  This paper introduces a comprehensive framework to adjust a discrete test
statistic for improving its hypothesis testing procedure. The adjustment
minimizes the Wasserstein distance to a null-approximating continuous
distribution, tackling some fundamental challenges inherent in combining
statistical significances derived from discrete distributions. The related
theory justifies Lancaster's mid-p and mean-value chi-squared statistics for
Fisher's combination as special cases. However, in order to counter the
conservative nature of Lancaster's testing procedures, we propose an updated
null-approximating distribution. It is achieved by further minimizing the
Wasserstein distance to the adjusted statistics within a proper distribution
family. Specifically, in the context of Fisher's combination, we propose an
optimal gamma distribution as a substitute for the traditionally used
chi-squared distribution. This new approach yields an asymptotically consistent
test that significantly improves type I error control and enhances statistical
power.
