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

z¼index of combined risk factors

1

threshold

– ∞ 0 + ∞

1/2

S-shape

z

III. The Logistic Model

z¼aþb 1 X 1 þb 2 X 2 þ...þbkXk

z = a + b 1 X 1 + b 2 X 2 + ... + bkXk

f z =

1 + e–z

1

=

1 + e–(a+

1

biXi)

The S-shape of the logistic function appeals to

epidemiologists if the variablezis viewed as

representing an index that combines contribu-

tions of several risk factors, andf(z) represents

the risk for a given value ofz.

Then, the S-shape off(z) indicates that the effect

ofzon an individual’s risk is minimal for lowzs

until somethresholdis reached. The risk then

rises rapidly over a certain range of intermedi-

atezvalues and then remains extremely high

around 1 oncezgets large enough.

Thisthresholdidea is thought by epidemiolo-

gists to apply to a variety of disease conditions.

In other words, an S-shaped model is consid-

ered to be widely applicable for considering the

multivariable nature of an epidemiologic research

question.

Now, let us go from the logisticfunctionto the

model, which is our primary focus.

To obtain the logisticmodelfrom the logistic

function, we writezas the linear sumaplusb 1

timesX 1 plusb 2 timesX 2 , and so on tobktimes

Xk, where theXs are independent variables of

interest andaand thebiare constant terms

representing unknown parameters.

In essence, then,z is an index that combines the Xs.

We now substitute the linear sum expression

forzin the right-hand side of the formula for

f(z) to get the expressionf(z) equals 1 over 1

plus e to minus the quantityaplus the sum of

biXiforiranging from 1 tok. Actually, to view

this expression as a mathematical model, we

must place it in an epidemiologic context.

SUMMARY So, the logisticmodelispopularbecause the
logisticfunction, on which the model is based,
provides the following:

 Estimates that must lie in the range

between zero and one

 An appealing S-shaped description of the

combined effect of several risk factors on

the risk for a disease.

Presentation: III. The Logistic Model 7