Introductory Biostatistics

11.5.1 Model

Consider a case–control design and suppose that each person in a large popu-
lation has been classified as exposed or not exposed to a certain factor, and as
having or not having some disease. The population may then be enumerated in
a22 table (Table 11.6), with entries being the proportions of the total pop-
ulation.
Using these proportions, the association (if any) between the factor and the
disease could be measured by the ratio of risks (or relative risk) of being disease
positive for those with or without the factor:

relative risk¼

P 1 =ðP 1 þP 3 Þ

P 2 =ðP 2 þP 4 Þ

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

OR¼

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 use of the odds ratio to measure
di¤erences, if any, in 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

TABLE 11.6

Disease

Factor þTotal

þ P 1 P 3 P 1 þP 3
 P 2 P 4 P 2 þP 4

Total P 1 þP 2 P 3 þP 4 1

406 ANALYSIS OF SURVIVAL DATA