KruskalWallis test  overview
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KruskalWallis test  Friedman test 


Independent variable  Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  One within subject factor ($\geq 2$ related groups)  
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 all $I$ populations:
Formulation 1:
 The scores in any of the related groups are not systematically higher or lower than the scores in any of the other related groups
Note: usually, the related groups are the different measurement points Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.  
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:
 The scores in some of the related groups are systematically higher or lower than the scores in other related groups  
Assumptions  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  Sample of 'blocks' (usually the subjects) is a simple random sample from the population. That is, blocks are independent of one another  
Test statistic  Test statistic  
$H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  $Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
Here $N$ is the number of 'blocks' (usually the subjects  so if you have 4 repeated measurements for 60 subjects, $N$ equals 60), $k$ is the number of related groups (usually the number of repeated measurements), and $R_i$ is the sum of ranks in group $i$. Remember that multiplication precedes addition, so first compute $\frac{12}{N \times k(k + 1)} \times \sum R^2_i$ and then subtract $3 \times N(k + 1)$. Note: if ties are present in the data, the formula for $Q$ is more complicated.  
Sampling distribution of $H$ if H0 were true  Sampling distribution of $Q$ 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.  If the number of blocks $N$ is large, approximately the chisquared distribution with $k  1$ degrees of freedom.
For small samples, the exact distribution of $Q$ should be used.  
Significant?  Significant?  
For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
Example context  Example context  
Do people from different religions tend to score differently on social economic status?  Is there a difference in depression level between measurement point 1 (preintervention), measurement point 2 (1 week postintervention), and measurement point 3 (6 weeks postintervention)?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  Jamovi  
ANOVA > One Way ANOVA  KruskalWallis
 ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  