Median of the p value under the alternative hypothesis

Due to absence of an universally acceptable magnitude of the Type I error in various fields, p values are used as a well-recognized tool in decision making in all areas of statistical practice. The distribution of p values under the null hypothesis is uniform. However, under the alternative hypothesis the distribution of the p values is skewed. The expected p value (EPV) has been proposed by authors to be used as a measure of the performance of the test. This article proposes the median of the p values (MPV) which is more appropriate for this purpose. We work out many examples to calculate the MPV's directly and also compare the MPV with the EPV. We consider testing equality of distributions against stochastic ordering in the multinomial case and compare the EPV's and MPV's by simulation. A second simulation study for general continuous data is also considered for two samples with different test statistics for the same hypotheses. In both cases MPV performs better than EPV.