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

1

f(z)

– ∞ 0 + ∞

1/2

z

f(–∞) = (^1) + e^1 –(–∞)
= 1 + e^1 ∞
= 0
f(+∞) =
1 + e–(+∞)
1
= 1 + e^1 – ∞
= 1
Range: 0f(z) 1
0 probability1 (individual risk)
Shape:

1

S-shape

– ∞ 0 + ∞

f(z

)^ increasing

f(z)≈ 0

f(z)≈ 1

z

Notice, in the balloon on the left side of the

graph, that whenzis1, the logistic function

f(z) equals 0.

On the right side, whenzis + 1 ,thenf(z)equals1.

Thus, as the graph describes, therangeoff(z)is

between 0 and 1, regardless of the value ofz.

The fact that the logistic functionf(z)ranges

between 0 and1 is the primary reason the logis-

tic model is so popular. The model is designed

to describe a probability, which is always some

number between 0 and 1. In epidemiologic

terms, such a probability gives theriskof an

individual getting a disease.

Thelogistic model, therefore, is set up to ensure

that whatever estimate of risk we get, it will

always be some number between 0 and 1.

Thus, for the logistic model, we can never get

a risk estimate either above 1 or below 0. This is

not always true for other possible models,

which is why the logistic model is often the

first choice when a probability is to be esti-

mated.

Another reason why the logistic model is pop-

ular derives from theshapeof the logistic func-

tion. As shown in the graph, it we start at

z¼1and move to the right, then as z

increases, the value off(z) hovers close to zero

for a while, then starts to increase dramatically

toward 1, and finally levels off around 1 asz

increases toward + 1. The result is an elon-

gated, S-shaped picture.

6 1. Introduction to Logistic Regression