A two-sided permutation p-value

Question

You compare two groups of two on the values

$1, \quad 2, \quad 3, \quad 4,$

where group A got the two lowest, $1$ and $2$ . The test statistic is the sum of group A, and you want a two-sided p-value (a difference in either direction counts). Enumerate every split of the four values into two groups of two.

What is the two-sided permutation p-value?

Accepted Answer

There are \binom{4}{2} = 6 ways to choose group A. Their sums: Group A Sum ------ \{1,2\} 3 \{1,3\} 4 \{1,4\} 5 \{2,3\} 5 \{2,4\} 6 \{3,4\} 7 The sums are symmetric around their center, the average group-A sum = 	frac{(1+2+3+4)\cdot 2}{4} = 5. The observed sum is 3, a distance of 2 below center. For a two-sided test, count splits that are at least that far from the center in either direction: sums \le 3 or \ge 7. Those are \{1,2\} (sum 3) and \{3,4\} (sum 7), two of the six. p-value. \frac{2}{6} = \frac{1}{3} \approx 0.333. Takeaway. A two-sided permutation p-value counts relabelings at leas…

A two-sided permutation p-value

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