Comparing Confidence Intervals for a Binomial Proportion with the Interval Score

There are over 55 different ways to construct a confidence respectively credible interval (CI) for the binomial proportion. Methods to compare them are necessary to decide which should be used in practice. The interval score has been suggested to compare prediction intervals. This score is a proper scoring rule that combines the coverage as a measure of calibration and the width as a measure of sharpness. We evaluate eleven CIs for the binomial proportion based on the expected interval score and propose a summary measure which can take into account different weighting of the underlying true proportion. Under uniform weighting, the expected interval score recommends the Wilson CI or Bayesian credible intervals with a uniform prior. If extremely low or high proportions receive more weight, the score recommends Bayesian credible intervals based on Jeffreys' prior. While more work is needed to theoretically justify the use of the interval score for the comparison of CIs, our results suggest that it constitutes a useful method to combine coverage and width in one measure. This novel approach could also be used in other applications.