148 CATEGORICAL DATA: ANALYSIS OF TWO-WAY FREQUENCY TABLESdiseased. The odds are estimated as
total diseased
total sample - total diseasedodds =
or- .212
21
120-21 99
odds =^21 - - --
If the odds are equal to .5, we say that we are equally likely to get the disease or not.
Odds can vary between 0 and cc (i.e., infinity).
Odds are frequently quoted by sports announcers. An announcer might say that the
odds are 3 to 1 that team A will defeat team B. If the teams had played together 100
times and team A had won 75 of the games, the odds would be 75/( 100 - 75) = 3,
or 3 to 1.
We can also compute the odds of low vital capacity separately for smokers and for
nonsmokers. The odds for smokers are 11/(30 - 11) = 11/19 = .579. Note that
1 1/19 is equivalent in symbols from Table 1 1.1 to dc. For nonsmokers the odds are
10/(90 - 10) = 10/80 = .125. Here, 10/80 is equivalent to b/d inTable 11.1B.
From the odds for smokers and for nonsmokers, we can compute the ratio of these
odds. The resulting statistic is call the odds ratio (OR) or cross-product ratio. From
Table 11.1A we can divide the odds for smokers by the odds for nonsmokers to obtain
.579
OR = - = 4.63
.125In symbols from Table 1 1.1 B ,
For the smoking and vital capacity survey results, we can say that the odds ratio of a
smoker having a low vital capacity is 4.63 times that of a nonsmoker. When the odds
ratio is 1, we say that being exposed has no effect on (is independent of) getting the
disease. If the odds ratio is < 1, the exposure results may lessen the chance of getting
the disease (negative association). If the odds ratio is > 1, we say that the exposure
factor may increase the chance of getting the disease (positive association). In the
smoking example, the chance of having low vital capacity appears to be positively
associated with smoking. The odds ratio can vary from 0 to 00.
The odds ratio is a commonly reported statistic in biomedical reports and journals
and has several desirable properties. The magnitude of the odds ratio does not change
if we multiply a row or a column of a table by a constant. Hence, it can be used in
case/control studies where the sample proportion is not indicative of the proportion
in the population. For rare diseases, the odds ratio can be used to approximate the
relative risk (see Schlesselman [1982] or van Belle et al. [2004]). The odds ratio
does not change if rows are switched to columns and/or columns to rows. It also
can be computed from more complex analyses (see Afifi et al. [2004]). For further
discussion of the odds ratio, see Rudas [1998].