A Short Note on P-Value Hacking

We present the expected values from p-value hacking as a choice of the minimum p-value among $m$ independents tests, which can be considerably lower than the "true" p-value, even with a single trial, owing to the extreme skewness of the meta-distribution. We first present an exact probability distribution (meta-distribution) for p-values across ensembles of statistically identical phenomena. We derive the distribution for small samples $2<n \leq n^*\approx 30$ as well as the limiting one as the sample size $n$ becomes large. We also look at the properties of the "power" of a test through the distribution of its inverse for a given p-value and parametrization. The formulas allow the investigation of the stability of the reproduction of results and "p-hacking" and other aspects of meta-analysis. P-values are shown to be extremely skewed and volatile, regardless of the sample size $n$, and vary greatly across repetitions of exactly same protocols under identical stochastic copies of the phenomenon; such volatility makes the minimum $p$ value diverge significantly from the "true" one. Setting the power is shown to offer little remedy unless sample size is increased markedly or the p-value is lowered by at least one order of magnitude.