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

Independent variables:
X 1 ,X 2 ,...,Xk

Xs may beEs,Cs, or combinations

The Multivariable Problem

X 1 , X 2 ,... , Xk D

The analysis:
mathematical model

Logistic model:
dichotomousD

Logistic is most popular

II. Why Is Logistic
Regression Popular?

1

Logistic

function:

• ∞ 0 + ∞

1/2
f(z) =1 + e–z

1

z

More generally, the independent variables can

be denoted asX 1 ,X 2 , and so on up toXk, where

kis the number of variables being considered.

We have aflexiblechoice for theXs, which can

represent any collection of exposure variables,

control variables, or even combinations of such

variables of interest.

For example, we may have the following:

X 1 equal to an exposure variableE

X 2 andX 3 equal to control variablesC 1 andC 2 ,

respectively

X 4 equal to the productEC 1

X 5 equal to the productC 1 C 2

X 6 equal toE^2

Whenever we wish to relate a set ofXstoa

dependent variable, likeD, we are considering

amultivariable problem. In the analysis of such

a problem, some kind ofmathematical modelis

typically used to deal with the complex inter-

relationships among many variables.

Logisticregression is a mathematical modeling

approach that can be used to describe the rela-

tionship of severalXstoadichotomousdepen-

dent variable, such asD.

Other modeling approaches are possible also,

but logistic regression is by far the mostpopu-

larmodeling procedure used to analyze epide-

miologic data when the illness measure is

dichotomous. We will show why this is true.

To explain the popularity of logistic regression,

we show here the logistic function, which

describes the mathematical form on which

thelogistic modelis based. This function, called

f(z), is given by 1 over 1 plus e to the minusz.

We have plotted the values of this function asz

varies from1to + 1.

EXAMPLE

X 1 ¼EX 4 ¼EC 1

X 2 ¼C 1 X 5 ¼C 1 C 2

X 2 ¼C 2 X 6 ¼E^2

Presentation: II. Why Is Logistic Regression Popular? 5