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The Wilcoxon Matched-Pairs Signed-Ranks Test

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

The observation pairs (either a pair each line or the difference itself)

A most usefull test to see whether the members of a pair differ in size. It resembles the Sign-Test in scope, but it is much more sensitive. In fact, for large numbers it is almost as sensitive as the Student t-test. For small numbers with unknown distributions this test is even more sensitive than the Student t-test.
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.

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.

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.

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

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.

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-.

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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