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

1 PðXÞ¼ 1 

1

1 þeðÞaþ~biXi

¼

eðÞaþ~biXi

1 þeðÞaþ~biXi

P(X)

1 – P(X)

1

1+e–(a+^ biXi)

1+e–(a+^ biXi)

e–(a+^ biXi)

=

¼e

aþ~biXi



lne

PðXÞ

1 PðXÞ



¼lne eðÞaþ~biXi

hi

¼ aþ~biXi



|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

linear sum

Logit form: logit PðXÞ¼aþ~b
iXi;
where

PðXÞ¼

1

1 þeðÞaþ~biXi

logit P(X)? OR

PðXÞ

1 PðXÞ

¼odds for individualX

odds¼

P

1 P

Also, using some algebra, we can write 1P(X)

as:

e to minus the quantityaplus the sum ofbiXi

divided by one over 1 plus e to minusaplus the

sum of thebiXi.

If we divide P(X)by1P(X), then the denomi-

nators cancel out,

and we obtain e to the quantityaplus the sum

of thebiXi.

We then compute the natural log of the for-

mula just derived to obtain:

the linear sumaplus the sum ofbiXi.

Thus, thelogitof P(X) simplifies to thelinear

sumfound in the denominator of the formula

for P(X).

For the sake of convenience, many authors

describe the logistic model in its logit form

rather than in its original form as P(X). Thus,

when someone describes a model aslogitP(X)

equal to a linear sum, we should recognize that

a logistic model is being used.

Now, having defined and expressed the for-

mula for the logit form of the logistic model,

we ask,where does the odds ratio come in?Asa

preliminary step to answering this question,

we first look more closely at the definition of

the logit function. In particular, the quantity

P(X) divided by 1P(X), whose log value gives

thelogit, describes theoddsfor developing the

disease for a person with independent vari-

ables specified byX.

In its simplest form, anoddsis the ratio of the

probability that some event will occur over the

probability that the same event will not occur.

The formula for an odds is, therefore, of the

formPdivided by 1P, wherePdenotes the

probability of the event of interest.

18 1. Introduction to Logistic Regression