KruskalWallis test  overview
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KruskalWallis test  Marginal Homogeneity test / StuartMaxwell test 


Independent/grouping variable  Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  2 paired groups  
Dependent variable  Dependent variable  
One of ordinal level  One categorical with $J$ independent groups ($J \geqslant 2$)  
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 all $I$ populations:
Formulation 1:
 H_{0}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$  
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 all $I$ populations:
Formulation 1:
 H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.  
Sampling distribution of $H$ if H_{0} were true  Sampling distribution of the test statistic if H_{0} 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.  Approximately the chisquared distribution with $J  1$ degrees of freedom  
Significant?  Significant?  
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 If we denote the test statistic as $X^2$:
 
Example context  Example context  
Do people from different religions tend to score differently on social economic status?  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  n.a.  
ANOVA > One Way ANOVA  KruskalWallis
   
Practice questions  Practice questions  