Mann-Whitney U Test
The Mann-Whitney U test is just one example of a class of inferential tests used when data are in the form of rankings (i.e., ordinal scale) or have been converted to ranks. In the latter case, ratio or interval data are collected in the study, but t or F tests cannot be used because these tests require certain assumptions to be met. For example, the data should normally distributed (or close) and the variance in each group ought to be similar. But sometimes data are skewed (e.g., too many high scores) or fail the “homogeneity of variance” criterion. In such situations, researchers will turn to a category of tests called “nonparametric tests.” The Mann-Whitney U is an example, to be used in a study involving independent samples of data (so it is the nonparametric equivalent of a t test for independent samples).
Consider a study in which behavior therapy is being evaluated as a procedure to eliminate snake phobia. The research recruits 14 snake phobic subjects and randomly assigns 7 to a therapy group and the remaining 7 to a waiting list control group. After therapy, an interval scale for snake avoidance is given. Scores range from 1 to 25, with high scores indicating high fear of snakes. Here are the scores for the two groups:
Therapy WL Control
420
717
13
1215
27
212
918
Both sets of scores appear to be skewed and the variance is much higher for the second group (38.48) than for the first (17.24), so the researcher decides to use the Mann-Whitney instead of a t test.
Formula for the Mann-Whitney U test:
A value for U is calculated for each group as follows:
N1(N1+ 1)
U1 = (N1)(N2) + ------– R1
2
N2(N2+ 1)
U2 = (N1)(N2) + ------– R2
2
Step 1.Convert the Data to Rankings
This means taking all 14 participants, lining their scores up from lowest to highest, and then converting the scores to ranks. Note what happens to tied scores (a mean rank is calculated). The numbers in boldface are the scores for those in the Therapy group.
score 1 2 2 3 4 7 7 9 12 12 15 17 18 20
rank 1 2.5 2.5 4 5 6.5 6.5 8 9.5 9.5 11 12 13 14
Step 2.Add Up the Ranks for Each Group
Therapy WL Control
14
2.56.5
2.59.5
511
6.512
813
9.514
R1 = 35 R2 = 70
Step 3.Apply the Formula for the Mann-Whitney U Test to Each Group
For the therapy group:
N1(N1+ 1)
U1 = (N1)(N2) + ------– R1
2
7(8)
U1 = (7)(7) + ------– 35 = 49 + 28 – 35 = 42
2
For the control group:
N2(N2+ 1)
U2 = (N1)(N2) + ------– R2
2
7(8)
U2 = (7)(7) + ------– 70 = 49 + 28 – 70 = 7
2
Step 4.Determine if the Calculated U is Significant
Your instructor will show you how to use a table of critical values for the Mann-Whitney U test. To evaluate the calculated value of U, first take the lower of the two calculated values (7 in this case—ignore the 42). If the calculated value of this lower U is less than or equal to the value in the table, then the null hypothesis can be rejected and it can be concluded that there is a significant difference in the rankings between the two groups. In our case, the difference is significant at the .05 level because the calculated value of 7 is less than the tabled critical value of 8. The research conclusion is that the therapy worked—participants in the therapy group showed a lower level of snake phobia than those in the control group.
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