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

odds :

P(X) P

1 – P(X)^1 – P

vs.

describes risk in

logistic model for

individual X

logit PðXÞ¼lne

PðXÞ

1 PðXÞ



¼log odds for individualX

¼aþ~biXi

For example, ifPequals 0.25, then 1P, the

probability of the opposite event, is 0.75 and

theoddsis 0.25 over 0.75, or one-third.

An oddsof one-third can be interpreted to

mean that the probability of the event occur-

ring is one-third the probability of the event not

occurring. Alternatively, we can state that the

oddsare 3to1 that the event will not happen.

The expression P(X) divided by 1P(X) has

essentially the same interpretation asPover

1 P, which ignoresX.

The main difference between the two formulae

is that the expression with theXis more spe-

cific. That is, the formula withXassumes that

the probabilities describe the risk for develop-

ing a disease, that this risk is determined by a

logistic model involving independent variables

summarized byX, and that we are interested in

the odds associated with a particular specifica-

tion ofX.

Thus, the logit form of the logistic model,

shown again here, gives an expression for the

log oddsof developing the disease for an indi-

vidual with a specific set ofXs.

And, mathematically, this expression equalsa

plus the sum of thebiXi.

As a simple example, consider what thelogit

becomes when all theXs are 0. To compute

this, we need to work with the mathematical

formula, which involves the unknown para-

meters and theXs.

If we plug in 0 for all theXs in the formula, we

find that the logit of P(X) reduces simply toa.

Because we have already seen that any logit

can be described in terms of anodds, we can

interpret this result to give some meaning to

the parametera.

One interpretation is thatagives thelog odds

for a person with zero values for allXs.

EXAMPLE

P= 0.25

odds¼

P

1 P¼

0 : 25

0 : 75 ¼

1

3

1

3

event occurs

event does not occur

3 to 1 event will not happen

EXAMPLE

all Xi = 0: logit P(X) =?

logit P(X) = a + biXi

logit P(X) ⇒ a

0

INTERPRETATION

(1) a¼log odds for individual with

allXi¼ 0

Presentation: VII. Logit Transformation 19