Confidence Interval for the Difference Between Two Independent Proportions

The Confidence Interval for the Difference Between Two Independent Proportions

This

page will calculate the lower and upper limits of the 95% confidence interval for the difference between two independent proportions, according to two methods described by Robert Newcombe, both derived from a procedure outlined by E.B.Wilson in 1927 (references below). The first method uses the Wilson procedure without a correction for continuity; the second uses the Wilson procedure with a correction for continuity.

For the notation used here, n_a and n_b represent the total numbers of observations in two samples, A and B; k_a and k_b represent the numbers of observations within each sample that are of particular interest; and p_a and p_b represent the proportions k_a/n_a and k_b/n_b, respectively. Thus, if sample A shows 23 recoveries among 60 patients, n_a=60, k_a=23, and the proportion is p_a=23/60=0.3833. If sample B shows 18 recoveries among 72 patients, n_b=72, k_b=18, and the proportion is p_b=18/72=0.2500. The difference between the two proportions is diff=p_a—p_b= 0.3833–0.2500 =0.1333.

Note that this method is recommended only for the case where p_a—p_b is equal to or greater than zero. Thus, the sample with the larger proportion should be designated as Sample A and the one with the smaller proportion should be designated as Sample B. To calculate the lower and upper limits of the confidence interval for a difference of this sort, enter the values of k and n for samples A and B in the designated places, then click the «Calculate» button.

Larger Proportion	Smaller Proportion
Sample A	Sample B
k_a =	k_b =
n_a =	n_b =
p_a =	p_b =
p_a— p_b =
Reset Calculate

95% confidence interval: no continuity correction
Lower limit =	Upper limit =

95% confidence interval: including continuity correction
Lower limit =	Upper limit =

References:

Newcombe, Robert G. "Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods," Statistics in Medicine, 17, 873-890 (1998).

Wilson, E. B. "Probable Inference, the Law of Succession, and Statistical Inference," Journal of the American Statistical Association, 22, 209-212 (1927).

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