Wilcoxon - Mann - Whitney U Test:

The Wilcoxon-Mann-Whitney test uses the ranks of data to test the hypothesis that two samples of sizes m and n might come from the same population. The procedure is as follows:

1. Combine the data from both samples
2. Rank each value
3. Take the ranks for the first sample and sum them
4. Compare this sum of ranks to all the possible rank sums that could result from random rearrangements of the data into two samples.

If step 4 reveals that the rank sum for the observed first sample is larger (or smaller) than nearly all the random orderings, this indicates that the first sample is significantly different from the second sample.

Note: Hollander and Wolfe suggest that ties be resolved by using the average rank of the tied observations.

Here´s an example in Excel (in step 4, rather than comparing the observed sum of ranks to ALL POSSIBLE randomly ordered sums, thousands of randomly shuffled sums are used for comparison):

Is there a difference in the transfer of titrated water across a placental membrane between human fetuses at 12-26 weeks and at term? The permeability constant Pd of the membrane is used as the measure. (Source: Hollander & Wolfe, Nonparametric Statistical Methods, John Wiley and Sons, 1973)

Here are the data:

Here are the ranks; note the rank sum for the first sample is 30:

Here are the ranks, randomly shuffled using the Resampling Stats add-in for Excel (30-day trial available for download at <a href="http://www.resample.com">http://www.resample.com):

Using the Resampling Stats add-in, the shuffling is repeated 10,000 times.

The column of 10,000 resulting values is sorted, and we see that 1291 of the 10,000 shufflings yielded a sum of ranks <= the observed value of 30. This translates into an estimated p-value of .1291 for a 1-sided test of the null hypothesis that the two samples might come from the same population (against the alternative that sample 1 is smaller than sample 2.)

Including the above method, there are several ways to determine this p-value:

1. Compare the observed rank sum to the distribution of rank sums resulting from all possible orderings of the ranks (an exact permutation test)
2. Compare the observed rank sum to repeatedly shuffled rank sums (an approximate or Monte Carlo permuitation test; the result approaches the result of #1 above as the number of repeats approaches infinity).
   This is the method described above.
3. Transform the observed test statistic into an equivalent statistic, which is approximately normally-distributed.
4. For certain sample sizes, you can consult tables of the exact distribution of the test statistic.

With the advent of high speed computing and the availability of resampling and permutation software, methods 1 and 2 have increasingly come to dominate 3 and 4.
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