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

r^j;jþ 1 ¼ 0 : 5254

r^j;jþ 2 ¼ð 0 : 5254 Þ^2 ¼ 0 : 2760

r^j;jþ 3 ¼ð 0 : 5254 Þ^3 ¼ 0 : 1450

Model 2:Exchangeable
correlation structure

Variable Coefficient

Empirical

Std Err

Wald

p-value

INTERCEPT 1.3987 1.2063 0.2463
BIRTHWGT 0.0005 0.0003 0.1237
GENDER 0.0262 0.5547 0.9623
DIARRHEA 0.6485 0.7553 0.3906

^b 3 for DIARRHEA¼0.6485

(vs. 0.2214 with Model 1)

Exchangeable working correlation
matrix

COL1 COL2 ... COL9

ROW1 1.0000 0.4381 ... 0.4381
ROW2 0.4381 1.0000 ... 0.4381
ROW3 0.4381 0.4381 ... 0.4381
ROW4 0.4381 0.4381 ... 0.4381
ROW5 0.4381 0.4381 ... 0.4381
ROW6 0.4381 0.4381 ... 0.4381
ROW7 0.4381 0.4381 ... 0.4381
ROW8 0.4381 0.4381 ... 0.4381
ROW9 0.4381 0.4381 ... 0.4381

Only one^r:r^¼ 0 : 4381

Model 3:Fixed correlation
structure

Variable Coefficient

Empirical

Std Err

Wald

p-value

INTERCEPT 1.3618 1.2009 0.2568
BIRTHWGT 0.0005 0.0003 0.1110
GENDER 0.0304 0.5457 0.9556
DIARRHEA 0.2562 0.8210 0.7550

Note that the correlation between measure-

ments 2 months apart (0.2760) is the square

of measurements 1 month apart (0.5254),

whereas the correlation between measure-

ments 3 months apart (0.1450) is the cube of

measurements 1 month apart. This is the key

property of the AR1 correlation structure.

Next we present the parameter estimates and

working correlation matrix for a GEE model

using the exchangeable correlation structure

(Model 2). The coefficient estimate for DIAR-

RHEA is 0.6485. This compares with the

parameter estimate of 0.2214 for the same

coefficient using the AR1 correlation structure

in Model 1.

There is only one correlation to estimate with

an exchangeable correlation structure. For this

model, this estimate is 0.4381. The interpreta-

tion is that the correlation between any two

outcome measures from the same infant is

estimated at 0.4381 regardless of which

months the measurements are taken.

Next we examine output from a model with a

fixed, or user-defined, correlation structure

(Model 3). The coefficient estimate and stan-

dard error for DIARRHEA are 0.2562 and

0.8210, respectively. These are similar to the

estimates in the AR1 model, which were

0.2214 and 0.8558, respectively.

546 15. GEE Examples