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

EXAMPLE (continued)

Options A and B(continued)

Confounding:

DoesOR meaningfully change^

when AGE and/or GENDER are

dropped?

GS model: no-interaction model A

above

ORGSðAÞ¼exp½b 1 ðE 1 *E 1 Þ

þb 2 ðE 2 *E 2 ފ;

whereX*¼(E 1 *,E 2 *) andX¼(E 1 ,

E 2 ) are two specifications of the twoEs

Our choices forE 1 andE 2 on two

subjects:

X*¼ðE 1 *¼ 1 ;

yes

E 2 *¼ 1 Þ

yes

vs:

X¼ðE 1 ¼ 0 ;

no

E 2 ¼no 0 Þ

ORGS(A)

¼exp½b 1 ð 1  0 Þþb 2 ð 1  0 ފ

¼exp½b 1 þb 2 Š

Table of ORs (check confounding)

Vs in model

OR ORI

AGE,GEN

ORII ORIII ORIV

AGE GEN Neither

Model #

I. Logit PI(X)¼aþb 1 E 1

þb 2 E 2 þg 1 V 1

þg 2 V 2

II. Logit PII(X)¼aþb 1 E 1

þb 2 E 2 þg 1 V 1

III. Logit PIII(X)¼aþb 1 E 1

þb 2 E 2 þg 2 V 2

IV. Logit PIV(X)¼aþb 1 E 1

þb 2 E 2

OR formula (E 1 *¼1, E 2 *¼1) vs.

(E 1 ¼0,E 2 ¼0) for all four models:

OR = exp [b 1 + b 2 ]

To assess confounding, we need to determine

whether the estimated OR meaningfully

changes (e.g., by more than 10%) when either

AGE or GENDER or both are dropped from the

model. Here, the gold standard (GS) model is

the no-interaction modelAjust shown.

The formula for the odds ratio for the GS

model is shown at the left, where (E 1 *,E 2 *)

and (E 1 ,E 2 ) denote two specifications of the

two exposures PREVHOSP (i.e.,E 1 ) and PAMU

(i.e.,E 2 ).

There are several ways to specifyX*andXfor

PREVHOSP and PAMU. Here, for convenience

and simplicity, we will choose to compare a

subjectX*who is positive (i.e., yes) for both

Es with a subjectXwho is negative (i.e., no) for

bothEs.

Based on the above choices, the OR formula

for our GS reduced model A simplifies, as

shown here.

To assess confounding, we must now deter-

mine whether estimates of our simplified

ORGS(A)meaningfully change when we drop

AGE and/or GENDER. This requires us to con-

sider a table of ORs, as shown at the left.

To complete the above table, we need to fit the

four models shown at the left. The first model,

which we have already described, is the GS(A)

model containing PREVHOSP, PAMU, AGE,

and GENDER. The other three models exclude

GENDER, AGE, or both from the model.

Since all four models involve the same twoEvari-

ables, the general formula for the OR that com-

pares a subject who is exposed on both Es

(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

the GS model.

250 8. Additional Modeling Strategy Issues