One class of statistical hypothesis testing procedures is the indisputable equivalence tests, whose main objective is to establish practical equivalence rather than the usual statistical significant difference. These hypothesis tests are prone in bioequivalence studies, where one would wish to show that, for example, an existing drug and a new one under development have the same therapeutic effect. In this article, we consider a two-stage randomized (RAND2) p-value utilizing the uniformly most powerful (UMP) p-value in the first stage when multiple two-one-sided hypotheses are of interest. We investigate the behavior of the distribution functions of the two p-values when there are changes in the boundaries of the null or alternative hypothesis or when the chosen parameters are too close to these boundaries. We also consider the behavior of the power functions to an increase in sample size. Specifically, we investigate the level of conservativity to the sample sizes to see if we control the type I error rate when using either of the two p-values for any sample size. In multiple tests, we evaluate the performance of the two p-values in estimating the proportion of true null hypotheses. We conduct a family-wise error rate control using an adaptive Bonferroni procedure with a plug-in estimator to account for the multiplicity that arises from the multiple hypotheses under consideration. We verify the various claims in this research using simulation study and real-world data analysis.

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