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

Adjusted odds ratio for effect ofE
adjusted forCs:

RORE¼ 1 vs:E¼ 0 ¼exp bþ~

p 2

j¼ 1

djWj

!

(giterms not in formula)

II. Odds Ratio for Other
Codings of a
DichotomousE

Need to modify OR formula if cod-
ing ofEis not (0, 1)

Focus: üdichotomous
ordinal
interval

E¼

a if exposed

b if unexposed

(

RORE¼avs:E¼b

¼exp ðÞabbþðÞab ~

p 2

j¼ 1

djWj

"

For this model, the formula for theadjusted

odds ratiofor the effect of the exposure variable

on disease status adjusted for the potential

confounding and interaction effects of theCs

is shown here. This formula takes the form

e to the quantitybplus the sum of terms of

the formdjtimesWj. Note that the coefficients

giof the main effect variablesVido not appear

in the odds ratio formula.

Note that this odds ratio formula assumes that

the dichotomous variable E is coded as a(0, 1)

variable with E equal to 1 when exposed and E

equal to 0 when unexposed. If the coding scheme

is different – for example, (1, 1) or (2, 1), or if

Eis an ordinal or interval variable – then the

odds ratio formula needs to be modified.

We now consider other coding schemes for

dichotomous variables. Later, we also con-

sider coding schemes for ordinal and interval

variables.

SupposeEis coded to take on the valueaif

exposed andbif unexposed. Then, it follows

from the general odds ratio formula that ROR

equals e to the quantity (ab) timesbplus

(ab) times the sum of thedjtimes theWj.

For example, if a equals 1 andb equals 0,

then we are using the (0, 1) coding scheme

described earlier. It follows thata minus b

equals 1 minus 0, or 1, so that the ROR expres-

sion is e to thebplus the sum of thedjtimes the

Wj. We have previously given this expression

for (0, 1) coding.

In contrast, ifaequals 1 andbequals1, then

aminusbequals 1 minus1, which is 2, so the

odds ratio expression changes to e to the quan-

tity 2 timesbplus 2 times the sum of thedj

times theWj.

As a third example, supposeaequals 100 and

bequals 0, thenaminusbequals 100, so the

odds ratio expression changes to e to the quan-

tity 100 timesb plus 100 times the sum of

thedjtimes theWj.

EXAMPLES

(A) a = 1, b = 0 ⇒ (a – b) = (1 – 0) = 1

ROR = exp(1b +1Σdj Wj)

(B) a = 1, b = – 1 ⇒ (a – b) = (1 – [–1]) = 2

ROR = exp(2b +2 ΣdjWj)

(C) a = 100, b = 0 ⇒ (a – b) = (100 – 0) = 100

ROR = exp(100b +100ΣdjWj)

Presentation: II. Odds Ratio for Other Codings of a DichotomousE 77