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Mann-Whitney Test
This
is one in a series of tutorials using examples from WINKS SDA.
Definition: A non-parametric test (distribution-free) used to compare two independent groups of sampled data.
Assumptions: Unlike the parametric t-test, this non-parametric makes no assumptions about the distribution of the data (e.g., normality).
Characteristics: This test is an alternative to the independent group t-test, when the assumption of normality or equality of variance is not met. This, like many non-parametric tests, uses the ranks of the data rather than their raw values to calculate the statistic. Since this test does not make a distribution assumption, it is not as powerful as the t-test.
Test: The hypotheses for the comparison of two independent groups are:
Ho: The two samples come from identical populations
Ha: The two samples come from different populations
Notice that the hypothesis makes no assumptions about the distribution of the populations. These hypotheses are also sometimes written as testing the equality of the central tendency of the populations.
The test statistic for the Mann-Whitney test is U. This value is compared to a table of critical values for U based on the sample size of each group. If U exceeds the critical value for U at some significance level (usually 0.05) it means that there is evidence to reject the null hypothesis in favor of the alternative hypothesis. (See the Zar reference for details.)
Note: Actually, there are two versions of the U statistic calculated, where U' = n1n2 - U where n1 and n2 are the sample sizes of the two groups. The largest of U or U' is compared to the critical value for the purpose of the test.
Note: For sample sizes greater than 8, a z-value can be used to approximate the significance level for the test. In this case, the calculated z is compared to the standard normal significance levels.
Note: The U test is usually perform as a two-tailed test, however some text will have tabled one-tailed significance levels for this purpose. If the sample size if large, the z-test can be used for a one-sided test.
Graphical comparison: The graphical comparison allows you to visually see the distribution of the two groups. If the p-value is low, chances are there will be little overlap between the two distributions. If the p-value is not low, there will be a fair amount of overlap between the two groups. There are a number of options available in the comparison graph to allow you to examine the two groups. These include box plots, means, medians, and error bars.
Location in KWIKSTAT and WINKS: The Mann-Whitney U test (independent group comparison test) is located in the Analyze/Non-parametric comparisons menu. When there are more than two groups in this comparison, the test becomes a Kruskal-Wallis test.
See Also: The independent group t-test.
Example: The Mann-Whitney U test
The FERTILIZ.DBF database contains information on the heights of plants that were grown using two different fertilizers. The Mann-Whitney test can be used to determine if there is evidence that one fertilizer that causes the plants to grow taller than the other. The results of running an analysis on this data is as follows:
----------------------------------------------------------------
Non-Parametric Independent Group Comparison
----------------------------------------------------------------
Results of Non- Parametric analysis:
Group variable = GROUP Observation variable = OBS
Mann- Whitney U' = 24. U = 18.
Rank sum group 1 = 46. N = 7 Mean Rank = 6.57
Rank sum group 2 = 45. N = 6 Mean Rank = 7.5
Significance estimated using the z statistic.
Z = .357 p = 0.721
(Note: This Z calculation uses a correction for continuity.)
The program reports the p-value based on the z approximation. Since the sample sizes for both groups are less than 8, you should can look up the critical value for n1 = 6 and n2 = 7 in the table contained in the KWIKSTAT or WINKS manual (or in a textbook) to see if the critical value is 34 for a test at the 0.05 level. Thus, if U' were 34 or greater, you could claim statistical significance at the 0.05 level. In this case, the conclusion is that there is no difference in the two groups. Note how this result agrees with the t-test for this same data set.
Exercise: Mann-Whitney U test
Professor Testum wondered if students tended to make better scores on his test depending if the test were taken in the morning or afternoon. From a group of 19 similarly talented students, he randomly selected some to take a test in the morning and some to take it in the afternoon. The scores by groups were:
Morning
Afternoon
89.8
87.3
90.2
87.6
98.1
87.3
91.2
91.8
88.9
86.4
90.3
86.4
99.2
93.1
94.0
89.2
88.7
90.1
83.9
1. Perform a Mann-Whitney U test on this data. Remember that the program expect two fields, a Group field and an Observation field.
2. What was the result?
3. From this evidence, does it appear that time of day makes a difference in performance on a test?
4. Change the value 83.9 in the second group to 11 and rerun the test. Does this change the statistic calculated or the conclusion? Also perform an independent group t-test on the original and changed data. Does it effect this statistic? Why?
5. Display a graphical comparison of the original and changed data.
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