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

Logit

logit PðXÞ¼lne

PðÞX

1 PðXÞ



;

where

PðXÞ¼

1

1 þeðaþ~biXiÞ

(1) P(X)

(2) 1P(X)

(3)

PðXÞ

1 PðXÞ

(4) lne

PðXÞ

1 PðXÞ



logit P(X) = lne

P(X)

P(X) =

1 – P(X)

=?

1

1 + e–(a+^ biXi)

To begin the description of the odds ratio in

logistic regression, we present an alternative

way to write the logistic model, called thelogit

formof the model. To get thelogitfrom the

logistic model, we make a transformation of

the model.

Thelogit transformation, denoted aslogitP(X),

is given by the natural log (i.e., to the base e) of

the quantity P(X) divided by one minus P(X),

where P(X) denotes the logistic model as previ-

ously defined.

This transformation allows us to compute a

number, calledlogitP(X), for an individual

with independent variables given byX.Wedo

so by:

(1) computing P(X) and

(2) 1 minus P(X) separately, then

(3) dividing one by the other, and finally

(4) taking the natural log of the ratio.

For example, if P(X) is 0.110, then

1 minus P(X) is 0.890,

the ratio of the two quantities is 0.123,

and the log of the ratio is2.096.

That is, thelogitof 0.110 is2.096.

Now we might ask,what general formula do we

get when we plug the logistic model form into the

logit function? What kind of interpretation can

we give to this formula? How does this relate to

an odds ratio?

Let us consider the formula for the logit func-

tion. We start with P(X), which is 1 over 1 plus

e to minus the quantityaplus the sum of the

biXi.

EXAMPLE

(1) P(X)¼0.110

(2) 1 P(X)¼0.890

(3) PðXÞ

1 PðXÞ

¼^0 :^110

0 : 890

¼ 0 : 123

(4) lne 1 PðPXðXÞÞ

hi

¼lnð 0 : 123 Þ¼ 2 : 096

i.e., logit (0.110)¼2.096

Presentation: VII. Logit Transformation 17