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

VI. The Model and Odds
Ratio for Several
Exposure Variables
with Confounders
and Interaction

As a second example, suppose we compare a

smoker who has a PAL score of 25 and a sys-

tolic blood pressure of 160 to a smoker who has

a PAL score of 5 and a systolic blood pressure

of 200, again controlling for AGE and SEX.

The ROR is then computed as e to the quantity

(11) times b 1 plus (255) times b 2 plus

(160200) timesb 3 , which equals e to 0 times

b 1 plus 20 timesb 2 plus40 timesb 3 , which

reduces to e to the quantity 20b 2 minus 40b 3.

We now consider a final situation involving

several exposure variables, confounders (i.e.,

Vs), andinteraction variables(i.e.,Ws), where

theWs go into the model as product terms with

one of theEs.

As an example, we again consider the three

exposures SMK, PAL, and SBP and the two

control variables AGE and SEX. We add to

this list product terms involving each exposure

with each control variable. These product

terms are shown here.

The corresponding model is given by logit P(X)

equalsaplusb 1 times SMK plusb 2 times PAL

plusb 3 times SBP plus the sum ofVterms

involving AGE and SEX plus SMK times the

sum ofdtimesWterms, where theWs are AGE

and SEX, plus PAL times the sum of additional

dtimesWterms, plus SBP times the sum of

additionaldtimesWterms. Here theds are

coefficients of interaction terms involving one

of the three exposure variables – either SMK,

PAL, or SEX – and one of the two control

variables – either AGE or SEX.

To obtain an odds ratio expression for this

model, we again must identify two specifications

of the collection of exposure variables to be com-

pared. We have referred to these specifications

generally by the bold termsE*andE**.Inthe

above example,E*is defined by SMK*¼0, PAL*

¼25, and SBP*¼160, whereasE**is defined

by SMK**¼1, PAL**¼10, and SBP**¼120.

ANOTHER EXAMPLE

E*¼(SMK*¼1, PAL*¼25,

SBP*¼160)

E**¼(SMK**¼1, PAL**¼5,

SBP**¼200)

controlling for AGE and SEX

RORE*vs:E**¼exp½ð 1  1 Þb 1 þð 25  5 Þb 2

þð 160  200 Þb 3 Š

¼exp½ð 0 Þb 1 þð 20 Þb 2

þð 40 Þb 3 Š

¼expð 20 b 2  40 b 3 Þ

EXAMPLE: The Variables

E 1 ¼SMK,E 2 ¼PAL,E 3 ¼SBP

V 1 ¼AGE¼W 1 ,V 2 ¼SEX¼W 2

E 1 W 1 ¼SMKAGE,E 1 W 2 ¼SMKSEX

E 2 W 1 ¼PALAGE,E 2 W 2 ¼PALSEX

E 3 W 1 ¼SBPAGE,E 3 W 2 ¼SBPSEX

EXAMPLE: The Model

logit PðÞ¼X aþb 1 SMKþb 2 PAL

þb 3 SBPþg 1 AGEþg 2 SEX

þSMKðÞd 11 AGEþd 12 SEX

þPALðÞd 21 AGEþd 22 SEX

þSBPðÞd 31 AGEþd 32 SEX

EXAMPLE: The Odds Ratio

E*vs.E**

E*¼(SMK*¼0, PAL*¼25,

SBP*¼160)

E**¼(SMK**¼1, PAL**¼10, SBP**¼

120)

Presentation: VI. The Model and Odds Ratio for Several Exposure Variables 87