Mann-Whitney U-Test: Exploring an Alternative to Independent Sample T-Test

Mann-Whitney U-Test. The independent sample T-test, popular among a pool of statistical tests, is widely used to develop statistical evidence for two populations average is significant or not. However, when the assumptions of this test are in doubt, it is the Mann- Whitney U test that comes in handy.

Mann- Whitney U-test is the alternative non-parametric test that compares conditions of two groups without making the assumption of normality. The test uses the ranks of values rather than comparing means. The added advantage of using ranks is that it only requires the data to be measured at the ordinal level. 

The main objective of the Mann- Whitney U-test is somewhat similar to Independent Sample T-test, i.e., to provide the statistical evidence to prove that the sampled population are different. Today, this test has turned into the most favoured test over that of independent sample T-test.  

The Mann-Whitney U- test considers assumptions for data to be used.

  1. The two sample groups should be randomly drawn from the population. Randomness implies the absence of sampling errors and measurement. 
  2. There should be independence between and within the groups.
  3. The data must be continuous. 
  4. Scale of measurement can be ordinal or interval continuous. The observation values can be interval, ordinal or ratio. 
  5. There exist no ties for maximum accuracy. 

Note: These assumptions are sufficient for determining if the two populations are different. In this test, one can take into consideration that the two populations used are identical. Considering all assumptions, it can be concluded that Mann- Whitney U-test on the chosen sample effectively.

It must be noted that Mann-Whitney U-test doesn’t hold good for all types of studies. It is best suited for: determining the differences in change scores, identifying the differences between interventions, and determining the difference between two independent groups. 

  1. Determining differences in change scores – if a study has two independent groups which have conducted different interventions, the same ordinal or continuous dependent variable is measured at the beginning and end of the test and change in score is calculated.
  2. Identifying the differences between interventions – This is similar to ‘determining the difference in change scores’ study design. The only difference here is, ordinal or continuous dependent variable is measured only at the end of the test.
  3. Determining the difference between two independent groups – If a study has the same ordinal or continuous dependent variable in two independent groups and the difference between the two groups is to be determined, we use Mann-Whitney U-test.

Calculating the Mann-Whitney U test 

In order to determine the U-test statistics, the mixed set of data is first ordered in ascending order by tied numbers getting a rank similar to the medium position of those numbers in the ordered sequence. The process involves combining all the observations from two samples into one combined sample. 

The steps involved in this test are as follows:

  1. Select samples for which ranks are smaller. Name them as sample 1 and sample 2.
  2. Take each observation in sample1, determine the number of observations in sample2 which are smaller than sample1.
  3. Organise the samples into a single ranked series.
  4. Add sample 1 and 2 observations. The sum of rank equals to N(N+1)⁄2, where ‘N’ is the total number of observations. 
  5. Calculate U for sample1 and sample2. The general formula used here is, U=R-n(n+1)/2 

Example, 

Sample a
Observation 22 34 45 22 12 19
Rank  15.5 15.5 3.5 12.4 12.5 3.5
Sample b
Observation 20 14 41 22 12 18
Rank  14.5 15.5 3.5 9.4 6.5 2.5

Sum of rank 1 = 62.9

Sum of rank 2 = 51.9

Calculate U1 and U2 and find the average U. 

By comparing two values (r1 and r2), we can reject or accept the null hypothesis. 

Some of the benefits of using this test are: provides a good approximation in the case of large samples, can be used for data sets of different sizes, presents the median between two data sets, explains if the difference occurred by chance or is significant, deals with skewed data, etc. 

Now that you know everything about this alternative test to Independent Sample T-test, use it in case of violation of assumptions in correlated-sample situations and perform the test accurately.

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