# Combining individually valid and conditionally i.i.d. P-variables - Statistics > Methodology

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Abstract: For a given testing problem, let $U 1,

.,U n$ be individually valid andconditionally on the data i.i.d.\ P-variables often called P-values. Forexample, the data could come in groups, and each $U i$ could be based onsubsampling just one datum from each group in order to satisfy an independenceassumption under the hypothesis. The problem is then to deterministicallycombine the $U i$ into a valid summary P-variable. Restricting here ourattention to functions of a given order statistic $U {k:n}$ of the $U i$, wecompute the function $f {n,k}$ which is smallest among all increasing functions$f$ such that $fU {k:n}$ is always a valid P-variable under the statedassumptions. Since $f {n,k}u\le 1\wedge \frac {n}{k} u$, with the righthand side being a good approximation for the left when $k$ is large, one may inparticular always take the minimum of 1 and twice the left sample median of thegiven P-variables.We sketch the original application of the above in a recent study ofassociations between various primate species by Astaras et al.

Author: ** Lutz Mattner**

Source: https://arxiv.org/