Tables with Critical Values for the Meta-Analysis of Genuine and Fake $\boldsymbol{p}$-Values

The classical theory for the meta-analysis of $p$-values is based on the assumption that if the overall null hypothesis is true, then all $p$-values used in a chosen combined test statistic are genuine, i.e., are observations from independent and identically distributed standard uniform random variables. However, the pressure felt by most researchers to publish, which is worsen by publication bias, can originate fake $p$-values to be reported, usually Beta(1,2) distributed. In general, the existence of fake $p$-values in a sample of $p$-values to be combined is unknown, and if, for some reason, there is information that they do exist, their number will most likely be unknown as well. Moreover, even if fake $p$-values are accounted for, the cumulative distribution function of classical combined test statistics does not have a closed-form expression that facilitates its practical usage. To overcome this problem, tables with estimated critical values are supplied for the commonly used combined tests for the meta-analysis of $p$-values when a few of them are fake ones, i.e., Beta(1,2) distributed.