suppose I have the following numbers.
y <- c(0.378, 0.347, 0.398, 2.05, .06, .29, 1.06, .14, 1.29)
x2 <- c(.1)
wilcox.test(y,x2)
x3 <- c(.1, .2,)
wilcox.test(y,x3)
I'm surprised that the first example with a sample of 1 yielded a p value. Does this p value mean anything? What about when its N=2?
thanks!
Repeating your R session---with a correction and some modifications.
y <- c(0.378, 0.347, 0.398, 2.05, .06, .29, 1.06, .14, 1.29)
x2 <- c(.1)
wilcox.test(y,x2)
Wilcoxon rank sum test
data: y and x2
W = 8, p-value = 0.4
alternative hypothesis: true location shift is not equal to 0
The P-value is $0.4 > 0.05,$ so there is no significant difference in
location at the 5% level. However, with only one observation in x1 you
will not get a significant result even if the value in x2 is greater
than any value in y.
xx2 = c(3)
wilcox.test(y,xx2)
Wilcoxon rank sum test
data: y and xx2
W = 0, p-value = 0.2
alternative hypothesis: true location shift is not equal to 0
By chance alone, there are two chances in ten that the one value in xx2 might have been the largest
or the smallest of all the observations.
x3 <- c(.1, .2) $ note: extra comma deleted
wilcox.test(y,x3)
Wilcoxon rank sum test
data: y and x3
W = 15, p-value = 0.2182
alternative hypothesis:
true location shift is not equal to 0
Again here, a non-significant P-value is computed.
The change here is that--if both values in xx3 are extreme--then it is
possible for the P-value to be significant at the 5% level.
xx3 = c(3,4)
wilcox.test(y,xx3)
Wilcoxon rank sum test
data: y and xx3
W = 0, p-value = 0.03636
alternative hypothesis:
true location shift is not equal to 0
By chance alone, there are 2 chances in ${11\choose 2}= 55$ that the two observations
in xx3 would be the two largest or the two smallest out of $9+2=11,$ then $2/55 = 0.03636.$
x3the P-value is not significant. However, in this case, it is possible for a second sample of size two to yield a significant result. $\endgroup$