Wilcoxon Signed RanksTest

An Alternative to the Paired Samples t

The Wilcoxon Signed RanksTest

The Wilcoxon test can be viewed as an alternative to the paired samples t test

The test uses data from one sample where each individual has been observed in two different treatment conditions (or a matchedsample where each sample is observedunder one treatment) to test for a significant difference between the two treatments.

PHowever, the Wilcoxon test does not requirethat the distribution of the difference scoresbe normal< Therefore, the Wilcoxon can be significantly more

powerful than the paired t when the distribution ofdifference scores is nonnormal

PThe null hypothesis for the Wilcoxon test simply states that there is no systematic or consistent difference between the two treatments being compared.

Wilcox Test, contd

Ranking Difference Scores

PThe calculation of the Wilcoxon T statisticrequires:< 1. Observe the difference between treatment 1

and treatment 2 for each subject.< 2. Rank order the absolute size of the differences

without regard to sign (increases are positive and decreases are negative).

< 3. Find the sum of the ranks for the positive differences and the sum of the ranks for the negative differences.

< 4. The Wilcoxon T is the smaller of the two sums

Calculating the Wilcoxon Test Statistic

P If there is a consistent difference betweenthe two treatments, the difference scoresshould be consistently positive (orconsistently negative). < At the extreme, all the differences will be in the

same direction and one of the two sums will be zero If there are no negative differences then G Ranks = 0 for

the negative differences.) PThus, a small value for T indicates a

difference between treatments.

Interpreting the Test Statistic

PTo determine whether the obtained T value is sufficiently small to be significant, you must consult the Wilcoxon table.

PFor large samples, the obtained T statistic can be converted to a z-score and the critical region can be determined using the unit normal table.

Evaluating the Wilcoxon TestStatistic

PDr. Brown wants to determine if reactiontimes increase (when solving simple mathproblems) under sleep deprivation

PStep 1: < Ho: nsd $ sd ; H1: nsd < sd< = .05

PStep 2:< From Appendix B.10, a value of T less than 3 is

statistically significant at = .05

Wilcoxon Signed Ranks Example

Example Data

P +ranks = 1, -ranks = 27PTherefore, T = 1PTherefore, since T obtained (1) is less than T

critical (3) there is a significant decrease inreaction times under no sleep deprivation,relative to under sleep deprivation

PWhen n is large, we can use a normalapproximation of the statistic< This z-statistic is reported by most software

packages

Wilcoxon Signed Ranks Example

1: Wilcoxon Signed Ranks Test 2: The Wilcoxon Signed Ranks Test 3: Wilcox Test, contd 4: Ranking Difference Scores 5: Calculating the Wilcoxon Test Statistic 6: Interpreting the Test Statistic 7: Evaluating the Wilcoxon Test Statistic 8: Wilcoxon Signed Ranks Example 9: Example Data 10: Wilcoxon Signed Ranks Example