear Mixed Models
E(Y|b) = X β + Z b
Responses Correlated
Correlation modeled in part by “random effects”
Generalized Linear Models
(Logistic regression, Poisson regression, etc.)
g(E(Y)) = X β
Responses Independent
Generalized Linear Mixed Models (GLMM)
g(E(Y|b)) = X β + Z b
Responses Correlated
Correlation modeled in part by “random effects”
Generalized Estimating Equations Approach (GEE)
g(E(Y)) = X β
Responses Correlated

4. Classes of Generalized Linear Models for Correlated Data
Linear Mixed Models
E(Y|b) = X β + Z b
Responses Correlated
Correlation modeled in part by “random effects”
Generalized Estimating Equations Approach (GEE)
g(E(Y)) = X β
Responses Correlated
Generalized Linear Mixed Models (GLMM)
g(E(Y|b)) = X β + Z b
Responses Correlated
Correlation modeled in part by “random effects”
Population Average Inference
Subject Specific Inference

5. Classes of Generalized Linear Models for Correlated Data
Population Average Inference
Subject Specific Inference
Generalized Estimating Equations Approach (GEE)
g(E(Y)) = X β
Responses Correlated
Generalized Linear Mixed Models (GLMM)
g(E(Y|b)) = X β + Z b
Responses Correlated
Analysis describes differences in the mean of Y across the entire population
Analysis describes differences in the mean of Y conditional on the
patient’s specific random effect b
Most relevant from an individual patient’s perspective
Often b represent a dimension of frailty – Hence, X β tells about the
relationship of Y to X among patients with the same frailty
Analysis informative from population perspective; most relevant from perspective of
Policy makers
Providers desiring to optimize outcomes across entire population

6. Extreme Example
Subject specific effects of X on Pr(Death), OR = 20 per 1 unit increase in X
Population average effect of X on Pr(Death), OR = 2.7 per 1 unit increase in X

8. Example: Toenail Data Toenail Dermatophyte Onychomycosis:
Common toenail infection, difficult to treat, affecting more than 2% of population.
Design: Randomized, double-blind, parallel group, multicenter study for the comparison of two new compounds (A and B) for oral treatment.
2 x189 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.
Research question: Severity relative to treatment of TDO ?

11. Review of Generalized Linear Models (Independent Responses)
Independent responses Yi, i = 1, 2, …, N
Yi, with distribution from exponential family
f(y;θ,ø) =
Mean model
μi = E(Yi|Xi1,Xi2,…,Xip)
g(μi) = β0 + β1Xi1 + β2Xi2+ βpXip
Variance function
Var(Yi) = øV(μi)
V(μi) is a known function determined by the assumed distribution of Y within the exponential family

12. Review of Generalized Linear Models (Independent Responses)

13. Review of Generalized Linear Models (Independent Responses)

14. Review of Generalized Linear Models (Independent Responses)
Independent responses Yi, i = 1, 2, …, N
Yi, with distribution from exponential family
f(y;θ,ø) =
Mean model
μi = E(Yi|Xi1,Xi2,…,Xip)
g(μi) = β0 + β1Xi1 + β2Xi2+ βJXiJ
Variance function
Var(Yi) = øV(μi)
vi = V(μi) is a known function determined by the assumed distribution of Y within the exponential family
The mean model is the only part we have to get right for valid large-sample inference!!!

