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

We assume the sameWjfor each
exposure variable

e.g., AGE and SEX areWs for

eachE.

Odds ratio for severalEs:

E*¼ E* 1 ;E* 2 ;...;E*q



E**¼ E** 1 ;E** 2 ;...;E**q



General Odds Ratio Formula:

RORE*vs:E**¼exp



E* 1 E** 1

   

b 1

þE* 2 E** 2

   

b 2

þþE*qE**q



bq

þE* 1 E** 1

   

~

p 2

j¼ 1

d 1 jWj

þE* 2 E** 2

   

~

p 2

j¼ 1

d 2 jWj

þ

þ E*qE**q



~

p 2

j¼ 1

dqjWj

#

Note that this model assumes that the same

effect modifying variables are being consid-

ered for each exposure variable in the model,

as illustrated in our preceding example above

with AGE and SEX.

A more general model can be written that allows

for different effect modifiers corresponding

to different exposure variables, but for conve-

nience, we limit our discussion to a model with

thesamemodifiersforeachexposurevariable.

To obtain an odds ratio expression for the

above model involving several exposures, con-

founders, and interaction terms, we again

must identify two specifications of the expo-

sure variables to be compared. We have

referred to these specifications generally by

the bold termsE*andE**. GroupE*is speci-

fied by the variable valuesE 1 *,E 2 *, and so on up

toEq*; groupE**is specified by a different col-

lection of valuesE 1 **,E 2 **, and so on up toEq**.

The general odds ratio formula for comparing

two such specifications,E*vs. E**, is given

by the formula ROR equals e to the quantity

(E 1 *E 1 **) timesb 1 plus (E 2 *E 2 **) timesb 2 ,

and so on up to (Eq*Eq**) timesbqplus the

sum of terms of the form (E*E**) times the

sum ofdtimesW, where each of these latter

terms correspond to interactions involving a

different exposure variable.

90 3. Computing the Odds Ratio in Logistic Regression