Mid-P confidence intervals: A brief review

Significance tests that are based on discrete probabilities are conservative in that the average value of the significance level, when the null hypothesis is true, always exceeds 0.5. An approach suggested by H. O. Lancaster over 40 years ago overcomes this problem. This is to calculate the mid-P value, where only half of the probability of the observed sample is included in the tail. The average value of the mid-P value is 0.5 and the variance is slightly less than that of a random variable uniformly distributed between 0 and 1. The mid-P concept has usually been advocated in the context of significance testing but it can be extended to the calculation of confidence intervals in an estimation approach by defining, for example, the 95% mid-P confidence limits as the values that have a one-sided mid-P value of 0.025. In this paper we review recent work supporting this approach.