Basic Statistics

146 CATEGORICAL DATA: ANALYSIS OF TWO-WAY FREQUENCY TABLES

(one of cases and one of controls) and the sample sizes of the cases and controls are
not indicative of the numbers in the population. The number of cases and controls
is considered fixed and is not a variable to be analyzed. Casekontrol studies are
sometimes designed with individually matched cases and controls (see Schlesselman
[ 19821). When this is done, the results are often displayed as given in Table 1 1.4 with
n pairs of subjects.
Two descriptive measures are defined and their use discussed in Section 11.2. The
use of the chi-square test is discussed in Section 11.3.

11.2 RELATIVE RISK AND ODDS RATIO

In this section, relative risk and odds ratios are defined and their use is discussed.

Relative risk is somewhat easier to understand, but the odds ratio is used more in

biomedical studies. References are given for the odds ratio for readers who desire

more information on this topic.

11.2.1 Relative Risk

In Section 10.5 we examined the effect of two different treatments upon the outcome of
a medical study or the relation of an exposure or risk factor on the outcome of disease
by looking at the difference between two proportions. But a difference between two
proportions may not always have the same meaning to an investigator. With pl = .43
and p2 = .40, the difference, or p1 - p2 = .03, may appear to be quite unimportant.
But with pl = .04 and pz = .01, the same difference may seem more striking. The
ratio of the proportions in the first instance it is .43/.40 = 1.075, and in the second
instance it is .04/.01 = 4. This ratio is called the relative risk (RR).
In the example shown in Table 11.3, the proportion of failures is pl = c/nl =
10/105 in the experimental group and p2 = d/n2 = 20/100 in the control group.
The relative risk of failure for the experimental and control group is

= ,476

RR=- 10/105 - - - .0952

20/1oo .20

When the numerical value of the relative risk is < 1, the relative risk is said to be
negative, and when the relative risk is > 1 the relative risk is said to be positive. In
this case we would say that the relative risk shows that the experimental treatment
may have a negative effect on the failure rate. A relative risk of 1 indicates that the
treatment may have no effect on the outcome.
When comparing the relative risk for an observational study, we compare the risk
of disease for the exposed group to the risk of disease for the nonexposed group. For
example, in the vital capacity survey (see Table 11.lA) we would compute

= 3.30

a/(a+c) - - 11/30 - .3667

RR = b/(b+ d) 10/90 .1111