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

Note: above table not for simple
analysis.

B¼ 1 B¼ 0

A¼ (^1) R 11 R 10
A¼ 0 R 01 R 00
B= 1
A= 1
A= (^0) referent cell
B= 0
OR 11 ¼odds(1, 1)/odds(0, 0)
OR 10 ¼odds(1, 0)/odds(0, 0)
OR 01 ¼odds(0, 1)/odds(0, 0)
odds (A,B)¼RAB/(1RAB)
OR 11 ¼
R 11 =ðÞ 1 R 11
R 00 =ðÞ 1 R 00

¼

R 11 ðÞ 1 R 00

R 00 ðÞ 1 R 11

OR 10 ¼

R 10 =ðÞ 1 R 10

R 00 =ðÞ 1 R 00

¼

R 10 ðÞ 1 R 00

R 00 ðÞ 1 R 10

OR 01 ¼

R 01 =ðÞ 1 R 01

R 00 =ðÞ 1 R 00

¼

R 01 ðÞ 1 R 00

R 00 ðÞ 1 R 01

ORAB¼

RABðÞ 1 R 00

R 00 ðÞ 1 RAB

A¼0, 1; B¼0, 1

Note that the two-way table presented here

does not describe a simple analysis because

the row and column headings of the table

denote two independent variables rather than

one independent variable and one disease vari-

able. Moreover, the information provided

within the table is a collection of four risks

corresponding to different combinations of

both independent variables, rather than four

cell frequencies corresponding to different

exposure-disease combinations.

Within this framework, odds ratios can be

defined to compare the odds for any one cell

in the two-way table of risks with the odds for

any other cell. In particular, three odds ratios

of typical interest compare each of three of the

cells to areferent cell. The referent cell is usually

selected to be the combinationAequals 0 andB

equals 0. The three odds ratios are then defined

as OR 11 ,OR 10 , and OR 01 , where OR 11 equals

the odds for cell 11 divided by the odds for cell

00, OR 10 equals the odds for cell 10 divided by

the odds for cell 00, and OR 01 equals the odds

for cell 01 divided by the odds for cell 00.

As the odds for any cellA,Bis defined in terms

of risks asRABdivided by 1 minusRAB, we can

obtain the following expressions for the three

odds ratios: OR 11 equals the product of R 11

times 1 minusR 00 divided by the product of

R 00 times 1 minus R 11. The corresponding

expressions for OR 10 and OR 01 are similar,

where the subscript 11 in the numerator and

denominator of the 11 formula is replaced by

10 and 01, respectively.

In general, without specifying the value ofA

andB, we can write the odds ratio formulae

as ORABequals the product ofRABand 1 minus

R 00 divided by the product ofR 00 and 1RAB,

whereAtakes on the values 0 and 1 andBtakes

on the values 0 and 1.

50 2. Important Special Cases of the Logistic Model