  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.
