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

EXAMPLE (continued)

Option C(continued)

Must specifyX*andX:

X*¼ðE 1 *¼ 1 ;

yes

E 2 *¼yes 1 Þ

vs:X¼ðE 1 ¼ 0 ;

no

E 2 ¼ 0

no

Þ

ORGSðBÞ¼exp½b 1 ð 1  0 Þþb 2 ð 1  0 Þ

þ^d*ð½ 1  1 Š½ 0  0 ŠÞŠ

¼exp½b 1 þb 2 þd*Š

Table of ORs (check confounding)

Vs in model

ORI*

AGE,GEN

ORII* ORIII* ORIV*

AGE GEN Neither

OR

Model choices:

I*.Logit PI*(X)¼aþb 1 E 1 þb 2 E 2

þg 1 V 1 þg 2 V 2

þd*E 1 E 2

II*. Logit PII*(X)¼aþb 1 E 1

þb 2 E 2

þg 1 V 1

þd*E 1 E 2

III*. Logit PIII*(X)¼aþb 1 E 1

þb 2 E 2

þg 2 V 2

þd*E 1 E 2

IV*. Logit PIV*(X)¼aþb 1 E 1

þb 2 E 2

þd*E 1 E 2

OR formula (E 1 *¼1,E 2 *¼1) vs.

(E 1 ¼0,E 2 ¼0) for all four

models:

OR¼exp½b 1 þb 2 þd*Š

However,^b 1 ;^b 2 , and^d*likely differ for

each model

As previously noted (forOptionA), there are

several ways to specifyX*andX. Here, again,

we will choose to compare a subjectX*who is

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

who is negative (i.e., no) for bothEs.

Based on the above choices, the OR formula

for our GS reduced model Bsimplifies, as

shown here.

To assess confounding, we must once again

(as withOptionA) consider a table ofdORs, as

shown at the left.

To complete the above table, we need to fit the

four models shown at the left. The first model

(I*), which we have already described, is the

Gold Standard (GS(B)) model containing

PREVHOSP, PAMU, AGE, GENDER, and

PRHPAM. The other three models exclude

GENDER, AGE, or both from the model.

Since all four models involve the same twoE

variables, the general formula for the OR that

compares a subject who is exposed on bothEs

(E 1 *¼1, E 2 *¼1) vs. a subject who is not

exposed on bothEs(E 1 ¼0,E 2 ¼0) has the

same algebraic form for each model, including

theGS(B) model.

However, since, the models do not all have the

same predictors, the estimates of the regres-

sion coefficients are likely to differ somewhat.

254 8. Additional Modeling Strategy Issues