Meta-Analysis of Mid-p-Values: Some New Results based on the Convex Order

Abstract

The mid- p -value is a proposed improvement on the ordinary p -value for the case where the test statistic is partially or completely discrete. In this case, the ordinary p -value is conservative, meaning that its null distribution is larger than a uniform distribution on the unit interval, in the usual stochastic order. The mid- p -value is not conservative. However, its null distribution is dominated by the uniform distribution in a different stochastic order, called the convex order. The property leads us to discover some new finite-sample and asymptotic bounds on functions of mid- p -values, which can be used to combine results from different hypothesis tests conservatively, yet more powerfully, using mid- p -values rather than p -values. Our methodology is demonstrated on real data from a cyber-security application.