Question

How do I prove the following: Show that for comparing two groups, the Kruskal-Wallis test is equivalent to the Mann-Whitney test. Please explain the meaning of each test statistic and identify the distribution each one follows.

Answer 1

Use the Kruskal–Wallis test when you have one nominal variable and one is that the variation within the groups is equal (homoscedasticity). The Mann–Whitney U-test (also known as the Mann–Whitney–Wilcoxon test, the same distribution, the Kruskal–Wallis test will show no difference among them.

Test statistics for Kruskal–Wallis test

1. Rank all data from all groups together; i.e., rank the data from 1 to N ignoring group membership. Assign any tied values the average of the ranks they would have received had they not been tied.
2. The test statistic is given by:

   where:

   • is the number of observations in group
   • is the rank (among all observations) of observation from group
   • is the total number of observations across all groups
   • is the average rank of all observations in group
   • is the average of all the .

we use the Mann Whitney U test. The test statistic is U, the smaller of

where R₁ and R₂ are the sums of the ranks in groups 1 and 2, respectively.