Wilcoxon signed-rank test
Wilcoxon signed-rank test
Comparison of central tendencies of two dependent samples.
Requirements:
Dependent variable must be interval scaled because differences are taken for the test. Distribution is free.
Idea:
Dependency of the two sample data is considered by taking the differences of paired values. The calculation of differences requires at least interval scaled data. In a next step the differences are brought into rank order. Under H0 the sum of positive differences should approximately equal the sum of negative differences. Zero-Differences are not considered in the test.
Measure:
T = Sum of ranks of less frequent sign.
Approximation with Normal Distribution:
For large sample sizes (n = Number of pairs > 25) T is approximately normal distributed with
and
Test:
If there are tied ranks, σT is corrected as follows:
Whereas
ti = Number of subjects sharing rank i
k = Number of tied ranks
Example of a Wilcoxon signed-rank test
A driving instructor is interested on the effect of alcohol on the fitness to drive. Learners had to drive the same test course with 0 per mill and with 0.8 per mill blood alcohol level. Driving errors of the learners were measured. The following table shows the raw data and rank scores for each learner.
| Perf. BAL 0.8 |
Perf. BAL 0.0 |
Diff. 0.8-0.0 |
abs. Diff. |
Rank of abs. Diff. |
| 3 |
1 |
2 |
2 |
16.5 |
| 2 |
2 |
0 |
0 |
|
| 0 |
1 |
-1 |
1 |
6.5 * |
| 3 |
0 |
3 |
3 |
21.5 |
| 1 |
0 |
1 |
1 |
6.5 |
| 2 |
2 |
0 |
0 |
|
| 1 |
0 |
1 |
1 |
6.5 |
| 3 |
1 |
2 |
2 |
16.5 |
| 1 |
0 |
1 |
1 |
6.5 |
| 0 |
0 |
0 |
0 |
|
| 1 |
0 |
1 |
1 |
6.5 |
| 3 |
1 |
2 |
2 |
16.5 |
| 2 |
1 |
1 |
1 |
6.5 |
| 4 |
2 |
2 |
2 |
16.5 |
| 2 |
0 |
2 |
2 |
16.5 |
| 2 | 3 |
-1 |
1 |
6.5 * |
| 0 |
1 |
-1 |
1 |
6.5 * |
| 1 |
0 |
1 |
1 |
6.5 |
| 2 |
0 |
2 |
2 |
16.5 |
| 2 |
1 |
1 |
1 |
6.5 |
| 3 |
1 |
2 |
2 |
16.5 |
| 1 |
0 |
1 |
1 |
6.5 |
| 4 |
0 |
4 |
4 |
24 |
| 5 |
1 |
4 |
4 |
24 |
| 2 |
3 |
-1 |
1 |
6.5 * |
| 5 |
2 |
3 |
3 |
21.5 |
| 1 |
1 |
0 |
0 |
|
| 3 |
1 |
2 |
2 |
16.5 |
| 4 |
0 |
4 |
4 |
24 |
Notice that 0 differences are left out. There are four negative differences (*). All other differences are positive.
T = Sum of ranks of less frequent sign
T = 26
The obtained z-Value is smaller than the critical z-Value (5% two-tailed). A blood alcohol level of 0.8 per mill leads to a significant degradation of driving performance.
BrightStat output of the Wilcoxon test example
This is a fictitious example.
How to do this example on BrightStat webapp
Wiki link Wilcoxon signed rank test
References
Bortz, J. (2005). Statistik für Human- und Sozialwissenschaftler (6th Edition). Heidelberg: Springer Medizin Verlag.
Conover, W.J. (1999). Practical nonparametric Statistics.(3rd edition). Wiley.
Schaich, H.E. & Hamerle, A. (1984). Verteilungsfreie statistische Prüfverfahren, Berlin.
Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin 1, (6), 80 – 83.
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