Table 3.

Minimum and Maximum Length for 95% Confidence Intervals for a difference in two proportions

Sample Size/Group (n)	Minimum	Maximum
10	0.82	1.07
20	0.54	0.71
30	0.42	0.57
40	0.36	0.48
50	0.32	0.43
60	0.29	0.39
70	0.26	0.36
80	0.24	0.33
90	0.23	0.31

Note: The Wald method with continuity correction was used to calculate 95% CI for the difference (d) in two proportions (p2 - p1 = d, set p1=0.1, 0.2, 0.3, 0.4, d=0.1, 0.2, 0.3, then p2 = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 based on the value of d). The proportions are selected based on clinically relevant estimates and their differences. Setting p1=0.9, 0.8, 0.7, 0.6, given the same d=p1-p2 and corresponding p2=0.8, 0.7, 0.6, 0.5, 0.4, 0.3, will yield the same estimates of the width of CI (differing only in the label of the events). The maximum width of a CI for a difference in two proportions can be as large as 0.57 for a group sample size of 30.

For example, given n=30, the maximum width occurs when p2=0.55 and p1=0.45 and the minimum width occurs when p2=0.2 and p1=0.1.

Note also that in this example, p1 and p2 were restricted to the less extreme values indicated above. If p1 and p2 are not limited, and any two proportions are selected, the maximum values occur when p1 and p2 are close to 0.5 and within the range of proportions we considered; thus the value is still very close to the numbers in the table. If we consider more extreme proportions close to 0 and 1 then the Wald method of calculating confidence intervals for their difference can underestimate the width of the interval. For example, for n=30, the maximum occurs when p1 and p2 are very close to 0.5; for p1=0.5001 and p2=0.4999, the width is 0.5727. The minimum occurs when p1 and p2 are very close to 0 or 1; for p1=0.0001 and p2=0.0002 (or p1 = 0.9999 and p2 = 0.9998), the width is 0.0791. Detailed values are provided in Appendix Table 2.