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

X. Special Case for (0, 1)

Variables

Adjusted OR = eb

whereb¼coefficient of (0, 1) variable

Xi(0, 1): adj.ROR¼ebi

controlling for otherXs

Our example illustrates an important special

case of the general odds ratio formula for logis-

tic regression that applies to (0, 1) variables.

That is, anadjusted odds ratiocan be obtained

by exponentiating the coefficient of a (0, 1)

variable in the model.

In our example, that variable is CAT, and the

other two variables, AGE and ECG, are the

ones for which we adjusted.

More generally, if the variable of interest isXi,

a (0, 1) variable, then e to thebi, wherebiis the

coefficient ofXi, gives an adjusted odds ratio

involving the effect ofXiadjusted or controlling

for the remainingXvariables in the model.

Suppose, for example, our focus had been on

ECG, also a (0, 1) variable, instead of on CAT in

a logistic model involving the same variables

CAT, AGE, and ECG.

Then e to theb 3 , whereb 3 is the coefficient of

ECG, would give the adjusted odds ratio for the

effect of ECG, controlling for CAT and AGE.

Note, however, that the example we have con-

sidered involves onlymain effect variables, like

CAT, AGE and ECG, and that the model does

not contain product terms like CATAGE or

AGEECG.

EXAMPLE

logit P(X) = a + b 1 CAT + b 2 AGE + b 3 ECG

adjusted

EXAMPLE

logit P(X) = a + b 1 CAT + b 2 AGE + b 3

adjusted

ECG

ECG (0, 1): adj.ROR¼eb^3

controlling for CAT and AGE

SUMMARY

Xiisð 0 ; 1 Þ:ROR¼ebi

General OR formula:

ROR¼e

~

k

i¼ 1

biðÞX 1 iX 0 i

Thus, we can obtain an adjusted odds ratio

for each (0, 1) variable in the logistic model by

exponentiating the coefficient corresponding

to that variable. This formula is much simpler

than the general formula for ROR described

earlier.

EXAMPLE

logit P(X) = a + b 1 CAT + b 2 AGE + b 3 ECG

main effect variables

Presentation: X. Special Case for (0, 1) Variables 27