Example:
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 &negative=
For N <= 20,
exact probabilities are calculated, for N > 20, the Normal
approximation is used.