Introduction to Categorical Data Analysis

28 CONTINGENCY TABLES
It can be any nonnegative real number. The proportions 0.010 and 0.001 have a relative
risk of 0.010/0.001 = 10.0, whereas the proportions 0.410 and 0.401 have a relative
risk of 0.410/0.401 = 1.02. A relative risk of 1.00 occurs when π1 = π2, that is,
when the response is independent of the group.
Two groups with sample proportions p1 and p2 have a sample relative risk of
p1/p2. For Table 2.3, the sample relative risk is p1/p2 = 0.0171/0.0094 = 1.82.
The sample proportion of MI cases was 82% higher for the group taking placebo. The
sample difference of proportions of 0.008 makes it seem as if the two groups differ
by a trivial amount. By contrast, the relative risk shows that the difference may have
important public health implications. Using the difference of proportions alone to
compare two groups can be misleading when the proportions are both close to zero.
The sampling distribution of the sample relative risk is highly skewed unless the
sample sizes are quite large. Because of this, its confidence interval formula is rather
complex (Exercise 2.15). For Table 2.3, software (e.g., SAS – PROC FREQ) reports
a 95% confidence interval for the true relative risk of (1.43, 2.30). We can be 95%
confident that, after 5 years, the proportion of MI cases for male physicians taking
placebo is between 1.43 and 2.30 times the proportion of MI cases for male physicians
taking aspirin. This indicates that the risk of MI is at least 43% higher for the
placebo group.
The ratio of failure probabilities, (1 − π1)/(1 − π2), takes a different value than the
ratio of the success probabilities. When one of the two outcomes has small probability,
normally one computes the ratio of the probabilities for that outcome.
2.3 THE ODDS RATIO
We will next study the odds ratio, another measure of association for 2 × 2 contingency
tables. It occurs as a parameter in the most important type of model for
categorical data.
For a probability of success π, the odds of success are defined to be
odds = π/(1 − π)
For instance, if π = 0.75, then the odds of success equal 0.75/0.25 = 3.
The odds are nonnegative, with value greater than 1.0 when a success is more
likely than a failure. When odds = 4.0, a success is four times as likely as a failure.
The probability of success is 0.8, the probability of failure is 0.2, and the odds equal
0.8/0.2 = 4.0. We then expect to observe four successes for every one failure. When
odds = 1/4, a failure is four times as likely as a success. We then expect to observe
one success for every four failures.
The success probability itself is the function of the odds,
π = odds/(odds + 1)
For instance, when odds = 4, then π = 4/(4 + 1) = 0.8.