Mann-Whitney U test: assumptions, steps to perform the Mann-Whitney U test, its calculation

• The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used in biostatistics and other fields to compare two independent samples.

• This test is particularly useful when the data do not meet the assumptions required for the parametric t-test, especially concerning normality.

Assumptions

1. Independence of Samples: The two samples being compared must be independent of each other.

2. Ordinal or Continuous Data: The test can be applied to ordinal data (ranked data) or continuous data.

3. Identical Shape and Scale: It is assumed that the distributions of both groups are the same shape and scale, though not necessarily normally distributed.

Steps to Perform the Mann-Whitney U Test

1. Prepare Your Data:

• Ensure your data meets the assumptions: two independent samples, and ordinal or continuous data.

2. Rank the Combined Data:

• Merge and rank the data from both groups, handling any ties appropriately.

3. Calculate 𝑈1 and 𝑈2:

• Use the formulas given to calculate the U statistic for both groups.

4. Determine the Test Statistic:

• Use the smaller of 𝑈1 or 𝑈2 as your test statistic.

5. Calculate the P-value:

• Depending on your sample size, either use the normal approximation or exact distribution tables to find the p-value.

6. Interpret the Results:

• Compare the p-value to your significance level (commonly 0.05) to decide whether to reject the null hypothesis (no difference between the groups).

Calculation

• The Mann-Whitney U test compares the ranks of the data from both groups, rather than the actual values.

1. Combine and Rank the Data:

• Merge the two samples into a single set, then rank all observations from the smallest to the largest.

• If there are ties (equal values), assign the average rank to the tied values.

2. Calculate 𝑈 for Each Group:

• Let 𝑛1 and 𝑛2 be the sizes of the two samples.

• Let 𝑅1 be the sum of the ranks in the first sample, and 𝑅2 be the sum of the ranks in the second sample.

• The U statistic for each sample can be computed as:
  

Alternatively, it can also be computed as:

3. Calculate the Test Statistic 𝑈:

• The Mann-Whitney U test statistic is the smaller of 𝑈1 and 𝑈2.

4. Significance Testing:

• For large samples, the distribution of 𝑈 can be approximated by a normal distribution.

• The mean and standard deviation of 𝑈 is used to standardize 𝑈U into a z-score, which is then used to determine the p-value.

• For small samples, use exact tables of the Mann-Whitney U distribution to determine the p-value.

This test is especially valuable in biostatistics for comparing the effects of treatments or conditions when the data cannot be assumed to be normally distributed.