Statistical Tests: Mann-Whitney U-test, Wilcoxon matched pairs signed rank test, Kruskal-Wallis test, Friedman test

Mann-Whitney U-test
Example 1: Mann-Whitney U test (2 independent samples) A researcher wants to test the
hypothesis that experienced podiatrists require less time than novices to make a particular
diagnosis. A group of experienced podiatrists is compared with a group of novices. Both groups
observe the same individuals. The data are stored in the SPSS file ‘Mann-Whitney.sav’. The two
variables are Obstime (time in seconds for the observer to make a diagnosis), and Experience (a
grouping variable denoting the experience of the observer). Should the researcher accept or reject
the hypothesis?
Test for normality
Table1: Descriptive analysis for Obstime time for novice and expert groups
Descriptive
Experience of observer Statistic Std. Error
Observation time to diagnosis Novice Mean 136.27 4.942
95% Confidence Interval for
Mean
Lower Bound 125.26
Upper Bound 147.28
5% Trimmed Mean 134.75
Median 136.00
Variance 268.618
Std. Deviation 16.390
Minimum 120
Maximum 180
Range 60
Interquartile Range 16
Skewness 2.064 .661
Kurtosis 5.463 1.279
Expert Mean 99.18 8.070
95% Confidence Interval for
Mean
Lower Bound 81.20
Upper Bound 117.16
5% Trimmed Mean 101.31
Median 110.00
Variance 716.364
Std. Deviation 26.765
Minimum 40
Maximum 120
Range 80
Interquartile Range 33

Skewness -1.499 .661
Kurtosis 1.310 1.279
From the analysis the skewness value for the novice group is given as 2.604 and a standard error
of 0.661. The quotient between the skewness and the standard error for the skewness i.e.
2.604/0.66 = 3.122, which is not present in range of -1.96 – 1.96, hence suggesting that the
novice Obstime measures are skewed. The skewness value for expert group is -1.49 with a
standard error of 0.661. The quotient value for the two statistical parameters skewness and
standard error -1.499/0.661 = 2.267, hence indicating that the expert Obstime values are not
normally distributed.
Table 1.1: Mean ranks for native and experts groups
Ranks
Experience of observer N Mean Rank Sum of Ranks
Observation time to diagnosis Novice 11 16.91 186.00
Expert 11 6.09 67.00
Total 22
Table 1.3: Mann-Whitney U-test summary output
Test Statisticsa
Observation time to diagnosis
Mann-Whitney U 1.000
Wilcoxon W 67.000
Z -3.911
Asymp. Sig. (2-tailed) .000
Exact Sig. [2*(1-tailed Sig.)] .000b
a. Grouping Variable: Experience of observer
b. Not corrected for ties.
From the resulst presented in table 1.3 above, the Mann-Whitney U-test is given as U = 1, p <
0.01 hence suggest that the test is sigfnificant at 0.05 level of significance and the null hypothesis
is rejected. The conclsuion for the test state that there is a sigificant difference in the mean
Obstime time for native and expert groups in the survey.

Figure 1: Boxplot summary output for native and expert Obstime time
Example 2: Wilcoxon matched pairs signed rank test
A sports physiotherapist investigating the effects of a new mobilization treatment regimen on range of
movement at the ankle, records before and after ranges of movement for a set of 15 female participants.
The data are contained in the SPSS data file ‘Wilcoxon.sav’. There are three variables, Before, After and
Diff, which are ankle range of motion (ROM) before the therapy, ankle ROM after therapy and change in
ankle ROM (after-before).
The research hypothesis for the test are given as follows
H0: The two underlying populations are equivalent
H1: The two underlying populations are not equivalent.