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

EXAMPLE (continued)

Option C(continued)

OR formula for Models I and III:

OR¼exp½b 1 ðE 1 *E 1 Þþb 2 ðE 2 *E 2 Þ

þd*ðE 1 *E 2 *E 1 E 2 ފ;

whereX¼(E 1 ,E 2 *) andX¼(E 1 ,E 2 )
are two specifications of the twoEs

Precision)computing CIs for the
OR for Models I
and III

CI depends on how we specifyX*
andX:
Our focus again:

X*¼ð 1 ; 1 Þvs:X¼ð 0 ; 0 Þ

+

OR¼exp½b 1 þb 2 þd*Š

Table of ORs and CIs for Models I
and III
OR 95% CI for OR
Model I (GS(B))
Model III (w/o AGE)

CI width
Model
I*

50.887110.0175¼40.8696

Model
III*

44.82509.4174¼ 35.4076

Better model: Model III
Logit PIIIðXÞ¼aþðb 1 E 1 þb 2 E 2 Þ
þg 1 V 2 þd*E 1 E 2
(same OR but better precision than
GS)

Model III* at this point contains

E 1 , E 2 , V2, and E 1 E2,

haven’t yet

addressed

Since both Models I* and III* include the inter-

action termE 1 E 2 , the OR formula has the same

structure for both models (shown again at the

left).

To evaluate precision for each odds ratio, we

must therefore compute (say, 95%) confidence

intervals (CIs) for the OR for each model.

The CI limits for each OR will depend on how

we specifyX*andX. As we did for confound-

ing, we again focus on comparing a subjectX*

who is positive (i.e., yes) for bothEs with a

subjectXwho is negative (i.e., no) for both

Es. The OR formula simplifies as shown at

the left.

To assess precision, therefore, we must now

consider a table that gives the (95%) CI for

the OR for each model, and then decide

whether or not precision is gained when AGE

is dropped from the GS model. The resulting

table is shown at the left.

From the above results, we can see that,

although both models give wide (i.e., impre-

cise) confidence intervals, Model III* has a

tighter confidence interval than Model I*.

Therefore, we suggest that Model III* be cho-

sen as the “better” model, since it gives the

“same” (within 10%) OR estimate and provides

more precision.

At this point, using model III*, we have decided

to dropV 1 ¼AGE from our initial model. Nev-

ertheless, we have not yet addressed theEsin

the model.

Presentation: II. Modeling Strategy for Several Exposure Variables 257