MannWhitneyWilcoxon test  overview
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MannWhitneyWilcoxon test 


Independent/grouping variable  
One categorical with 2 independent groups  
Dependent variable  
One of ordinal level  
Null hypothesis  
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
Alternative hypothesis  
If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
Assumptions  
 
Test statistic  
Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.  
Sampling distribution of $W$ and of $U$ if H_{0} were true  
Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated.  
Significant?  
For large samples, the table for standard normal probabilities can be used: Two sided:
 
Equivalent to  
If there are no ties in the data, the two sided MannWhitneyWilcoxon test is equivalent to the KruskalWallis test with an independent variable with 2 levels ($I$ = 2).  
Example context  
Do men tend to score higher on social economic status than women?  
SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 
Jamovi  
TTests > Independent Samples TTest
 
Practice questions  