Introductory Biostatistics

4.4 ESTIMATION OF ODDS RATIOS

So far we have relied heavily on the central limit theorem in forming confidence
intervals for the means (Section 4.2) and the proportions (Section 4.3). The
central limit theorem stipulates that as a sample size increases, the means of
samples drawn from a population of any distribution will approach the normal
distribution; and a proportion can be seen as a special case of the means. Even
when the sample sizes are not large, since many real-life distributions are
approximately normal, we still can form confidence intervals for the means (see
Section 4.2.2 on the uses of small samples).
Besides the mean and the proportion, we have had two other statistics of
interest, theodds ratioand Pearson’scoe‰cient of correlation. However, the
method used to form confidence intervals for the means and proportions does
not apply directly to the case of these two new parameters. The sole reason is
that they do not have the backing of the central limit theorem. The sampling
distributions of the (sample) odds ratio and (sample) coe‰cient of correlation
are positively skewed. Fortunately, these sampling distributions can be almost
normalized by an appropriate data transformation: in these cases, by taking the
logarithm. Therefore, we learn to form confidence intervals on the log scale;
then taking antilogs of the two endpoints, a method used in Chapter 2 to
obtain thegeometric mean. In this section we present in detail such a method
for the calculation of confidence intervals for odds ratios.
Data from a case–control study, for example, may be summarized in a 2 2
table (Table 4.7). We have:

(a) The odds that a case was exposed is

odds for cases¼

a

b

(b) The odds that a control was exposed is

odds for controls¼

c

d

Therefore, the (observed) odds ratio from the samples is

OR¼

a=b

c=d

¼

ad

bc

TABLE 4.7

Exposure Cases Controls

Exposed ac

Unexposed bd

ESTIMATION OF ODDS RATIOS 165