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

General:

RORX 1 ,X 0 ¼e
~

k

biðÞX 1 iX 0 i

i¼ 1

Simple analysis:

k¼1,X¼(X 1 ),bi¼b 1

group 1: X 1 ¼E¼ 1

group 0: X 0 ¼E¼ 0

X 1 ¼(X 11 )¼(1)

X 0 ¼(X 01 )¼(0)

RORX 1 ,X 0 ¼eb^1 ðÞX^11 X^01

¼eb^1 ðÞ^1 ^0

¼eb^1

RORdX
1 ,X 0 ¼e

^b 1

The general formula computes ROR as e to the

sum of eachbitimes the difference betweenX 1 i

andX 0 i, whereX 1 idenotes the value of theithX

variable for group 1 persons andX 0 idenotes

the value of theithXvariable for group 0 per-

sons. In a simple analysis, we have only oneX

and oneb; in other words,k, the number of

variables in the model, equals 1.

For a simple analysis model, group 1 corre-

sponds to exposed persons, for whom the

variableX 1 , in this caseE, equals 1. Group

0 corresponds to unexposed persons, for

whom the variableX 1 orEequals 0. Stated

another way, for group 1, the collection ofXs

denoted by theboldXcan be written asX 1 and

equals the collection of one valueX 11 , which

equals 1. For group 0, the collection of Xs

denoted by theboldXis written asX 0 and

equals the collection of one valueX 01 , which

equals 0.

Substituting the particular values of the oneX

variable into the general odds ratio formula

then gives e to theb 1 times the quantityX 11

minusX 01 , which becomes e to theb 1 times

1 minus 0, which reduces to e to theb 1.

We can estimate this odds ratio by fitting the

simple analysis model to a set of data. The

estimate of the parameter b 1 is typically

denoted as^b 1. The odds ratio estimate then

becomes e to theb^ 1.

SIMPLE ANALYSIS

SUMMARY

PðÞ¼X

1

1 þeðÞaþb^1 E

ROR¼eb^1

In summary, for the simple analysis model

involving a (0, 1) exposure variable, the logis-

tic model P(X) equals 1 over 1 plus e to minus

the quantityaplusb 1 timesE, and the odds

ratio that describes the effect of the exposure

variable is given by e to theb 1 , whereb 1 is the

coefficient of the exposure variable.

48 2. Important Special Cases of the Logistic Model