Confidence interval calculator for a proportion

Let x be the number of participants with the event, n the number in the denominator, p = x/n the observed proportion, and z = 1.96 for a 95% interval. The Wald interval treats the sampling distribution of p as normal with variance p(1 − p)/n and returns p ± z √(p(1 − p)/n). It is simple, it is what most introductory courses teach, and it is the default in more spreadsheets than it should be. Its flaw is that the standard error is computed at the observed p. When p is 0 or 1 the standard error is zero, and the interval has no width however small n is.

The Wilson score interval (1927) solves the problem the other way round: it asks which true proportions would not be rejected by a score test given x of n. The result is centre (p + z²/2n) / (1 + z²/n) and half-width z √(p(1 − p)/n + z²/4n²) / (1 + z²/n). It pulls the centre toward 0.5, widens the interval for small n, and stays inside [0, 1] automatically. Brown, Cai and DasGupta (Stat Sci 2001) recommend it, or the Agresti-Coull interval, over Wald for essentially every n.

The Clopper-Pearson interval (1934) is exact. The lower limit is the proportion at which the probability of seeing x or more events is 2.5%, and the upper limit is the one at which the probability of seeing x or fewer is 2.5%. For 3 of 20 that gives 3.2% to 37.9%. With x = 0 the lower limit is 0 and the upper is 1 − 0.025^(1/n); with x = n the upper limit is 1. Because the binomial is discrete, the exact interval is wider than needed on average, with coverage at or above 95% rather than at it. That conservatism is a feature when the cost of understating a risk is high.

A 95% interval should contain the true proportion in 95% of repeated studies. That is its coverage, and it can be computed exactly for the binomial by summing the probabilities of the outcomes whose interval contains the true p. For a true rate of 5% with n = 30, the Wald interval has an actual coverage of 78.2%, the Wilson interval 93.9% and the Clopper-Pearson interval 98.4%. For a true rate of 2% with n = 50, Wald covers only 63.5%, Wilson 92.2% and Clopper-Pearson 98.2%. For a moderate rate of 30% with n = 40 the gap closes: 93.0%, 94.4% and 96.1%.

Two things follow. First, Wald is the worst where safety work needs the best: small samples and rare events. The reason is simple: with n = 30 and a true rate of 5%, about a fifth of studies observe zero events, and Wald reports a point with no uncertainty. Second, coverage of every discrete interval wobbles from one n to the next, so one favourable example proves little. The ranking, though, is stable: exact is conservative, Wilson is close to nominal and Wald is anti-conservative for small n or extreme p. A reasonable rule: use Wald only when both the expected events (np) and non-events (n(1 − p)) are at least about 10, and even then Wilson costs nothing extra.

The commonest use of these intervals in a clinical trial is a rate with a small denominator: the proportion of participants with a grade 3 or higher event, a response rate in a phase 2 cohort, a dropout rate in a pilot. Take a safety cohort of 48 participants in which 2 had an event: 4.2%, with a 95% Clopper-Pearson interval of 0.5% to 14.3% (Wilson 1.2% to 14.0%). Reporting “4.2%” alone suggests precision the data do not have, and the Wald interval would have given a lower limit of −1.5%. For 0 of 30, the honest statement is “no events were observed; the 95% upper confidence limit for the true rate is 11.6%”. A safety conclusion that 30 uneventful participants prove a rate below 5% is not supported: the data cannot rule out 10%.

Report the method with the interval. Clopper-Pearson is common in safety tables and in single-arm phase 2 reports because it cannot understate the uncertainty, and Wilson is widely used where a tighter, near-nominal interval is wanted. Either way the statistical analysis plan should name the method before the data are seen. For the clinical side of recording and grading events see the adverse event and serious adverse event glossary entries, the adverse event reporting software page and the adverse event eCRF template. Counts like x and n only mean something if the denominator is defined: everyone treated, everyone evaluable, or everyone followed for a minimum time. The interval is only as good as that definition.

These intervals assume independent participants and a single fixed denominator. They do not apply to rates per unit of exposure time, to participants who contribute multiple events, to clustered data (see the cluster-randomised sample size calculator for the design effect), or to time-to-event outcomes where follow-up differs. Planning the sample size for a given interval width is a different calculation: n = z² p(1 − p) / h² for a Wald-type half-width h, so a 10% rate known to within ±5 points needs 139 participants.

The interval is a one-line calculation once the counts are right, and getting the counts right is the data management work. In Capture, adverse events are recorded on a structured eCRF with onset, severity and seriousness; edit checks raise an auto-query when a site enters an inconsistent value; every change is recorded in the field-level audit trail with the old value, new value, user, time and reason; and the monitoring dashboard shows enrolment against target so the denominator is visible. Data export as CSV or Excel with a data dictionary, or as SDTM datasets in SAS XPT files with Define-XML, so the statistician can compute the counts and intervals in the package of their choice. The two-proportions sample size calculator covers planning a comparison of two rates, and the power analysis calculator states what a given n can detect.