Database Reference
In-Depth Information
3.3.3 Wilcoxon Rank-Sum Test
A
t
-test represents a
parametric test
in that it makes assumptions about the
population distributions from which the samples are drawn. If the populations
cannot be assumed or transformed to follow a normal distribution, a
nonparametric test
can be used. The
Wilcoxon rank-sum test
[15] is a
nonparametric hypothesis test that checks whether two populations are identically
distributed. Assuming the two populations are identically distributed, one would
expect that the ordering of any sampled observations would be evenly intermixed
among themselves. For example, in ordering the observations, one would not
expect to see a large number of observations from one population grouped
together, especially at the beginning or the end of ordering.
Let the two populations again be
pop1
and
pop2
, with independently random
samples of size and respectively. The total number of observations is then
. The first step of the Wilcoxon test is to rank the set of observations
from the two groups as if they came from one large group. The smallest observation
receives a rank of 1, the second smallest observation receives a rank of 2, and
so on with the largest observation being assigned the rank of
N
. Ties among the
observations receive a rank equal to the average of the ranks they span. The test
uses ranks instead of numerical outcomes to avoid specific assumptions about the
shape of the distribution.
After ranking all the observations, the assigned ranks are summed for at least one
population's sample. If the distribution of
pop1
is shifted to the right of the other
distribution, the rank-sum corresponding to
pop1
's sample should be larger than
the rank-sum of
pop2
. The Wilcoxon rank-sum test determines the significance
of the observed rank-sums. The following R code performs the test on the same
dataset used for the previous
t
-test.
wilcox.test(x, y, conf.int = TRUE)
Wilcoxon rank sum test
data: x and y
W = 55, p-value = 0.04903
alternative hypothesis: true location shift is not equal to
0
95 percent confidence interval:
-6.2596774 -0.1240618
sample estimates:
