Why the order of application of properties of Wilcoxon Rank-Sum makes a difference

Question

Suppose I am calculating the p-value of a Wilcoxon Rank-Sum test where $W_s = 23$, $n=4$ and $m=5$ suppose that we have:

$pvalue = P(W_s \ge 23 | H_0)$

The way I manage to get to the right answer is to use the property that $W_s$ is simetrical around $n*((n+m)+1)/2$ and get:

• As $n=4$ and $m=5$, $n*((n+m)+1)/2 = 20$

$P(W_s \ge 20 + 3 | H_0 ) = P(W_s \le 20 - 3 | H_0 )$

and then find $W_{x,y}$ using that $W_{x,y} = W_s - n*(n+1)/2$:

• As $n=4$, $n*(n+1)/2 = 10$

$P(W_s \le 17 | H_0 ) = P(W_x,y \le 7) = 0.2778$

However, if I just apply the second property directly I get a different, wrong result:

• As $n=4$, $n*(n+1)/2 = 10$

$pvalue = P(W_s \ge 23 | H_0) = P(W_{x,y} \ge 13) = 0.20$

Why?

Answer 1

Turns out that I made a mistake, I was calculating $P(Wx,y>13)$ and not $P(Wx,y \ge 13)$. The properties pointed out by me are valid regardless of the order that you apply them.

Notice that the big problem was that I was comparing

pwilcox(13,4,5,lower.tail=FALSE)

With:

pwilcox(7,4,5,lower.tail=TRUE)

In R, and as lower.tail = FALSE implies strictly greater, and not greater or equals to, I was getting a mistake.