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

logit? OR

VIII. Derivation of OR Formula

OR¼

odds 1

odds 0

X¼(X 1 ,X 2 ,...,Xk)

(1) X 1 ¼(X 11 ,X 12 ,...,X 1 k)

(0) X 0 ¼(X 01 ,X 02 ,...,X 0 k)

NOTATION

ORX 1 ;X 0 ¼

odds forX 1

odds forX 0

Now, how can we use this information about

logits to obtain anodds ratio, rather than an

odds? After all, we are typically interested in

measures of association, like odds ratios, when

we carry out epidemiologic research.

Anyodds ratio, by definition, is a ratio of two

odds, written here asodds 1 divided byodds 0 ,in

which the subscripts indicate two individuals

or two groups of individuals being compared.

Now we give an example of an odds ratio in

which we compare two groups, called group 1

and group 0. Using our CHD example involving

independent variables CAT, AGE, and ECG,

group 1 might denote persons with CAT¼1,

AGE¼40, and ECG¼0, whereas group

0 might denote persons with CAT¼0, AGE

¼40, and ECG¼0.

More generally, when we describe an odds

ratio, the two groups being compared can be

defined in terms of the boldXsymbol, which

denotes a general collection of Xvariables,

from 1 tok.

LetX 1 denote the collection ofXs that specify

group 1 and letX 0 denote the collection ofXs

that specify group 0.

In our example, then,k, the number of vari-

ables, equals 3, and

Xis the collection of variables CAT, AGE, and

ECG,

X 1 corresponds to CAT¼1, AGE¼40, and

ECG¼0, whereas

X 0 corresponds to CAT¼0, AGE¼40, and

ECG¼0.

Notationally, to distinguish the two groupsX 1

andX 0 in anodds ratio, we can write ORX 1 ,X 0

equals theodds forX 1 divided by theodds

forX 0.

We will now apply the logistic model to this

expression to obtain a general odds ratio for-

mula involving the logistic model parameters.

EXAMPLE

X¼(CAT, AGE, ECG)

(1)X 1 ¼(CAT¼1,AGE¼40,ECG¼0)

(0)X 0 ¼(CAT¼0,AGE¼40,ECG¼0)

EXAMPLE

(1) CAT¼1, AGE = 40, ECG = 0

(0) CAT¼0, AGE = 40, ECG = 0

22 1. Introduction to Logistic Regression