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SPSS dialogs, help and algorithms make (at
least up to version 20) imprecise statements about what certain
nonparametric tests do test. I would be happy to attract Rick
Oliver's attention to that.
1) In Related-Samples Nonparametric Tests Wilcoxon and
Sign tests are absurdly said to test the same thing: "Compare Median Difference to Hypothesized", "test
of whether the median difference between two continuous fields is
different from 0", the same thing is stated formally in the
algorithms.
For Sign test, it is indeed so. H0: in population, the
differences have median 0 (= equal probability for two randomly
chosen differences to be of same or opposite sign); H1: median not
0.
For Wilcoxon test, it is not true. H0: in population, the
differences are distributed symmetrically about 0 (= equal
probability for the sum of two randomly chosen differences
to be "+" or "-"); H1: the distribution is not symmetric about 0 -
it has more "gravity" on one side. (Note that H1 does not
necessarily mean that the mean or median difference departure from
0.)
For Wilcoxon with additional assumption that the differences
in population are distributed symmetrically (the "restricted
formulation" of the test) H0: the differences have mean (and median)
0; H1: mean/median not 0. It is this restricted Wilcoxon which is
the direct nonparametric counterpart of paired-sample t-test (which
tests the same hypothesis, only under the assumption that the
differences in population are distributed normally).
2) In One-Sample Nonparametric Tests Wilcoxon is
said to be testing "Compare median to
hypothesized". This is again incorrect. One-sample Wilcoxon isn't
any special: it is just a paired-sample Wilcoxon test with one of
the two variables being that hypothesized constant. Thus,
one-sample Wilcoxon tests whether the variable is distributed symmetrically
about the hypothesized value, and not that it has median equal
to that value. It is one-sample Sign test that would test
if the median is equal to the value. "Restricted" Wilcoxon (with
assumption added that the variable is distributed
symmetrically) will test if the mean is equal to the value.
3) There is also a long-lasting misuderstanding and controversy
about what do Mann-Whitney and Kruskal-Wallis (its direst
extension to multiple groups) test. And the controversy is not
resolved in SPSS documentation. However, I won't touch this topic
in this post.
4) I think that regarding Friedman test, SPSS should make a
special teacher's warning, that this test is not the
extension of Wilcoxon to multiple groups. Instead, it is much closer
to be the extension of Sign test to multiple groups. While
indeed close, it is nevertheless not the direct (exact) extension
of it (whereas, for example, Kruskal-Wallis is the direct
extension of Mann-Whitney): in situation of 2 samples, Friedman is
somewhat more conservative (this difference vanishes with the
growth of N).
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