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

P

X



¼

1

1 þe

aþ~biXi



ðÞ 1 odds:

P

X 1



1 P

X 1



ðÞ 0 odds:

P

X 0



1 P

X 0



odds forX 1

odds forX 0

¼

P

X 1



1 P

X 1



P

X 0



1 P

X 0



¼RORX 1 ;X 0

P(X 1 )

P(X) =

1 – P(X 1 )

P(X 1 )

1 – P(X 1 )

P(X 0 )

1 – P(X 0 )

P(X 0 )

1 – P(X 0 )

ROR =

(1)

(0)

1

1 + e–(a+^ biXi)

= e(a+^ biX^1 i)

= e(a+^ biX 0 i)

RORX 1 ;X 0 ¼

odds forX 1

odds forX 0

¼

e

aþ~biX 1 i



e

aþ~biX 0 i



Algebraic theory :

ea

= ea–b

eb

a = a+ biX 1 i, b = a+ biX 0 i

Given a logistic model of the general form P(X),

we can write theoddsfor group1asP(X 1 )

divided by 1P(X 1 )

and theoddsforgroup0asP(X 0 ) divided by

1 P(X 0 ).

To get an odds ratio, we then divide the first

odds by the second odds. The result is an

expression for the odds ratio written in terms

of the two risks P(X 1 ) and P(X 0 ), that is, P(X 1 )

over 1P(X 1 ) divided by P(X 0 ) over 1P(X 0 ).

We denote this ratio asROR, forrisk odds ratio,

as the probabilities in the odds ratio are all

defined as risks. However, we still do not have

a convenient formula.

Now, to obtain a convenient computational

formula, we can substitute the mathematical

expression 1 over 1 plus e to minus the quantity

(aþ~biXi) for P(X) into therisk odds ratio

formula above.

For group 1, theoddsP(X 1 ) over 1P(X 1 )

reduces algebraically to e to the linear suma

plus the sum ofbitimesX 1 i, whereX 1 idenotes

the value of the variableXifor group 1.

Similarly, the odds for group 0 reduces to e to the

linear sumaplus the sum ofbitimesX 0 i,where

X 0 idenotes the value of variableXifor group 0.

To obtain theROR, we now substitute in the

numerator and denominator the exponential

quantities just derived to obtain e to the

group 1 linear sum divided by e to the group

0 linear sum.

The above expression is of the form e to thea

divided by e to theb, whereaandbare linear

sums for groups 1 and 0, respectively. From

algebraic theory, it then follows that this ratio

of two exponentials is equivalent to e to the

difference in exponents, or e to theaminusb.

Presentation: VIII. Derivation of OR Formula 23