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

General model
Several exposures
Confounders
Effect modifiers

logit PðÞ¼X aþb 1 E 1 þb 2 E 2 þ...

þbqEqþ~

p 1

i¼ 1

giVi

þE 1 ~

p 2

j¼ 1

d 1 jWi

þE 2 ~

p 2

j¼ 1

d 2 jWjþ...

þEq~

p 2

j¼ 1

dqjWj

Note that this expression tells us that once

we have fitted the model to the data to obtain

estimates of thebanddcoefficients, we must

specify values for the effect modifiers AGE and

SEX before we can get a numerical value for

the odds ratio. In other words, the odds ratio

will give a different numerical value depending

on which values we specify for the effect modi-

fiers AGE and SEX.

For instance, if we choose AGE equals 35 and

SEX equals 1 say, for females, then the esti-

mated odds ratio becomes the expression

shown here.

This odds ratio expression can alternatively

be written as e to the quantity minusb^ 1 plus

15 times^b 2 plus 40 timesb^ 3 minus 35 times

^d 11 plus 525 times ^d 21 plus 1,400 times ^d 31

minus^d 12 plus 15 times^d 22 plus 40 times^d 32.

This expression will give us a single numerical

value for 35-year-old females once the model is

fitted and estimated coefficients are obtained.

We have just worked through a specific exam-

ple of the odds ratio formula for a model involv-

ing several exposure variables and controlling

for both confounders and effect modifiers. To

obtain a general odds ratio formula for this

situation, we first need to write the model in

general form.

This expression is given by the logit of P(X)

equalsaplusb 1 timesE 1 plusb 2 timesE 2 , and

so on up tobqtimesEqplus the usual set ofV

terms of the formgiViplus the sum of addi-

tional terms, each having the form of an expo-

sure variable times the sum ofdtimesWterms.

The first of these interaction expressions is

given by E 1 times the sum of d 1 jtimes Wj,

whereE 1 is the first exposure variable,d 1 jis

an unknown coefficient, andWjis thejth effect

modifying variable. The last of these terms is

Eqtimes the sum ofdqjtimesWj, whereEqis

the last exposure variable,dqjis an unknown

coefficient, andWjis thejth effect modifying

variable.

EXAMPLE (continued)

Note. Specify AGE and SEX to get a

numerical value.

e.g., AGE¼35, SEX¼1:

ROR = exp[–b 1 + 15b 2 + 40b 3

+ 35(–d 11 + 15d 21 + 40d 31 )

+ 1(–d 12 + 15d 22 + 40d 32 )]

AGE

SEX

RORd ¼exp^b 1 þ 15 ^b 2 þ 40 ^b 3

 35 ^d 11 þ 525 ^d 21 þ 1400 ^d 31

^d 12 þ 15 ^d 22 þ 40 ^d 32

Presentation: VI. The Model and Odds Ratio for Several Exposure Variables 89