15. Extension to GEE for Longitudinal Data
GEE: Generalized Estimating Equations (Liang & Zeger, 1986;
Zeger & Liang, 1986)
• Method is semi-parametric
– estimating equations are derived without full specification
of the joint distribution of a subject’s observations
• Instead, specification of
The mean model for the marginal distributions of the yij
The variance function of yij given µij
The “working” correlation matrix for the vector of repeated observations from each subject
Relies on the independence across subjects (or clusters) to estimate consistently the variance of the regression coefficients

16. GEE Method Outline
1. Relate the marginal response μij = E(yij) to a linear combination of the covariates g(μij) = Xtijβ
• yij is the response for subject i at time j, j = 1,2, .., J
• Xij is a p × 1 vector of covariates
β is a p × 1 vector of regression coefficients
• g(·) is the link function
2. Describe the variance of yij as a function of the mean
V(yij) = v(μij)ø
• ø is possibly unknown scale parameter
• v(·) is a known variance function

17. Link and Variance Functions
• Normally-distributed response
g(μij) = μij “Identity link”
v(μij) = 1
V(yij) = ø
• Binary response (Bernoulli)
g(μij) = log[μij/(1 − μij)] “Logit link”
v(μij) = μij(1 − μij)
ø = 1
• Poisson response
g(μij) = log(μij) “Log link”
v(μij) = μij
ø = 1

18. GEE Method Outline
3. Choose the form of a n × n “working” correlation matrix Ri for each Yi

19. Working Correlation Structures

20. Working Correlation Structures

21. Working Correlation Structures
(AR(1)

22. Working Correlation Structures

23. GEE Estimation
• Define Ai = n × n diagonal matrix with V(μij) as the jth diagonal element
• Define Ri(α) = n × n “working” correlation matrix (of the n repeated measures)
Working variance–covariance matrix for Yi equals
Vi(α) = øAi1/2 Ri(α) Ai1/2

29. GEE vs. GLMM 1) Target of Inference: GEE: Population Average
GLMM: Subject Specific
Notes: Recent work on perform population average inference under GLMM models

30. GEE vs. GLMM 2) Outputs: GEE: GLMM: Coefficients relating Y to X
Coefficients relating Y to X conditional on b
Estimates of subject specific random effects
Variance of subject specific random effects

31. GEE vs. GLMM 3) Robustness: GEE (with robust variance estimates):
Inference valid in large samples even if distribution of Y and/or variance of Y are incorrectly specified
GLMM (with model-based estimates)
Valid inference generally requires correct specification of distribution of Y and of variance of Y
Notes:
Recent proposals for robust variance estimates under GLMM
Inference for Linear Mixed Models remains valid if Y is not normal for large N
Caveat to GEE robustness: GEE can be biased if time dependent covariates are used unless an independent working correlation matrix is used

32. GEE vs. GLMM 4) Efficiency (power and width of confidence intervals)
Usually fairly efficient if variance function is correctly specified
Between subject comparisons are nearly efficient if an independence covariance structure is used for balanced data
GLMM:
Maximum likelihood estimates are asymptotically efficient as long as the model is correctly specified

33. GEE vs. GLMM 5) Missing Data:
“Classical” GEE (with robust variance estimates)
Valid inference if data are Missing Completely At Random (MCAR) even if variance model is wrong
If variance model is correct, estimate of β is still consistent if data are MAR but not MCAR (but standard errors are not correct)
GLMM (with model-based estimates)
Valid inference if data are Missing At Random (MAR)
Notes:
Various strategies for valid GEE inference if data are MAR

34. Missing data
Three general approaches to dealing with missing data under GEE which assume MAR but not MCAR
Inverse probability weighting (Robins, Rotnitzky and Zhao, JASA, 1995)
Multiple imputation
Inverse probability weighting with augmentation, or doubly robust estimation
Each method can incorporate covariate information not included in the GEE model itself. This can make the MAR assumption much more plausible.
Methods 2 and 3 can be considerably more efficient than standard inverse probability weighting

36. GEE vs. GLMM 6) Small to Moderate Samples:
GEE (with robust variance estimates):
Estimated standard errors are unstable and biased downwards
Inefficient estimating equation for estimating variance
Effectively uses fully unstructured variance model
“Sample size” means the number of independent units
Various corrections have been proposed (available in PROC GLIMMIX)
GLMM (with model-based estimates)
Large-sample approximations are often invoked, but performance usually better than GEE with small to moderate N if model is correctly specified.

