Basic Statistics

RELATIVE RISK AND ODDS RATIO 149

One difficulty in interpreting the odds ratio is that negative relationships are mea-

sured along a scale going from 0 to l and positive relationships are measured along a

scale going from 1 to m. This lack of symmetry on both sides of 1 can be removed

by calculating the natural logarithm (logarithm to the base e = 2.718.. .) of the odds

ratio [ln(OR)]. The ln(0R) varies from --x to fm, with 0 indicating independence.

The ln(0R) is used in computing confidence intervals for the odds ratio since the

distribution of ln(0R) is closer to a normal distribution than is the distribution of the

odds ratio. The population odds ratio is denoted by w. To compute an approximate

confidence interval that has a 95% chance of including the true w from the data in

Table 1 1.1, we would first compute

ln(0R) = ln(4.63) = 1.5326

Second, we compute an estimate of the standard error (se) of ln(0R) as

1111

or numerically,

seln(0R) = /-

or

seln(0R) = dw = .495

Next, the confidence interval for ln(0R) is given by

ln(0R) f z[1 - a/2] [se ln(OR)]

or for a 95% confidence interval

1.5326 f 1.96(.495) = 1.5326 f .9702

or

.562 < ln(w) < 2.503

The final step is to take the antilogarithm of the endpoints of the confidence limit (S62
and 2.503) to get a confidence interval in terms of the original odds ratio rather than
ln(0R). This is accomplished by computing e.562 and e2.503. The 95% confidence
interval about w is
1.755 < w < 12.219
The odds for a smoker having low vital capacity are greater than those for a nonsmoker,
so the ratio of the odds is > 1. Further, the lower limit of the confidence limit does
not include 1.
If we had a matched sample such as given in Table 11.4 for a caselcontrol study,
the paired odds ratio is estimated as

OR = b/c = 23/8 = 2.875