Abstract
A novel credible interval of the binomial proportion is proposed by improving the Highest Posterior Density (HPD) interval using the logit transformation. It is constructed in two steps: first the HPD interval for the logit transformation of the binomial proportion is driven and then the corresponding credible interval of the binomial proportion is calculated by the inverse logit transformation of that interval. Two characteristics of the proposed credible interval are: (i) the lower limit is over 0% when the zero events are obtained and (ii) the error probability is not large for any population binomial proportion. The characteristic in (i) corresponds to the claims in the Rule of Three, which means that even if zero events are obtained from n trials, the events might occur three times in other n trials. The characteristic in (ii) is important for medical research. This is because the error probabilities of all groups do not increase even if there is a high population binomial proportion among some groups. The proposed credible interval is compared with the existing confidence and credible intervals. We verified using numerical and practical examples to confirm the potential usefulness of the proposed credible interval.
Highlights
Since the confidence or credible interval of the binomial proportion p is routinely used in medical research, further detailed studies are important under various practical settings
The existing Highest Posterior Density (HPD) interval was narrower than the equal-tailed credible interval, the lower limit was 0% for zero events
The equal-tailed credible interval was over 0% for zero events
Summary
Since the confidence or credible interval of the binomial proportion p is routinely used in medical research, further detailed studies are important under various practical settings. The error probabilities of the equal-tailed credible interval under Jeffreys' and the reverse J-shaped prior densities are occasionally over 0.1, in the range of p > 0.2, they are less than the existing HPD interval. From the figures of the error probability in the sample sizes n = 8, 20, the proposed HPD interval under Jeffreys' prior density is the most stable near for the Clopper-Pearson confidence and three credible intervals under the three prior densities. Since this example is discussed under p > 0, there is no concern that the error probability of the proposed HPD interval will become too large
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