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# Generalized Estimating Equations

The generalized estimating equations (GEE), introduced by Liang and Zeger (1986), is a method of analyzing correlated data that otherwise could be modeled as a generalized linear model. GEEs have become an important strategy in the analysis of correlated data. These data sets can arise from longitudinal studies, in which subjects are measured at different points in time, or from clustering, in which measurements are taken on subjects who share a common characteristic such as belonging to the same litter.

SAS/STAT® software's GENMOD procedure enables you to perform GEE analysis by specifying a REPEATED statement in which you provide clustering information and a working correlation matrix. The generalized linear model estimates are used as the starting values. Both model-based and empirical standard errors of the parameter estimates are produced. Many correlation structures are available, including autoregressive(1), exchangeable, independent, m-dependent, and unstructured. You can also input your own correlation structures. The GENMOD procedure also provides the following:

• Type III tests for model effects

• CONTRAST, LSMEANS, and ESTIMATE statements

• alternating logistic regression estimation

• models for ordinal data

The proportional odds model is a popular method for GEE analysis of ordinal data and is based on modeling cumulative logit functions. The GENMOD procedure also models cumulative probits and cumulative complementary log-log functions.

Example

A study on the effects of pollution on children produced the following data. The binary response indicates whether children exhibited symptoms during the period of study at ages 8, 9, 10, and 11. A logistic regression is fit to the data with explanatory variables age, city of residence, and a passive smoking index. The correlations among the binary outcomes are modeled as exchangeable.

```  data children;
input id city\$ @@;
do i=1 to 4;
input age smoke symptom @@;
output;
end;
datalines;
1 steelcity  8 0 1  9 0 1  10 0 1  11 0 0
2 steelcity  8 2 1  9 2 1  10 2 1  11 1 0
3 steelcity  8 2 1  9 2 0  10 1 0  11 0 0
4 greenhills 8 0 0  9 1 1  10 1 1  11 0 0
5 steelcity  8 0 0  9 1 0  10 1 0  11 1 0
6 greenhills 8 0 1  9 0 0  10 0 0  11 0 1
7 steelcity  8 1 1  9 1 1  10 0 1  11 0 0
8 greenhills 8 1 0  9 1 0  10 1 0  11 2 0
9 greenhills 8 2 1  9 2 0  10 1 1  11 1 0
10 steelcity  8 0 0  9 0 0  10 0 0  11 1 0
11 steelcity  8 1 1  9 0 0  10 0 0  11 0 1
12 greenhills 8 0 0  9 0 0  10 0 0  11 0 0
13 steelcity  8 2 1  9 2 1  10 1 0  11 0 1
14 greenhills 8 0 1  9 0 1  10 0 0  11 0 0
15 steelcity  8 2 0  9 0 0  10 0 0  11 2 1
16 greenhills 8 1 0  9 1 0  10 0 0  11 1 0
17 greenhills 8 0 0  9 0 1  10 0 1  11 1 1
18 steelcity  8 1 1  9 2 1  10 0 0  11 1 0
19 steelcity  8 2 1  9 1 0  10 0 1  11 0 0
20 greenhills 8 0 0  9 0 1  10 0 1  11 0 0
21 steelcity  8 1 0  9 1 0  10 1 0  11 2 1
22 greenhills 8 0 1  9 0 1  10 0 0  11 0 0
23 steelcity  8 1 1  9 1 0  10 0 1  11 0 0
24 greenhills 8 1 0  9 1 1  10 1 1  11 2 1
25 greenhills 8 0 1  9 0 0  10 0 0  11 0 0
;
run;

proc genmod data=children;
class id city smoke;
model  symptom = city age smoke / dist=bin type3;
repeated  subject=id / type=exch covb corrw;
contrast 'Smoke=0 vs Smoke=1' smoke 1 –1 0;
run;
```

The REPEATED statement requests a GEE analysis. The SUBJECT=ID option identifies ID as the clustering variable, and the TYPE=EXCH option specifies an exchangeable correlation structure. The TYPE3 option in the MODEL statement requests Type III statistics for each effect in the model. The CONTRAST statement requests a test that compares the first and second levels of the SMOKE effect.

 GEE Model Information Correlation Structure Exchangeable Subject Effect id (25 levels) Number of Clusters 25 Correlation Matrix Dimension 4 Maximum Cluster Size 4 Minimum Cluster Size 4
 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Parameter Estimate Standard Error 95% Confidence Limits Z Pr > |Z| Lower Upper Intercept 4.2569 1.9577 0.4199 8.0938 2.17 0.0297 city greenhil 0.0287 0.5365 -1.0227 1.0802 0.05 0.9573 city steelcit 0.0000 0.0000 0.0000 0.0000 . . age -0.3330 0.1937 -0.7126 0.0467 -1.72 0.0856 smoke 0 -1.6781 0.6123 -2.8783 -0.4780 -2.74 0.0061 smoke 1 -1.7418 0.6588 -3.0330 -0.4507 -2.64 0.0082 smoke 2 0.0000 0.0000 0.0000 0.0000 . .