Computation and Measurement of Risk - Mathematics and Statistics for the Environment - page 207

8.6.3 o dds r atio
An odds ratio (OR) is another measure of association that quantifies the relationship between an
exposure with two categories and health outcomes. Referring to the four cells in Table 8.11, the odds
ratio is calculated as

a
b

c
d

Odds ratio =

=

ad / bc

where
a = Number of persons exposed and with disease.
b = Number of persons exposed but without disease.
c = Number of persons unexposed but with disease.
d = Number of persons unexposed and without disease.
a + c = Total number of persons with disease (case-patients).
b + d = Total number of persons without disease (controls).

The odds ratio is sometimes called the cross-product ratio because the numerator is based on mul-
tiplying the value in cell a times the value in cell d , whereas the denominator is the product of cell
b and cell c . A line from cell a to cell d (for the numerator) and another from cell b to cell c (for the
denominator) creates an X, or cross, on the two-by-two table.

■ EXAMPLE 8.23
Problem: Use the data in Table 8.11 to calculate the risk and odds ratios.

Solution:

1. Risk ratio

5.0/1.0 = 5.0

2. Odds ratio

(100 × 7920)/(1900 × 80) = 5.2

Notice that the odds ratio of 5.2 is close to the risk ratio of 5.0. That is one of the attractive features
of the odds ratio—when the health outcome is uncommon, the odds ratio provides a reasonable
approximation of the risk ratio. Another attractive feature is that the odds ratio can be calculated
with data from a case-control study, whereas neither a risk ratio nor a rate ratio can be calculated.
The odds ratio is a case-control study , which is based on enrolling a group of persons with dis-
ease (“case-patients”) and a comparable group without disease (“controls”). The number of persons
in the control group is usually decided by the investigator. Often, the size of the population from

TABLE 8.11
Exposure and Disease in a Hypothetical Population
of 10,000 Persons

Disease No Disease Total Risk

Exposed a = 100 b = 1900 2000 5.0%

Not exposed c = 80 d = 7920 8000 1.0%

Total 180 9820 10,000 —