Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Model 1

logit PðD¼ 1 jXÞ¼b 0 þb 1 RX

ni¼ 2 :½AR 1 ;exchangeable;

or unstructuredŠ

)same 2  2 Ci

Ci¼

1 r

r 1

"

Exchangeablecorrelation structure

Variable Coefficient

Empirical

Std Err

Waldp-

value

INTERCEPT 0.2007 0.3178 0.5278
RX 0.3008 0.3868 0.4368
Scale 1.0127 

ORd¼expð 0 : 3008 Þ¼ 1 : 35

95 %CI¼ð 0 : 63 ; 2 : 88 Þ

ExchangeableCi
COL1 COL2

ROW1 1.0000 0.2634
ROW2 0.2634 1.0000

SLR (naive) model

Variable Coefficient

Model-

based Std Err

Wald

p-value

INTERCEPT 0.2007 0.3178 0.5278
RX 0.3008 0.4486 0.5826
Scale 1.0000 

ORd¼expð 0 : 3008 Þ¼ 1 : 35

95 %CI¼ð 0 : 56 ; 3 : 25 Þ

For this analysis, RX is the only independent

variable considered. The model is stated as

shown on the left. With exactly two observa-

tions per subject, the only correlation to con-

sider is the correlation between the two

responses for the same subject. Thus, there is

only one estimated correlation parameter,

which is the same for each cluster. As a result,

using an AR1, exchangeable, or unstructured

correlation structure yields the same 2 2

working correlation matrix (Ci).

The output for a GEE model with an exchange-

able correlation structure is presented on the

left.

The odds ratio estimate for the effect of

treatment for relieving heartburn is

exp(0.3008)¼1.35 with the 95% confidence

interval of (0.63, 2.88). The working correlation

matrix shows that the correlation between

responses from the same subject is estimated

at 0.2634.

A standard logistic regression is presented

for comparison. The odds ratio estimate at

exp(0.3008)¼1.35 is exactly the same as was

obtained from the GEE model with the

exchangeable correlation structure; however,

the standard error is larger, yielding a larger

95% confidence interval of (0.56, 3.25).

Although an odds ratio of 1.35 suggests that

the active treatment provides greater relief for

heartburn, the null value of 1.00 is contained in

the 95% confidence intervals for both models.

556 15. GEE Examples