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The randomized $p$-value, (nonrandomized) mid-$p$-value and abstractrandomized $p$-value have all been recommended for testing a null hypothesiswhenever the test statistic has a discrete distribution. This paper provides aunifying framework for these approaches and extends it to the multiple testingsetting. In particular, multiplicity adjusted versions of the aforementioned$p$-values and multiple test functions are developed. It is demonstrated that,whenever the usual nonrandomized and randomized decisions to reject or retainthe null hypothesis may differ, the (adjusted) abstract randomized $p$-valueand test function should be reported, especially when the number of tests islarge. It is shown that the proposed approach dominates the traditionalrandomized and nonrandomized approaches in terms of bias and variability. Toolsfor plotting adjusted abstract randomized $p$-values and for computing multipletest functions are developed. Examples are used to illustrate the method and tomotivate a new type of multiplicity adjusted mid-$p$-value.

Multiple Test Functions and Adjusted p-Values for Test Statistics with Discrete Distributions