MannWhitneyWilcoxon test  overview
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MannWhitneyWilcoxon test 


Independent 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  
Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another  
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 H0 were true  
Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated if ties are present in the data.  
Significant?  
For large samples, the table for standard normal probabilities can be used: Two sided:
 
Equivalent to  
If no ties in the data: two sided MannWhitneyWilcoxon test is equivalent to 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  