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

logit P(X)¼aþ~biXi

i¼L:

bL¼¶ln (odds)

when¶XL¼1, otherXs fixed

The first expression below this model shows

that when CAT¼1, AGE¼40, and ECG¼0,

this logit reduces toaþb 1 þ 40 b 2.

The second expression shows that when

CAT¼0, but AGE and ECG remain fixed at

40 and 0, respectively, the logit reduces to

aþ 40 b 2.

If we subtract thelogit for CAT¼0 from the

logit for CAT¼1, after a little arithmetic, we

find that the difference isb 1 , the coefficient of

the variable CAT.

Thus, letting the symbol¶denote change, we

see thatb 1 represents the change in the logit

that would result from a unit change in CAT,

when the other variables are fixed.

An equivalent explanation is thatb 1 represents

thechange in the log odds that would result from

a one unit changein the variable CAT when the

other variables are fixed. These two statements

are equivalent because, by definition, alogitis

alog odds, so that the difference between two

logits is the same as the difference between two

log odds.

More generally, using the logit expression, if

we focus on any coefficient, saybL, fori¼L,

we can provide the following interpretation:

bLrepresents the change in the log odds that

would result from a one unit change in the

variableXL, when all otherXs are fixed.

EXAMPLE (continued)

(1) CAT¼1, AGE¼40, ECG¼ 0

logit P(X)¼aþb 11 þb 240

þb 30

=a+b 1 + 40 b 2

(2) CAT¼0, AGE¼40, ECG¼ 0

logit P(X)¼aþb 10 þb 240

þb 30

=a + 40 b 2

logit P 1 (X)logit P 0 (X)

¼(aþb 1 þ 40 b 2 )

(aþ 40 b 2 )

¼ b 1

NOTATION

= change

= log odds

when CAT = 1

AGE and ECG fixed

b 1 = logit

SUMMARY

a = background
log odds

bi = change in

log odds

logit P(X)

In summary, by looking closely at the expres-

sion for the logit function, we provide some

interpretation for the parametersaandbiin

terms of odds, actuallylog odds.

Presentation: VII. Logit Transformation 21