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

RORE¼ 1 vs:E¼ 0 ¼

R 1

1 R 1

R 0

1 R 0

Substitute PðÞ¼X

1

1 þeðÞaþ~biXi
into ROR formula:

E¼ 1 :R 1 ¼

1

1 þeðÞaþ½b^1 ^1 Š

¼

1

1 þeðÞaþb^1

E¼ 0 :R 0 ¼

1

1 þeðÞaþ½b^1 ^0 Š

¼

1

1 þea

ROR¼

R 1

1 R 1

R 0

1 R 0

¼

1

1 þeðÞaþb^1

1

1 þea

algebra
= eb^1

General ROR formula used for
other special cases

We would like to use the above model for sim-

ple analysis to obtain an expression for the

odds ratio that compares exposed persons

with unexposed persons. Using the termsR 1

and R 0 , we can write this odds ratio asR 1

divided by 1 minusR 1 overR 0 divided by 1

minusR 0.

To compute the odds ratio in terms of the para-

meters of the logistic model, we substitute the

logistic model expression into the odds ratio

formula.

ForEequal to 1, we can writeR 1 by substitut-

ing the valueEequals 1 into the model formula

for P(X). We then obtain 1 over 1 plus e to

minus the quantityaplusb 1 times 1, or simply

1 over 1 plus e to minusaplusb 1.

ForEequal to zero, we writeR 0 by substituting

Eequal to 0 into the model formula, and we

obtain 1 over 1 plus e to minusa.

To obtain ROR then, we replaceR 1 with 1 over

1 plus e to minusaplusb 1 , and we replaceR 0

with 1 over 1 plus e to minus a. The ROR

formula then simplifies algebraically to e to

theb 1 , whereb 1 is the coefficient of the expo-

sure variable.

We could have obtained this expression for the

odds ratio using the general formula for the

ROR that we gave during our review. We will

use the general formula now. Also, for other

special cases of the logistic model, we will use

the general formula rather than derive an odds

ratio expression separately for each case.

Presentation: II. Special Case – Simple Analysis 47