Introductory Biostatistics

relative risk¼

P 1

P 1 þP 3

o

P 2

P 2 þP 4

¼

P 1 ðP 2 þP 4 Þ

P 2 ðP 1 þP 3 Þ

since in many (although not all) situations, the proportions of subjects classified
as disease positive will be small. That is,P 1 is small in comparison withP 3 , and
P 2 will be small in comparison withP 4. In such a case, the relative risk is
almost equal to

y¼

P 1 P 4

P 2 P 3

¼

P 1 =P 3

P 2 =P 4

the odds ratio of being disease positive, or

¼

P 1 =P 2

P 3 =P 4

the odds ratio of being exposed. This justifies the use of an odds ratio to deter-
mine di¤erences, if any, in the exposure to a suspected risk factor.
As a technique to control confounding factors in a designed study, individ-
ual cases are matched, often one to one, to a set of controls chosen to have
similar values for the important confounding variables. The simplest example
of pair-matched data occurs with a single binary exposure (e.g., smoking versus
nonsmoking). The data for outcomes can be represented by a 22 table
(Table 3.13) whereðþ;
Þdenotes (exposed, unexposed).
For example,n 10 denotes the number of pairs where the case is exposed, but
the matched control is unexposed. The most suitable statistical model for
making inferences about the odds ratioyis to use the conditional probability of
the number of exposed cases among the discordant pairs. Givenn¼n 10 þn 01
being fixed, it can be seen thatn 10 hasBðn;pÞ, where

p¼

y

1 þy

TABLE 3.13

Case

Control þ


þ n 11 n 01


 n 10 n 00

BRIEF NOTES ON THE FUNDAMENTALS 139