*Example:*

**W+ = 5, W- = 40, N = 9, p <= 0**

A

As it is only on rare occasions that we do know that values are Normal distributed, this test is to be preferred over the Student t-test.

*H0:*

The difference (d = x - y) between the members of each pair (x, y) has
*median* value zero. To be complete, x an y have identical distributions.

*Assumptions:*

The distribution of the difference (d) between the values within each pair
(x, y) must be symmetrical, the median difference must be identical to the mean
difference.

As members of a pair are assumed to have identical distributions, their
differences (under *H0*) should *always* have a symmetrical
distribution, so this assumption is not very restrictive.

*Scale:*

Between ordinal and interval (also called an *ordered metric* scale). It
must be possible to rank the differences.

*Procedure:*

Rank the differences without regard to the sign of the difference (i.e.,
rank order the *absolute* differences). Ignore all zero differences
(i.e., pairs with equal members, x=y). Affix the original signs to the rank
numbers. All pairs with equal absolute differences (*ties*) get the same
rank: all are ranked with the mean of the rank numbers that would have been
assigned if they would have been different.

Sum all positve ranks (*W+*) and all negative ranks (*W-*) and
determine the total number of pairs (*N*).

*Level of Significance:*

The level of significance is calculated by dividing the number of all
distributions of signs over the ranks that have a SUM(+ranks) <= *W+*
(if *W+* < *W-*) by 2***N* (i.e., the total number of possible
distributions of signs).

These values are tabulated and the level of significance can be looked up.

*Approximation:*

If N > 15, then

Z = (*W* - 0.5 - *N* * ( *N* + 1 ) / 4 ) /
sqrt( *N** ( *N* + 1 ) * ( 2 * *N* + 1 ) / 24 )

has an approximate
Standard Normal distribution with W the larger of *W+* and *W-*.

*Remarks:*

It is not quite clear why this test is so impopular. It could be due to the fact that
ranking the differences (and calculating the tables) cannot be done with a desk
calculator or be programmed in C easily. When this is not a problem, this test
should *realy* be preferred over the Student t-test (at least when
*N* < 50). The Student t-test
is much too vulnerable to deviations from the normal distribution.

Note that the Wilcoxon Matched-Pairs Signed-Ranks Test uses the *sizes*
of the differences. The result can differ from that of the Sign-test, which
uses the *number* of + and - signs of the differences.

For *N* <= 20,
exact probabilities are calculated, for *N* > 20, the Normal
approximation is used.
A perl script of the test is available
here.
A minimalist Windows version (with dosperl interpreter) is available
here
(<500 kB).

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