Friedman test  overview
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Friedman test  MannWhitneyWilcoxon test 


Independent/grouping variable  Independent/grouping variable  
One within subject factor ($\geq 2$ related groups)  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One of ordinal level  One of ordinal level  
Null hypothesis  Null hypothesis  
H_{0}: the population scores in any of the related groups are not systematically higher or lower than the population scores in any of the other related groups
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.  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  Alternative hypothesis  
H_{1}: the population scores in some of the related groups are systematically higher or lower than the population scores in other related groups  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  Assumptions  

 
Test statistic  Test statistic  
$Q = \dfrac{12}{N \times k(k + 1)} \sum R^2_i  3 \times N(k + 1)$
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.  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 $Q$ if H_{0} were true  Sampling distribution of $W$ and of $U$ if H_{0} were true  
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.  Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated.  
Significant?  Significant?  
If the number of blocks $N$ is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 For large samples, the table for standard normal probabilities can be used: Two sided:
 
n.a.  Equivalent to  
  If there are no ties in the data, the two sided MannWhitneyWilcoxon test is equivalent to the KruskalWallis test with an independent variable with 2 levels ($I$ = 2).  
Example context  Example context  
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)?  Do men tend to score higher on social economic status than women?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 
Jamovi  Jamovi  
ANOVA > Repeated Measures ANOVA  Friedman
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  