9.1.11, 2.2.12, D.1.8, D.1.13, D.2.2 - Wilcoxon signed-rank decisions

Choosing Wilcoxon for paired scores

Wilcoxon signed-rank tests a difference when each score in one condition belongs with a particular score in the other.

In a repeated-measures design, the same people contribute both scores, such as recall before and after learning a memory strategy. In a matched-pairs design, each participant is paired with a different participant matched on relevant characteristics. In either case, keep the pair identities. Comparing the group totals alone throws away information about how each pair differs.

Scores need to be at least ordinal, with meaningful differences that can be ordered in size. Numerical recall scores fit this requirement. Unordered categories cannot be subtracted meaningfully; independent groups require a different test. Wilcoxon does not require raw scores to be normally distributed, but it is not assumption-free. Pairs should be independent of other pairs. A location-change interpretation also depends on the distribution of differences being reasonably symmetric; strongly asymmetric differences need caution or another justified analysis.

Using each person as their own comparison reduces variation caused by stable individual differences. It does not remove order effects. For example, improved recall after a strategy might reflect practice with the test. Counterbalancing conditions where feasible, equivalent word lists and appropriate controls address design problems; statistical significance does not repair them.

Excluding zeros and ranking absolute differences

The signs of paired differences show their direction, while their absolute values show the sizes to rank.

These illustrative recall scores out of 20 were created to demonstrate the calculation. Nine people were measured before and after a strategy. Subtract after minus before for every pair. Reversing that convention consistently would reverse all signs without changing T; changing convention between rows would invalidate the calculation.

First remove person I's zero difference from the ranking. An unchanged pair does not provide a positive or negative change, so it also does not count towards N, the number of nonzero pairs. Here $N=8$, although nine people were measured and there are 18 raw scores.

Rank the remaining absolute differences from smallest to largest, ignoring signs at this stage. The two absolute differences of 1 occupy positions 1 and 2: both receive $(1+2)/2=1.5$. This tie includes +1 and −1 because their sizes are equal. The two absolute differences of 2 occupy positions 3 and 4: both receive 3.5. The next absolute difference, 3, receives rank 5. Ranking signed values directly would make −1 appear smaller than +1 and would fail to compare the sizes of change.

Forming signed-rank totals and T

Wilcoxon compares how much of the total rank weight belongs to changes in each direction.

Restore each nonzero difference's sign after ranking. Add ranks attached to positive differences to form $W_+$, and add the magnitudes of ranks attached to negative differences to form $W_-$. The negative total is a nonnegative total of rank weights, not a negative number.