37. More Toenail Data
Multicenter trial comparing active vs. control oral treatments for toenail infection
Repeated measurements of binary outcome:
0 = none or mild separation
1 = severe separation
1908 observations in 294 patients, mostly over 1 year

38. **** Standard GENMOD GEE program using Robust SEs *****;
**** Binary outcome leads to default logistic link function ****;
proc genmod descending;
Class id;
model outcome = treatment month treatment*month/ dist=bin;
repeated subject=id/type=exch covb corrw;
estimate 'Control Slope' month 1/exp;
estimate 'Treartment Slope' month 1 treatment*month 1/exp;
run;
Working Correlation Matrix
Col1 Col2 Col3 Col4 Col5 Col6 Col7
Row
Row
Row
Row
Row
Row
Row

39. Analysis Of GEE Parameter Estimates
**** Standard GENMOD GEE program using Robust SEs;
**** Binary outcome leads to default logistic link function;
proc genmod descending;
Class id;
model outcome = treatment month treatment*month/ dist=bin;
repeated subject=id/type=exch covb corrw;
estimate 'Control Slope' month 1/exp;
estimate 'Treatment Slope' month 1 treatment*month 1/exp;
run;
Analysis Of GEE Parameter Estimates
Empirical Standard Error Estimates
Standard 95% Confidence
Parameter Estimate Error Limits Z Pr > |Z|
Intercept
treatment
month <.0001
treatment*month

40. **** Standard GENMOD GEE program using Robust SEs *****;
**** Binary outcome leads to default logistic link function ****;
proc genmod descending;
Class id;
model outcome = treatment month treatment*month/ dist=bin;
repeated subject=id/type=exch covb corrw;
estimate 'Control Slope' month 1/exp;
estimate 'Treatment Slope' month 1 treatment*month 1/exp;
run;
Can ignore in this case
Contrast Estimate Results
Mean Mean L'Beta Standard
Label Estimate Confidence Limits Estimate Error
Control Slope
Exp(Control Slope)
Treatment Slope
Exp(Treatment Slope)
L'Beta Chi-
Label Alpha Confidence Limits Square Pr > ChiSq
Control Slope <.0001
Exp(Control Slope)
Treatment Slope <.0001
Exp(Treatment Slope)

41. Solutions for Fixed Effects
**** GLIMMIX GLMM Estimating Subject Specific Effects ****;
**** Binary outcome leading to default logistic link function ****;
proc glimmix method=RSPL data=toenail;
Class id;
model outcome (event="1") = treatment month treatment*month/ s dist=binary;
random int / subject=id;
estimate 'Control Slope' month 1/or;
estimate 'Treartment Slope' month 1 treatment*month 1/or cl; run;
Solutions for Fixed Effects
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept
treatment
month <.0001
treatment*month

42. data small; set toenail; if id <= 20;
*** Small Sample;
data small; set toenail; if id <= 20;
** Standard GENMOD GEE with Robust SEs: 17 Patients Only ***;
** Binary outcome leading to default logistic link function **;
proc genmod descending;
Class id;
model outcome = treatment month treatment*month/ dist=bin;
repeated subject=id/type=exch covb corrw; run;
Standard 95% Confidence
Parameter Estimate Error Limits Z Pr > |Z|
Intercept
treatment
month
treatment*month

43. Solutions for Fixed Effects
**** GLIMMIX GEE program using Robust SEs;
**** Binary outcome leads to default logistic link function;
**** Restricted to 17 patients;
**** Small N Adjustment of Morel, Bokossa, and Neerchal (2003);
proc glimmix method=RSPL empirical=mbn data=small;
Class id;
model outcome (event="1") = treatment month treatment*month/ s dist=binary ddfm=kenwardroger;
random _residual_ / subject=id type=cs;
run;
Solutions for Fixed Effects
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept
treatment
month
treatment*month

44. THAT’s ALL