Basic Statistics

RELATIVE RISK AND ODDS RATIO 147

Table 11.5 Doubling the Control Group Size in (A) and (B)

A. Exposure to Risk Factor for an Unmatched Case/Control Study

Exposure

Outcome Yes No Total

Case 70=a 30=b 100

Control 55=c 45=d 100

Total 125 75 200 = n

B. Exposure to Risk Factor When Controls Doubled

Exuosure

Outcome Yes No Total

Case 70=a 30=b 100

Control 110=c 90=d 200

Total 180 120 300 = n

The relative risk of low vital capacity for smokers is positive and 3.30 times that for
nonsmokers. The relative risk measures how many times as likely the disease occurs
in the exposed group as in the unexposed group.
For prospective studies using a single sample, the resulting data would appear
similar to that in Table 1 1.1. The difference is that the disease outcome is measured
at a later time and we would probably start the study with disease-free participants.
If in a prospective study, we took a sample of smokers and a sample of nonsmokers,
the relative risk would be computed in the same way. The relative risk can be used
for both of these types of prospective studies.
But there are difficulties with relative risk in casekontrol studies. The following
example illustrates the problem. Table 1 1.5A presents hypothetical data from an
unpaired casekontrol study. Table 1 1.5B presents the same results except that we have
doubled the size of the control group. Note that all the numbers in the control row have
simply been multiplied by 2. But if we compute the relative risk from Table 1 lSA,
we get RR = (70/125)/(30/75) = 1.40 and RR = (70/180)/(30/120) = 1.56
from Table 11 SB. This unfortunate result is due to the way the sample is taken. For
a further explanation of this problem, see Schlesselman [1982].
Next, we discuss the odds ratio, a measure of the strength of a relationship that
can be used for all the types of studies discussed here for two-way tables with two
rows and two columns.

11.2.2 Odds Ratios

In Table 11.1 the exposure was smoking and the disease outcome was low vital
capacity. From Table 11.1 we can estimate from our sample the odds of getting