Biostatistics

12.7 RELATIVE RISK, ODDS RATIO, AND THE MANTEL–HAENSZEL STATISTIC 645
Odds Ratio and Relative Risk Section
Common Original Iterated Log Odds Relative
Parameter Odds Ratio Odds Ratio Odds Ratio Ratio Risk
Upper 95% C.L.
2.1350
2.2683 0.7585 2.1192
Estimate 1.1260 1.1207 1.1207 0.1140 1.1144
Lower
95% C.L.
0.5883
0.5606 0.5305 0.5896
FIGURE 12.7.1 NCSS output for the data in Example 12.7.1.
These data indicate that the risk of experiencing preterm labor when a woman
exercises heavily is 1.1 times as great as it is among women who do not
exercise at all.
We compute the 95 percent confidence interval for RR as follows. By
Equation 12.4.1, we compute from the data in Table 12.7.2:
X 2 455 22
¼ ½ð
Þð199Þ
ð216Þð18Þ
ð40Þð415Þð238Þð217Þ
Š2
¼ .1274
By Equation 12.7.2, the lower and upper confidence limits are, respectively,
pffiffiffiffiffiffiffiffi
1 1:96= :1274
1:1 ¼ :65 and 1:1 ffiffiffiffiffiffiffiffi
1þ1:96=
p
:1274
¼ 1:86. Since the interval includes
1, we conclude, at the .05 level of significance, that the population risk may
be 1. In other words, we conclude that, in the population, there may not be
an increased risk of experiencing preterm labor when a pregnant woman
exercises extensively.
The data were processed by NCSS. The results are shown in Figure
12.7.1. The relative risk calculation is shown in the column at the far right of
the output, along with the 95% confidence limits. Because of rounding errors,
these values differ slightly from those given in the example.
&
Odds Ratio When the data to be analyzed come from a retrospective study, relative
risk is not a meaningful measure for comparing two groups. As we have seen, a
retrospective study is based on a sample of subjects with the disease (cases) and a separate
sample of subjects without the disease (controls or noncases). We then retrospectively
determine the distribution of the risk factor among the cases and controls. Given the results
of a retrospective study involving two samples of subjects, cases, and controls, we may
display the data in a 2 2 table such as Table 12.7.3, in which subjects are dichotomized
with respect to the presence and absence of the risk factor. Note that the column headings in
Table 12.7.3 differ from those in Table 12.7.1 to emphasize the fact that the data are from a
retrospective study and that the subjects were selected because they were either cases or
controls. When the data from a retrospective study are displayed as in Table 12.7.3,
the ratio a=ða þ bÞ, for example, is not an estimate of the risk of disease for subjects with
the risk factor. The appropriate measure for comparing cases and controls in a retrospective
study is the odds ratio. As noted in Chapter 11, in order to understand the concept of