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

ROR¼eðaþ~biX^1 iÞðaþ~biX^0 iÞ

¼e½Šaaþ~biðX^1 iX^0 iÞ

¼e~biðX^1 iX^0 iÞ

 RORX 1 ;X 0 ¼e

~

k

i¼ 1

biðÞX 1 iX 0 i

ea+b =^ ea ×^ eb

e

~

k

i¼ 1

zi

¼ez^1 ez^2 ezk

NOTATION

zi^ =^ bi(X 1 i–X 0 i)

ezi

i=1

=

k

Õ

 RORX 1 ;X 0 ¼

Qk

i¼ 1

ebiðÞX^1 iX^0 i

Yk

i¼ 1

ebiðÞX^1 iX^0 i

¼eb^1 ðÞX^11 X^01 eb^2 ðÞX^12 X^02 ...e

bkðÞX 1 kX 0 k

We then find that theRORequals e to the

difference between the two linear sums.

In computing this difference, theas cancel out

and thebis can be factored for theith variable.

Thus, the expression forRORsimplifies to the

quantity e to the sumbitimes the difference

betweenX 1 iandX 0 i.

We thus have a general exponential formula for

the risk odds ratio from a logistic model com-

paring any two groups of individuals, as speci-

fied in terms of X 1 and X 0. Note that the

formula involves thebis but nota.

We can give an equivalent alternative to our

ROR formula by using the algebraic rule that

says that the exponential of a sum is the same

as the product of the exponentials of each term

in the sum. That is, e to theaplusbequals e to

theatimes e to theb.

More generally, e to the sum ofziequals the

product of e to theziover alli, where thezi’s

denote any set of values.

We can alternatively write this expression

using the product symbolP, whereP is a

mathematical notation which denotes the

product of a collection of terms.

Thus, using algebraic theory and lettingzicor-

respond to the termbitimes (X 1 iX 0 i),

we obtain thealternative formulaforRORas

the product fromi¼1tokof e to thebitimes

the difference (X 1 iX 0 i).

That is,Pof e to thebitimes (X 1 iX 0 i) equals

e to theb 1 times (X 11 X 01 ) multiplied by e to

theb 2 times (X 12 X 02 ) multiplied by addi-

tional terms, the final term

being e to thebktimes (X 1 kX 0 k).

24 1. Introduction to Logistic Regression