Meta-Analysis of Genuine and Fake p-Values

For Sir Ronald Fisher, it is important to consistently obtain significant p-values to support an experimental hypothesis. So, replicating experiments to obtain independent p-values is a legitimate and desirable research practice. Several simple statistics have been proposed to meta-analyze p-values, all assuming that they are genuine, i.e. observations from independent standard Uniform random variables. But, as publication bias favors the studies that report "significant" p-values, when a p>0.05 is obtained for the outcome of an experiment, some researchers will "fall into temptation" and decide to replicate the experiment in the hope of getting a smaller second p-value, ideally a significant one. Consequently, if the smallest of two p-values is reported, this is a Beta(1,2) distributed "fake" p-value, not a uniformly distributed genuine p-value. This is an unacceptable scientific research practice, and moreover the detection of fake p-values is unpractical. Even when it is possible, the analytic results to accommodate their existence in combined tests are cumbersome. For an informed decision, inclusive when the presence of fake p-values in a sample of p-values to be meta-analyzed is probable, tables with simulated critical values for the usual combined testing are supplied. This will also allow comparisons to be made between several combined tests.