MannWhitneyWilcoxon test  overview
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MannWhitneyWilcoxon test  Spearman's rho 


Independent variable  Independent variable  
One categorical with 2 independent groups  One of ordinal level  
Dependent variable  Dependent variable  
One of ordinal level  One of ordinal level  
Null hypothesis  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:
 $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. In words: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  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:
 Two sided: $\rho_s \neq 0$ Right sided: $\rho_s > 0$ Left sided: $\rho_s < 0$  
Assumptions  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  Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.  
Test statistic  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.  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ where $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.  
Sampling distribution of $W$ and of $U$ if H0 were true  Sampling distribution of $t$ 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.  Approximately a $t$ distribution with $N  2$ degrees of freedom  
Significant?  Significant?  
For large samples, the table for standard normal probabilities can be used: Two sided:
 Two sided:
 
Equivalent to  n.a.  
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  Example context  
Do men tend to score higher on social economic status than women?  Is there a monotonic relationship between physical health and mental health?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 Analyze > Correlate > Bivariate...
 
Jamovi  Jamovi  
TTests > Independent Samples TTest
 Regression > Correlation Matrix
 
Practice questions  Practice questions  