The selection of which test to execute is automatically based on the number of distinct values of the independent variable. The Mann-Whitney is used for two groups, the Kruskal-Wallis for three or more groups.
A special version of the Mann-Whitney/Kruskal-Wallis test performs a separate, independent test for each independent variable, and displays the result of each test with its accompanying column name. Under the primary version of the Mann-Whitney/Kruskal-Wallis test, all independent variable value combinations are used, often forcing the Kruskal-Wallis test, since the number of value combinations exceeds two. When a variable which has more than two distinct values is included in the set of independent variables, then the Kruskal-Wallis test is performed for all variables. Since Kruskal-Wallis is a generalization of Mann-Whitney, the Kruskal-Wallis results are valid for all the variables, including two-valued ones. In the discussion below, both types of Mann-Whitney/Kruskal-Wallis are referred to as Mann-Whitney/Kruskal-Wallis tests, since the only difference is the way the independent variable is treated.
The Mann-Whitney test, AKA Wilcoxon Two Sample Test, is the nonparametric analog of the 2-sample t test. It is used to compare two independent groups of sampled data, and tests whether they are from the same population or from different populations (i.e., whether the samples have the same distribution function). Unlike the parametric t-test, this non-parametric test makes no assumptions about the distribution of the data (e.g., normality). It is to be used as an alternative to the independent group t-test, when the assumption of normality or equality of variance is not met. Like many non-parametric tests, it uses the ranks of the data rather than the data itself to calculate the U statistic. But since the Mann-Whitney test makes no distribution assumption, it is less powerful than the t-test. On the other hand, the Mann-Whitney is more powerful than the t-test when parametric assumptions are not met. Another advantage is that it will provide the same results under any monotonic transformation of the data so the results of the test are more generalizable.
The Mann-Whitney is used when the independent variable is nominal or ordinal and the dependent variable is ordinal (or treated as ordinal). The main assumption is that the variable on which the 2 groups are to be compared is continuously distributed. This variable may be non-numeric, and if so, is converted to a rank based on alphanumeric precedence.
The null hypothesis is that both samples have the same distribution. The alternative hypotheses are that the distributions differ from each other in either direction (two-tailed test), or in a specific direction (upper-tailed or lower-tailed tests). Output is a p-value, which when compared to the user’s threshold, determines whether the null hypothesis should be rejected. Given one or more columns (independent variables) whose values define two independent groups of sampled data, and a column (dependent variable) whose distribution is of interest from the same input table, the Mann-Whitney test is performed for each set of unique values of the group-by variables (GBVs), if any.
The Kruskal-Wallis test is the nonparametric analog of the one-way analysis of variance or F-test used to compare three or more independent groups of sampled data. When there are only two groups, it reduces to the Mann-Whitney test (above). The Kruskal-Wallis test tests whether multiple samples of data are from the same population or from different populations (i.e., whether the samples have the same distribution function). Unlike the parametric independent group ANOVA (one way ANOVA), this non-parametric test makes no assumptions about the distribution of the data (e.g., normality). Since this test does not make a distributional assumption, it is not as powerful as ANOVA.
Given k independent samples of numeric values, a Kruskal-Wallis test is produced for each set of unique values of the GBVs, testing whether all the populations are identical. This test variable may be non-numeric, and if so, is converted to a rank based on alphanumeric precedence. The null hypothesis is that all samples have the same distribution. The alternative hypotheses are that the distributions differ from each other. Output for each unique set of values of the GBVs is a statistic H, and a p-value, which when compared to the user’s threshold, determines whether the null hypothesis should be rejected for the unique set of values of the GBVs.