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

Exchangeablecorrelation structure

Variable Coefficient

Empirical Std

Err

Wald

p-

value

INTERCEPT 1.1583 2.3950 0.6286
ASPIRIN 0.3934 3.2027 0.9022
AGE 0.0104 0.0118 0.3777
GENDER 0.9377 0.3216 0.0035
WEIGHT 0.0061 0.0088 0.4939
HEIGHT 0.0116 0.0151 0.4421
ASPIRIN
AGE

0.0069 0.0185 0.7087

ASPIRIN
GENDER

0.9836 0.5848 0.0926

ASPIRIN
WEIGHT

0.0147 0.0137 0.2848

ASPIRIN
HEIGHT

0.0107 0.0218 0.6225

Odds ratio(ASPIRIN¼1 vs. ASPIRIN¼0)

odds¼expðb 0 þb 1 ASPIRINþb 2 AGE

þb 3 GENDERþb 4 WEIGHT

þb 5 HEIGHTþb 6 ASPIRINAGE

þb 7 ASPIRINGENDER

þb 8 ASPIRINWEIGHT

þb 9 ASPIRINHEIGHTÞ

Separate OR for each pattern of
covariates:

OR¼expðb 1 þb 6 AGEþb 7 GENDER

þb 8 WEIGHTþb 9 HEIGHTÞ

AGE¼60, GENDER¼1, WEIGHT

¼75 kg, HEIGHT¼170 cm

ORdASPIRIN¼ 1 vs:ASPIRIN¼ 0 Þ

¼exp½ 0 : 3934 þð 0 : 0069 Þð 60 Þ

þð 0 : 9836 Þð 1 Þþð 0 : 0147 Þð 75 Þ

þð 0 : 0107 Þð 170 ފ¼ 0 : 32

Chunk test

H 0 :b 6 ¼b 7 ¼b 8 ¼b 9 ¼ 0

Likelihood ratio test

for GEE models

Notice that the model contains a term for ASPI-

RIN, terms for the four covariates, and four

product terms containing ASPIRIN. An

exchangeable correlation structure is speci-

fied. The parameter estimates are shown on

the left.

The output can be used to estimate the odds

ratio for ASPIRIN¼1 vs. ASPIRIN¼0. If

interaction is assumed, then adifferentodds

ratio estimate is allowed for each pattern of

covariates where the covariates interacting

with ASPIRIN change values.

The odds ratio estimates can be obtained by

separately inserting the values ASPIRIN¼ 1

and ASPIRIN¼0 in the expression of the

odds shown on the left and then dividing one

odds by the other.

This yields the expression for theodds ratio,

also shown on the left.

The odds ratio (comparing ASPIRIN status) for

a 60-year-old male who weighs 75 kg and is

170 cm tall can be estimated using the output

as 0.32.

A chunk test can be performed to determine if

the four product terms can be dropped from

the model. The null hypothesis is that the betas

for the interaction terms are all equal to zero.

Recall for a standard logistic regression that

the likelihood ratio test can be used to simulta-

neously test the statistical significance of sev-

eral parameters. For GEE models, however, a

likelihood is never formulated, which means

that the likelihood ratio test cannot be used.

552 15. GEE Examples