KruskalWallis test  overview
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KruskalWallis test 


Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  
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 all $I$ 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 all $I$ 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, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another  
Test statistic  
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  
Sampling distribution of $H$ if H0 were true  
For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  
Significant?  
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 
Example context  
Do people from different religions tend to score differently on social economic status?  
SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 
Jamovi  
ANOVA > One Way ANOVA  KruskalWallis
 
Practice questions  