Proportions: Correlated and Paired
Calculate a two-sided confidence interval for the difference between two correlated or paired proportions using Newcombe’s Score method. Enter the observed 2×2 paired table and obtain the estimated difference, confidence limits, and interval width directly in your browser.
| Yt \ Ys | Ys = 1 | Ys = 0 |
|---|---|---|
| Yt = 1 | ||
| Yt = 0 |
This calculator estimates the difference between two correlated binary proportions, δ = Pt − Ps, where both binary outcomes are observed on the same subjects. PASS identifies Newcombe’s Score method (Newcombe, 1998, Method 10) as one of its confidence-interval methods for the difference between correlated proportions.
The method first calculates Wilson score intervals for each marginal proportion. It then combines those marginal score limits with a continuity-corrected estimate of the within-subject correlation to obtain asymmetric lower and upper distances from the observed difference.
For each marginal proportion p, the Wilson score limits are calculated as:
Here z is the standard normal quantile corresponding to the requested two-sided confidence level.
Newcombe’s Score method uses a continuity-corrected estimate of the within-subject correlation. For the observed 2×2 table, PASS defines A, B, and C from the four cell counts and calculates ρcc = C / √A. When the relevant marginal totals make the correlation undefined, the corrected correlation is set to zero.
Let Lt, Ut be the Wilson limits for pt, and Ls, Us the corresponding limits for ps. The lower and upper confidence limits are:
PASS’s validation example based on Newcombe (1998, p. 2641) uses P11=0.40, P10=0.24, P01=0.04, P00=0.32, and N=50. Multiplying these probabilities by 50 gives f11=20, f10=12, f01=2, and f00=16. For a 95% interval, PASS reports a confidence interval of approximately 0.0562 to 0.3292, with width 0.27305.
Validation target: Difference = 0.20; 95% CI = 0.0562 to 0.3292; CI width = 0.27305.
Newcombe, R. G. (1998). Improved confidence intervals for the difference between binomial proportions based on paired data. Statistics in Medicine, 17, 2635–2650. DOI: 10.1002/(SICI)1097-0258(19981130)17:22<2635::AID-SIM954>3.0.CO;2-C.
NCSS, LLC. PASS Sample Size Software: Confidence Intervals for the Difference Between Two Correlated Proportions. PASS procedure documentation, Chapter 102. The PASS documentation identifies Newcombe’s Score as Method 10 and provides the validation example used above.
Altman, D. G., Machin, D., Bryant, T. N., & Gardner, M. J. (2000). Statistics with Confidence, 2nd Edition. BMJ Publishing.