Introductory Biostatistics

both scales of measurement, discrete and continuous, even though in most
practical applications, the independent variable under investigation is often on
a continuous scale.

9.1.1 Simple Logistic Regression Model

The usual regression analysis goal, as seen in various sections of Chapter 8, is
to describe the mean of a dependent variableYas a function of a set of pre-
dictor variables. The logistic regression, however, deals with the case where the
basic random variableYof interest is a dichotomous variable taking the value
1 with probabilitypand the value 0 with probabilityð 1
pÞ. Such a random
variable is called apoint-binomialorBernouilli variable, and it has the simple
discrete probability distribution

PrðY¼yÞ¼pyð 1 
pÞ^1 
y y¼ 0 ; 1

Suppose that for theith individual of a sampleði¼ 1 ; 2 ;...;nÞ,Yiis a Ber-
nouilli variable with

PrðYi¼yiÞ¼piyið 1 
piÞ^1 
yi yi¼ 0 ; 1

The logistic regression analysis assumes that the relationship betweenpiand
the covariate valuexiof the same person is described by the logistic function

pi¼

1

1 þexp½
ðb 0 þb 1 xiފ

i¼ 1 ; 2 ;...;n;

The basic logistic function is given by

fðzÞ¼

1

1 þe
z

where, as in this simple regression model,

zi¼b 0 þb 1 xi

or, in the multiple regression model of subsequent sections,

zi¼b 0 þ

Xk

j¼ 1

bjxji

representing an index of combined risk factors. There are two important rea-
sons that make logistic regression popular:

SIMPLE REGRESSION ANALYSIS 317