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Suppose an experiment is done and the following is presented in a paper: median for experimental, median for control, and a WRS p-value, and (n) for EXP and CTRL arms. Now suppose experiment 2 is ongoing, and (n) for EXP and CTRL is known. I am trying to compute highest p-value could be in Trial 2, such that the pooled analysis of experiment 1 and 2 would be p<0.05. Any suggestions?

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First you need to choose a method for combining $p$-values. There are many of these but two which are often used are known as Fisher's method and Stouffer's method.

Now you need to search for the value of the as yet unknown value $p_2$ to combine with the value you know $p_1$. To do this by hand choose values at the limits, say 0.001 and 0.999 as values of $p_2$ and see what combining them with $p_1$ does. Then change one of them at a time and recalculate. Eventually you will home in on the value for $p_2$ which gives 0.05.

There is a programming solution for this using R but you do not specify which language you want to use.

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