Introduction to Categorical Data Analysis

26 CONTINGENCY TABLES
compare two groups on a binary response, Y . The data can be displayed in a 2 × 2
contingency table, in which the rows are the two groups and the columns are the
response levels of Y . This section presents measures for comparing groups on binary
responses.
2.2.1 Difference of Proportions
As in the discussion of the binomial distribution in Section 1.2, we use the generic
terms success and failure for the outcome categories. For subjects in row 1, let π1
denote the probability of a success, so 1 − π1 is the probability of a failure. For
subjects in row 2, let π2 denote the probability of success. These are conditional
probabilities.
The difference of proportions π1 − π2 compares the success probabilities in the
two rows. This difference falls between −1 and +1. It equals zero when π1 = π2,
that is, when the response is independent of the group classification. Let p1 and p2
denote the sample proportions of successes. The sample difference p1 − p2 estimates
π1 − π2.
For simplicity, we denote the sample sizes for the two groups (that is, the row
totals n1+ and n2+) byn1 and n2. When the counts in the two rows are independent
binomial samples, the estimated standard error of p1 − p2 is
SE =
�
p1(1 − p1)
n1
+ p2(1 − p2)
n2
(2.1)
The standard error decreases, and hence the estimate of π1 − π2 improves, as the
sample sizes increase.
A large-sample 100(1 − α)% (Wald) confidence interval for π1 − π2 is
(p1 − p2) ± zα/2(SE)
For small samples the actual coverage probability is closer to the nominal confidence
level if you add 1.0 to every cell of the 2 × 2 table before applying this formula. 1 For
a significance test of H0: π1 = π2, az test statistic divides (p1 − p2) by a pooled
SE that applies under H0. Because z 2 is the Pearson chi-squared statistic presented
in Section 2.4.3, we will not discuss this test here.
2.2.2 Example: Aspirin and Heart Attacks
Table 2.3 is from a report on the relationship between aspirin use and myocardial
infarction (heart attacks) by the Physicians’ Health Study Research Group at Harvard
1 A. Agresti and B. Caffo, Am. Statist., 54: 280–288, 2000.