Permutation p-value when the split is not the most extreme

Question

You compare two groups drawn from five values:

$5, \quad 10, \quad 15, \quad 20, \quad 25.$

Group A has $2$ units, group B has $3$ , and group A's observed values are $15$ and $25$ . The statistic is the sum of group A (larger favors A). Enumerate every way to choose which $2$ of the $5$ values are group A.

What is the one-sided permutation p-value?

Accepted Answer

There are \binom{5}{2} = 10 possible group-A pairs. The observed sum is 15 + 25 = 40. We count pairs with sum \ge 40. Check the large pairs: \{20, 25\} = 45 and \{15, 25\} = 40 both qualify. The next candidates, \{10, 25\} = 35 and \{15, 20\} = 35, fall short. So exactly two of the ten pairs reach 40 or more. p-value. \frac{\{	ext{sums} \ge 40\}}{10} = \frac{2}{10} = 0.20. Notice the observed split was not the single most extreme (that would be \{20,25\}), so the count includes it plus every split beyond it. Takeaway. The permutation p-value tallies every relabeling at least as extreme as o…

Permutation p-value when the split is not the most extreme

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