Purpose
This procedure
compares two independent samples.
Preparations
Run Statistics→Nonparametric
Statistics →Comparing two
independent samples.... command.
Results
MannWhitney U test or Wilcoxon RankSum
Test.
This test is the nonparametric substitute for the equalvariance ttest when
the assumption of normality is not valid. When in doubt about normality, play it
safe and use this test. The MannWhitney U test assumes that the variable under
consideration was measured on at least an ordinal (rank order) scale. The
interpretation of the test is essentially identical to the interpretation of the
result of a ttest for independent samples, except that the U test is computed
based on rank sums rather than means. The U test is the most powerful (or
sensitive) nonparametric alternative to the ttest for independent samples; in
fact, in some instances it may offer even greater power to reject the null
hypothesis than the ttest.
With samples larger than 20, the sampling distribution of the
U statistic rapidly approaches the normal distribution (see Siegel, 1956).
Hence, the U statistic (adjusted for ties) will be accompanied by a z value
(normal distribution variate value), and the respective pvalue.
U (Mann Whitney U) 
criterion statistic. U  is defined as the total number of times a
Y precedes an X in the configuration of combined samples Gibbons (1985). It is
directly related to the sum of ranks. This is why this test is sometimes called
the MannWhitney U test and other times called the Wilcoxon Rank Sum testThe
MannWhitney U test calculates Ux and Uy.
U
= W  n_{x }(n_{x}  1) / 2
Mean W_{x} = n_{x} (n_{x} +
n_{y} + 1) / 2
For the Wilcoxon ranksum test, the
null and alternative hypotheses relate to the equality or nonequality of two
medians..
KolmogorovSmirnov Test.
This is a twosample test for differences between two
samples or distributions. If a statistical difference is found between the
distributions of X and Y, the test provides no insight as to what caused the
difference. For example, the difference could be due to differences in location
(mean), variation (standard deviation), presence of outliers, type of skewness,
type of kurtosis, number of modes, and so on.
The assumptions for this nonparametric test are: (1) there are two independent
random samples; (2) the two population distributions are continuous; and (3) the
data are at least ordinal in scale. This test is frequently preferred over the
Wilcoxon sumrank test when there are a lot of ties. The test statistic is the
maximum distance between the empirical distribution functions (EDF) of the two
samples.
WaldWolfowitz Runs
Test.
The
WaldWolfowitz runs test is a nonparametric alternative to the ttest for
independent samples.
The WaldWolfowitz
runs test works as follows: Imagine you want to compare male and female subjects
on some variable. You can sort the data by that variable and look for cases
when, in the sorted data, samegender subjects are adjacent to each other. If
there are no differences between male and female subjects, then the number and
"lengths" of such adjacent "runs" of subjects of the same gender will be more or
less random. If not, the two groups (genders in our example) are somehow
different from each other. This test assumes that the variable under
consideration is continuous, and that it was measured on at least an ordinal
scale (i.e., rank order). The WaldWolfowitz runs test assesses the hypothesis that two independent samples were drawn from two
populations that differ in some respect, i.e., not just with respect to the
mean, but also with respect to the general shape of the distribution. The null
hypothesis is that the two samples were drawn from the same population. In this
respect, this test is different from the parametric
ttest which strictly tests for
differences in locations (means) of two samples.
Rosenbaum Test.
For the Rosenbaum test, the null and
alternative hypotheses relate to the nonequality or equality of two medians..
Criterion statistic
Q = S_{1} +
S_{2} where
S_{1 }
 number of variants from the first sample, bigger than maximum of the the
second sample
S_{2 }
 number of variants from the
second sample, bigger than maximum of the the
first sample.
The criterion cannot be applied to samples with size less than 11.
