Wilcoxon Match-Pairs Signed Ranks Test, W
{From the Institute of Phonetic Sciences (IFA): http://www.fon.hum.uva.nl/}

This is a. useful test to see whether the members of a pair differ. It resembles the Sign-Test in scope, but is much more sensitive to treatment effects. In fact, for large numbers of cases this test 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.

Since it is only on rare occasions that we can be sure that data values are normally distributed, this test may be preferred over the t test.

As an example (this is the same example used for the Sign-test), suppose we are interested in whether situational anxiety affects test performance. We might give a test (actually parallel forms of a test) to a group of individuals under two types of anxiety-inducing instructions: one (Treatment A) designed to induce low situational anxiety, the other (Treatment B) designed to induce high situational anxiety. Of course we would want to counter-balance the order in which the anxiety-inducing situation were applied–half would receive the low-anxiety instructions first and the other half, the high-anxiety instructions first. In each case an achievement test would be administered immediately following the anxiety-inducing instructions.

The (null) hypothesis to be tested is that the test scores under the two anxiety situations are due to sampling error, i.e., that there is no reliable difference between the mean test scores under the two conditions. The data array, following the experiment, would look something like that shown in Table X. In the column, "Sign of (A-B)," are recorded the sign of the difference obtained by subtracting each pairs score under condition B from his or her score under condition A. When the difference is zero, as it is for the 3ed individual, the sign is discarded.

 Table X Ind. Treatment Un-signed Difference Ranked Difference Signed Ranked Difference A B 1 51 50 1 1.5 +1.5 2 49 48 1 1.5 +1.5 3 46 46 0 (Unranked) 4 45 43 2 4.5 +4.5 5 46 44 2 4.5 +4.5 6 39 41 2 4.5 -4.5 7 41 39 2 4.5 +4.5 8 42 39 3 7.5 +7.5 9 41 38 3 7.5 +7.5 Sum of "+" Ranks (W+) 31.5 Sum of "-" Ranks (W-) 4.5

H0: The difference (d = x - y) between the members of each pair (x, y) has a median value zero.

Assumptions: The distribution of the differences (d = x - y) between the values within each pair (x, y) is symmetrical, i.e., the median difference must is 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.

Procedure: Rank-order, from smallest to largest, the differences (between the paired observations) for each pair without regard to the sign of the difference (i.e., rank order the absolute differences). Ignore all zero differences (i.e., pairs with equal scores). Affix the original signs to the ranks. All pairs with equal absolute differences (ties) get the same rank. Sum all positive ranks (W+) and all negative ranks (W-) and determine the total number of pairs (N).

In the example, the sum of the positive ranks (W+) is 31; the sum of negative ranks (W-) is 5. N, the number of pairs, is 9.

The Wilcoxon statistic, W, is the smaller of W+ or W-. Hence, in our example W is 4.5. This value is compared with tabled critical values of the Wilcoxon statistic. An abbreviated of critical values is given below. For 9 pairs, W is less than the critical value given for both alpha = .1 and alpha  = .05. Hence, we reject the null hypothesis.

 Selected Critical Values of the Wilcoxon Statistic (W) for alpha = .1 and .05 Number of Pairs Alpha Level .1 .05 5 0 - 6 2 0 7 3 2 8 5 3 9 8 5 10 10 8 12 17 13 15 30 25 20 60 52 25 100 89 Note: W is significant at the chosen alpha level if it is smaller than the critical value given in the  table.

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).

You can also compute the sign test by clicking here: Wilcoxon Signed Ranks Test