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

Epidemiologic framework

X 1 ,X 2 ,...,Xkmeasured atT 0

Time: T 0 T 1

X 1 , X 2 ,... , Xk D(0,1)

P(D¼1|X 1 ,X 2 ,...,Xk)

DEFINITION

Logistic model:

PðÞD¼ 1 jX 1 ;X 2 ;...;Xk

¼

1

1 þeðÞaþ~biXi

""

unknown parameters

NOTATION

P(D¼1|X 1 ,X 2 ,...,Xk)

¼P(X)

Model formula:

P

X



¼

1

1 þeðaþ~biXiÞ

The logistic model considers the following gen-

eralepidemiologic study framework: We have

observed independent variablesX 1 ,X 2 , and so

on up toXkon a group of subjects, for whom we

have also determined disease status, as either 1

if “with disease” or 0 if “without disease”.

We wish to use this information to describe the

probability that the disease will develop during

a defined study period, sayT 0 toT 1 , in a disease-

free individual with independent variable values

X 1 ,X 2 ,uptoXk, which are measured atT 0.

The probability being modeled can be denoted

by the conditional probability statement

P(D¼1|X 1 ,X 2 ,...,Xk).

The model is defined aslogisticif the expres-

sion for the probability of developing the dis-

ease, given theXs, is 1 over 1 plus e to minus

the quantityaplus the sum fromiequals 1 tok

ofbitimesXi.

The terms aand bi in this model represent

unknown parametersthat we need to estimate

based on data obtained on theXs and onD

(disease outcome) for a group of subjects.

Thus, if we knew the parametersaand thebi

and we had determined the values of X 1

throughXkfor a particular disease-free individ-

ual, we could use this formula to plug in these

values and obtain the probability that this indi-

vidual would develop the disease over some

defined follow-up time interval.

For notational convenience, we will denote the

probability statement P(D¼1|X 1 ,X 2 ,...,Xk)as

simply P(X) where theboldXis a shortcut

notation for the collection of variables X 1

throughXk.

Thus, the logistic model may be written as P(X)

equals 1 over 1 plus e to minus the quantitya

plus the sumbiXi.

8 1. Introduction to Logistic Regression