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 The meaning of “95% confidence” when the numerator is zero

Interpreting a confidence interval is usually straightforward. But if the numerator of a proportion is zero, the interpretation is not so clear. In fact, the “95% confidence interval” really gives you 97.5% confidence. Here's why:

When the proportion does not equal zero, Prism reports the 95% confidence interval so that there is a 2.5% chance that the true proportion is less than the lower limit of the interval, and a 2.5% chance that the true proportion is higher than the upper limit. This leaves a 95% chance (100% -2.5% - 2.5%) that the interval includes the true proportion. When the numerator is zero, we know that the true proportion cannot be less than zero, so we only need to compute an upper confidence limit. Prism still calculates the upper limit so that there is a 2.5% chance that the true proportion is higher. Since the uncertainty only goes one way you'll actually have a 97.5% CI (100% - 2.5%). The advantage of calculating the “95%” confidence interval this way is that it is consistent with 95% CIs computed for proportions where the numerator is not zero.

If you don't care about consistency with other data, but want to really calculate a 95% CI, you can do that by computing a “90% CI”. This is computed so that there is a 5% chance that the true proportion is higher than the upper limit. If the numerator is zero, there is no chance of the proportion being less than zero, so the “90% CI” really gives you 95% confidence.