Wilcoxon signedrank test
Wilcoxon signedrank 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. ZeroDifferences 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, σ_{T} is corrected as follows:
Whereas
t_{i} = Number of subjects sharing rank i
k = Number of tied ranks
Example of a Wilcoxon signedrank 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. 0.80.0 
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 zValue is smaller than the critical zValue (5% twotailed). 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.
Wiki link Wilcoxon signed rank test
References
Bortz, J. (2005). Statistik für Human und Sozialwissenschaftler (6^{th} Edition). Heidelberg: Springer Medizin Verlag.
Conover, W.J. (1999). Practical nonparametric Statistics.(3^{rd} 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.